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<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"/><title>3 Special Functions</title><link rel="stylesheet" type="text/css" href="../scribble.css" title="default"/><link rel="stylesheet" type="text/css" href="../racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-style.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../doc-site.css" title="default"/><script type="text/javascript" src="../scribble-common.js"></script><script type="text/javascript" src="../manual-racket.js"></script><script type="text/javascript" src="../manual-racket.js"></script><script type="text/javascript" src="../doc-site.js"></script><script type="text/javascript" src="../local-redirect/local-redirect.js"></script><script type="text/javascript" src="../local-redirect/local-user-redirect.js"></script><!--[if IE 6]><style type="text/css">.SIEHidden { overflow: hidden; }</style><![endif]--></head><body id="doc-racket-lang-org"><div class="tocset"><div class="tocview"><div class="tocviewlist tocviewlisttopspace"><div class="tocviewtitle"><table cellspacing="0" cellpadding="0"><tr><td style="width: 1em;"><a href="javascript:void(0);" title="Expand/Collapse" class="tocviewtoggle" onclick="TocviewToggle(this,"tocview_0");">▼</a></td><td></td><td><a href="index.html" class="tocviewlink" data-pltdoc="x">Math Library</a></td></tr></table></div><div class="tocviewsublisttop" style="display: block;" id="tocview_0"><table cellspacing="0" cellpadding="0"><tr><td align="right">1 </td><td><a href="base.html" class="tocviewlink" data-pltdoc="x">Constants and Elementary Functions</a></td></tr><tr><td align="right">2 </td><td><a href="flonum.html" class="tocviewlink" data-pltdoc="x">Flonums</a></td></tr><tr><td align="right">3 </td><td><a href="" class="tocviewselflink" data-pltdoc="x">Special Functions</a></td></tr><tr><td align="right">4 </td><td><a href="number-theory.html" class="tocviewlink" data-pltdoc="x">Number Theory</a></td></tr><tr><td align="right">5 </td><td><a href="bigfloat.html" class="tocviewlink" data-pltdoc="x">Arbitrary-<wbr></wbr>Precision Floating-<wbr></wbr>Point Numbers (<a name="(tech._bigfloat)"></a><span style="font-style: italic">Bigfloats</span>)</a></td></tr><tr><td align="right">6 </td><td><a href="array.html" class="tocviewlink" data-pltdoc="x">Arrays</a></td></tr><tr><td align="right">7 </td><td><a href="matrices.html" class="tocviewlink" data-pltdoc="x">Matrices and Linear Algebra</a></td></tr><tr><td align="right">8 </td><td><a href="stats.html" class="tocviewlink" data-pltdoc="x">Statistics Functions</a></td></tr><tr><td align="right">9 </td><td><a href="dist.html" class="tocviewlink" data-pltdoc="x">Probability Distributions</a></td></tr><tr><td align="right">10 </td><td><a href="utils.html" class="tocviewlink" data-pltdoc="x">Stuff That Doesn’t Belong Anywhere Else</a></td></tr></table></div></div><div class="tocviewlist"><table cellspacing="0" cellpadding="0"><tr><td style="width: 1em;"><a href="javascript:void(0);" title="Expand/Collapse" class="tocviewtoggle" onclick="TocviewToggle(this,"tocview_1");">►</a></td><td>3 </td><td><a href="" class="tocviewselflink" data-pltdoc="x">Special Functions</a></td></tr></table><div class="tocviewsublistbottom" style="display: none;" id="tocview_1"><table cellspacing="0" cellpadding="0"><tr><td align="right">3.1 </td><td><a href="#%28part._real-functions%29" class="tocviewlink" data-pltdoc="x">Real Functions</a></td></tr><tr><td align="right">3.2 </td><td><a href="#%28part._flonum-functions%29" class="tocviewlink" data-pltdoc="x">Flonum Functions</a></td></tr></table></div></div></div><div class="tocsub"><div class="tocsubtitle">On this page:</div><table class="tocsublist" cellspacing="0"><tr><td><span class="tocsublinknumber">3.1<tt> </tt></span><a href="#%28part._real-functions%29" class="tocsubseclink" data-pltdoc="x">Real Functions</a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">gamma</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">log-<wbr></wbr>gamma</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi0%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">psi0</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">psi</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erf%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">erf</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">erfc</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">lambert</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert-%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">lambert-</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">zeta</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._eta%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">eta</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._hurwitz-zeta%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">hurwitz-<wbr></wbr>zeta</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">beta</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-beta%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">log-<wbr></wbr>beta</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">gamma-<wbr></wbr>inc</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma-inc%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">log-<wbr></wbr>gamma-<wbr></wbr>inc</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta-inc%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">beta-<wbr></wbr>inc</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-beta-inc%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">log-<wbr></wbr>beta-<wbr></wbr>inc</span></span></a></td></tr><tr><td><span class="tocsublinknumber">3.2<tt> </tt></span><a href="#%28part._flonum-functions%29" class="tocsubseclink" data-pltdoc="x">Flonum Functions</a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flgamma%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">flgamma</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllog-gamma%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">fllog-<wbr></wbr>gamma</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flpsi0%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">flpsi0</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flpsi%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">flpsi</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flerf%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">flerf</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flerfc%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">flerfc</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllambert%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">fllambert</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllambert-%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">fllambert-</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flzeta%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">flzeta</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fleta%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">fleta</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flhurwitz-zeta%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">flhurwitz-<wbr></wbr>zeta</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flbeta%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">flbeta</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllog-beta%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">fllog-<wbr></wbr>beta</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flgamma-inc%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">flgamma-<wbr></wbr>inc</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllog-gamma-inc%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">fllog-<wbr></wbr>gamma-<wbr></wbr>inc</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flbeta-inc%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">flbeta-<wbr></wbr>inc</span></span></a></td></tr><tr><td><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllog-beta-inc%29%29" class="tocsubnonseclink" data-pltdoc="x"><span class="RktSym"><span class="RktValLink">fllog-<wbr></wbr>beta-<wbr></wbr>inc</span></span></a></td></tr></table></div></div><div class="maincolumn"><div class="main"><div class="versionbox"><span class="version">6.7</span></div><div class="navsettop"><span class="navleft"><form class="searchform"><input class="searchbox" style="color: #888;" type="text" value="...search manuals..." title="Enter a search string to search the manuals" onkeypress="return DoSearchKey(event, this, "6.7", "../");" onfocus="this.style.color="black"; this.style.textAlign="left"; if (this.value == "...search manuals...") this.value="";" onblur="if (this.value.match(/^ *$/)) { this.style.color="#888"; this.style.textAlign="center"; this.value="...search manuals..."; }"/></form> <a href="../index.html" title="up to the documentation top" data-pltdoc="x" onclick="return GotoPLTRoot("6.7");">top</a></span><span class="navright"> <a href="flonum.html" title="backward to "2 Flonums"" data-pltdoc="x">← prev</a> <a href="index.html" title="up to "Math Library"" data-pltdoc="x">up</a> <a href="number-theory.html" title="forward to "4 Number Theory"" data-pltdoc="x">next →</a></span> </div><h3 x-source-module="(lib "math/scribblings/math.scrbl")" x-source-pkg="math-doc" x-part-tag=""special"">3<tt> </tt><a name="(part._special)"></a><a name="(mod-path._math/special-functions)"></a>Special Functions</h3><div class="SAuthorListBox"><span class="SAuthorList"><p class="author">Neil Toronto <<a href="mailto:ntoronto@racket-lang.org">ntoronto@racket-lang.org</a>></p></span></div><p><table cellspacing="0" cellpadding="0" class="defmodule"><tr><td align="left"><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=require.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._require%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">require</a></span><span class="stt"> </span><a href="" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/special-functions</span></a><span class="RktPn">)</span></td><td align="right"><span class="RpackageSpec"><span class="Smaller"> package:</span> <span class="stt">math-lib</span></span></td></tr></table></p><p>The term “special function” has no formal definition. However, for the purposes of the
<a href="index.html" class="RktModLink" data-pltdoc="x"><span class="RktSym">math</span></a> library, a <a name="(tech._special._function)"></a><span style="font-style: italic">special function</span> is one that is not <a href="base.html#%28tech._elementary%29" class="techoutside" data-pltdoc="x"><span class="techinside">elementary</span></a>.</p><p><div class="SIntrapara">The special functions are split into two groups: <a href="#%28part._real-functions%29" data-pltdoc="x">Real Functions</a> and
<a href="#%28part._flonum-functions%29" data-pltdoc="x">Flonum Functions</a>. Functions that accept real arguments are usually defined
in terms of their flonum counterparts, but are different in two crucial ways:
</div><div class="SIntrapara"><ul><li><p>Many return exact values for certain exact arguments.</p></li><li><p>When applied to exact arguments outside their domains, they raise an
<span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=exns.html%23%2528def._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._exn%7E3afail%7E3acontract%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exn:fail:contract</a></span> instead of returning <span class="RktVal">+nan.0</span>.</p></li></ul></div></p><p>Currently, <a href="" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/special-functions</span></a> does not export any functions that accept
or return complex numbers. Mathematically, some of them could return complex numbers given
real numbers, such <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._hurwitz-zeta%29%29" class="RktValLink" data-pltdoc="x">hurwitz-zeta</a></span> when given a negative second argument. In
these cases, they raise an <span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=exns.html%23%2528def._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._exn%7E3afail%7E3acontract%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exn:fail:contract</a></span> (for an exact argument) or return
<span class="RktVal">+nan.0</span> (for an inexact argument).</p><p>Most real functions have more than one type, but they are documented as having only
one. The documented type is the most general type, which is used to generate a contract for
uses in untyped code. Use <span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=Exploring_Types.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fprims..rkt%2529._%7E3aprint-type%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">:print-type</a></span> to see all of a function’s types.</p><p><div class="SIntrapara">A function’s types state theorems about its behavior in a way that Typed Racket can understand
and check. For example, <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span> has these types:
</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._case-%7E3e%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">case-></a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Zero%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Zero</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._-%7E3e%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x"><span class="nobreak">-></span></a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Zero%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Zero</a></span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._-%7E3e%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x"><span class="nobreak">-></span></a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._-%7E3e%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x"><span class="nobreak">-></span></a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._.U%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">U</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Zero%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Zero</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr></table></blockquote></div></p><p>Because <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="stt"> </span><span class="RktSym">:</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Zero%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Zero</a></span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._-%7E3e%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x"><span class="nobreak">-></span></a></span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Zero%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Zero</a></span>, Typed Racket proves during typechecking that one
of its exact cases is <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="stt"> </span><span class="RktVal">0</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">0</span>.</p><p>Because the theorem <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="stt"> </span><span class="RktSym">:</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._-%7E3e%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x"><span class="nobreak">-></span></a></span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span> is stated as a type and proved
by typechecking, Typed Racket’s optimizer can transform the expressions around its use
into bare-metal floating-point operations. For example, <span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252B%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">+</a></span><span class="stt"> </span><span class="RktVal">2.0</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="stt"> </span><span class="RktVal">3.0</span><span class="RktPn">)</span><span class="RktPn">)</span> is
transformed into <span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=unsafe.html%23%2528def._%2528%2528lib._racket%252Funsafe%252Fops..rkt%2529._unsafe-fl%252B%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">unsafe-fl+</a></span><span class="stt"> </span><span class="RktVal">2.0</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="stt"> </span><span class="RktVal">3.0</span><span class="RktPn">)</span><span class="RktPn">)</span>.</p><p>The most general type <span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._-%7E3e%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x"><span class="nobreak">-></span></a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._.U%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">U</a></span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Zero%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Zero</a></span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span><span class="RktPn">)</span> is used to generate
<span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span>’s contract when it is used in untyped code. Except for this discussion,
this the only type documented for <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span>.</p><h4 x-source-module="(lib "math/scribblings/math.scrbl")" x-source-pkg="math-doc" x-part-tag=""real-functions"">3.1<tt> </tt><a name="(part._real-functions)"></a>Real Functions</h4><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._gamma))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValDef RktValLink" data-pltdoc="x">gamma</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._.U%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">U</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Positive-.Integer%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Positive-Integer</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span><span class="RktPn">)</span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></blockquote></div><div class="SIntrapara">Computes the <a href="http://en.wikipedia.org/wiki/Gamma_function">gamma function</a>,
a generalization of the factorial function to the entire real line, except nonpositive integers.
When <span class="RktVar">x</span> is an exact integer, <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="stt"> </span><span class="RktVar">x</span><span class="RktPn">)</span> is exact.</div></p><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=pairs.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._list%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">list</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=lambda.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._%7Ece%7Ebb%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">λ</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252B%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">+</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal">0</span><span class="hspace"> </span><span class="RktVal">4.5</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:label</span><span class="hspace"> </span><span class="RktVal">"gamma(x+1)"</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=lambda.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._%7Ece%7Ebb%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">λ</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="number-theory.html#%28def._%28%28lib._math%2Fnumber-theory..rkt%29._factorial%29%29" class="RktValLink" data-pltdoc="x">factorial</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._truncate%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">truncate</a></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">#:color</span><span class="hspace"> </span><span class="RktVal">2</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:label</span><span class="hspace"> </span><span class="RktVal">"factorial(floor(x))"</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_7.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-2</span>.5</span><span class="hspace"> </span><span class="RktVal">5.5</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">#:y-min</span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-5</span>0</span><span class="hspace"> </span><span class="RktPn">#:y-max</span><span class="hspace"> </span><span class="RktVal">50</span><span class="RktPn">)</span></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_8.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal">5</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">24</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal">5.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">24.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="number-theory.html#%28def._%28%28lib._math%2Fnumber-theory..rkt%29._factorial%29%29" class="RktValLink" data-pltdoc="x">factorial</a></span><span class="hspace"> </span><span class="RktVal">4</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">24</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">gamma: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: Real, not Zero or Negative-Integer</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: -1</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span>.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">+nan.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal">0.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">+inf.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-0</span>.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-inf.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal">172.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">+inf.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="bigfloat.html#%28def._%28%28lib._math%2Fbigfloat..rkt%29._bf%29%29" class="RktValLink" data-pltdoc="x">bf</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal">172</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">(bf "1.241018070217667823424840524103103992618e309")</span></p></td></tr></table></blockquote></div></p><p>Error is no more than 10 <a href="flonum.html#%28tech._ulp%29" class="techoutside" data-pltdoc="x"><span class="techinside">ulps</span></a> everywhere that has been tested, and is usually no more than 4
ulps.</p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._log-gamma))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma%29%29" class="RktValDef RktValLink" data-pltdoc="x">log-gamma</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._.U%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">U</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Zero%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Zero</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span><span class="RktPn">)</span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></blockquote></div><div class="SIntrapara">Like <span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._abs%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">abs</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="stt"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span>, but more accurate and without unnecessary overflow.
The only exact cases are <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma%29%29" class="RktValLink" data-pltdoc="x">log-gamma</a></span><span class="stt"> </span><span class="RktVal">1</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">0</span> and <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma%29%29" class="RktValLink" data-pltdoc="x">log-gamma</a></span><span class="stt"> </span><span class="RktVal">2</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">0</span>.</div></p><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=pairs.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._list%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">list</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma%29%29" class="RktValLink" data-pltdoc="x">log-gamma</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-5</span>.5</span><span class="hspace"> </span><span class="RktVal">10.5</span><span class="hspace"> </span><span class="RktPn">#:label</span><span class="hspace"> </span><span class="RktVal">"log-gamma(x)"</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=lambda.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._%7Ece%7Ebb%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">λ</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._abs%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">abs</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:color</span><span class="hspace"> </span><span class="RktVal">2</span><span class="hspace"> </span><span class="RktPn">#:style</span><span class="hspace"> </span><span class="RktVal">'</span><span class="RktVal">long-dash</span><span class="hspace"> </span><span class="RktPn">#:width</span><span class="hspace"> </span><span class="RktVal">2</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:label</span><span class="hspace"> </span><span class="RktVal">"log(abs(gamma(x)))"</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_9.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma%29%29" class="RktValLink" data-pltdoc="x">log-gamma</a></span><span class="hspace"> </span><span class="RktVal">5</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">3.1780538303479458</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._abs%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">abs</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal">5</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">3.1780538303479458</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma%29%29" class="RktValLink" data-pltdoc="x">log-gamma</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">log-gamma: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: Real, not Zero or Negative-Integer</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: -1</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma%29%29" class="RktValLink" data-pltdoc="x">log-gamma</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span>.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">+inf.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma%29%29" class="RktValLink" data-pltdoc="x">log-gamma</a></span><span class="hspace"> </span><span class="RktVal">0.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">+inf.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._abs%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">abs</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal">172.0</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">+inf.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma%29%29" class="RktValLink" data-pltdoc="x">log-gamma</a></span><span class="hspace"> </span><span class="RktVal">172.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">711.71472580229</span></p></td></tr></table></blockquote></div></p><p>Error is no more than 11 <a href="flonum.html#%28tech._ulp%29" class="techoutside" data-pltdoc="x"><span class="techinside">ulps</span></a> everywhere that has been tested, and is usually no more than 2
ulps. Error reaches its maximum near negative roots.</p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._psi0))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi0%29%29" class="RktValDef RktValLink" data-pltdoc="x">psi0</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></blockquote></div><div class="SIntrapara">Computes the <a href="http://en.wikipedia.org/wiki/Digamma_function">digamma function</a>,
the logarithmic derivative of the gamma function.</div></p><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi0%29%29" class="RktValLink" data-pltdoc="x">psi0</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-2</span>.5</span><span class="hspace"> </span><span class="RktVal">4.5</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">#:y-min</span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-5</span></span><span class="hspace"> </span><span class="RktPn">#:y-max</span><span class="hspace"> </span><span class="RktVal">5</span><span class="RktPn">)</span></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_10.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi0%29%29" class="RktValLink" data-pltdoc="x">psi0</a></span><span class="hspace"> </span><span class="RktVal">0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">psi0: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: Real, not Zero or Negative-Integer</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: 0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi0%29%29" class="RktValLink" data-pltdoc="x">psi0</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-0.5772156649015329</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="hspace"> </span><span class="RktSym"><a href="base.html#%28def._%28%28lib._math%2Fbase..rkt%29._gamma..0%29%29" class="RktValLink" data-pltdoc="x">gamma.0</a></span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-0.5772156649015329</span></p></td></tr></table></blockquote></div></p><p>Except near negative roots, maximum observed error is 2 <a href="flonum.html#%28tech._ulp%29" class="techoutside" data-pltdoc="x"><span class="techinside">ulps</span></a>, but is usually no more
than 1.</p><p>Near negative roots, which occur singly between each pair of negative integers, <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi0%29%29" class="RktValLink" data-pltdoc="x">psi0</a></span>
exhibits <a href="flonum.html#%28tech._catastrophic._cancellation%29" class="techoutside" data-pltdoc="x"><span class="techinside">catastrophic cancellation</span></a> from using the reflection formula, meaning that
relative error is effectively unbounded. However, maximum observed <span class="RktSym"><a href="base.html#%28def._%28%28lib._math%2Fbase..rkt%29._absolute-error%29%29" class="RktValLink" data-pltdoc="x">absolute-error</a></span>
is <span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252A%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">*</a></span><span class="stt"> </span><span class="RktVal">5</span><span class="stt"> </span><span class="RktSym"><a href="flonum.html#%28def._%28%28lib._math%2Fflonum..rkt%29._epsilon..0%29%29" class="RktValLink" data-pltdoc="x">epsilon.0</a></span><span class="RktPn">)</span>. This is the best we can do for now, because there are currently
no reasonably fast algorithms for computing <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi0%29%29" class="RktValLink" data-pltdoc="x">psi0</a></span> near negative roots with low relative
error.</p><p>If you need low relative error near negative roots, use <span class="RktSym"><a href="bigfloat.html#%28def._%28%28lib._math%2Fbigfloat..rkt%29._bfpsi0%29%29" class="RktValLink" data-pltdoc="x">bfpsi0</a></span>.</p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._psi))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi%29%29" class="RktValDef RktValLink" data-pltdoc="x">psi</a></span></span><span class="hspace"> </span><span class="RktVar">m</span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">m</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Integer%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Integer</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></blockquote></div><div class="SIntrapara">Computes a <a href="http://en.wikipedia.org/wiki/Polygamma_function">polygamma function</a>,
or the <span class="RktVar">m</span>th logarithmic derivative of the gamma function. The order <span class="RktVar">m</span> must be
a natural number, and <span class="RktVar">x</span> may not be zero or a negative integer. Note that
<span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi%29%29" class="RktValLink" data-pltdoc="x">psi</a></span><span class="stt"> </span><span class="RktVal">0</span><span class="stt"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi0%29%29" class="RktValLink" data-pltdoc="x">psi0</a></span><span class="stt"> </span><span class="RktVar">x</span><span class="RktPn">)</span>.</div></p><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=for.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._for%252Flist%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">for/list</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktPn">[</span><span class="RktVar">m</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=sequences.html%23%2528def._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._in-range%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">in-range</a></span><span class="hspace"> </span><span class="RktVal">4</span><span class="RktPn">)</span><span class="RktPn">]</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=lambda.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._%7Ece%7Ebb%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">λ</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi%29%29" class="RktValLink" data-pltdoc="x">psi</a></span><span class="hspace"> </span><span class="RktVar">m</span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-2</span>.5</span><span class="hspace"> </span><span class="RktVal">2.5</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:color</span><span class="hspace"> </span><span class="RktVar">m</span><span class="hspace"> </span><span class="RktPn">#:style</span><span class="hspace"> </span><span class="RktVar">m</span><span class="hspace"> </span><span class="RktPn">#:label</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=Writing.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._format%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">format</a></span><span class="hspace"> </span><span class="RktVal">"psi~a(x)"</span><span class="hspace"> </span><span class="RktVar">m</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:y-min</span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-3</span>00</span><span class="hspace"> </span><span class="RktPn">#:y-max</span><span class="hspace"> </span><span class="RktVal">300</span><span class="hspace"> </span><span class="RktPn">#:legend-anchor</span><span class="hspace"> </span><span class="RktVal">'</span><span class="RktVal">top-right</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_11.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi%29%29" class="RktValLink" data-pltdoc="x">psi</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="hspace"> </span><span class="RktVal">2.3</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">psi: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: Natural</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: -1</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">argument position: 1st</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">other arguments...:</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">2.3</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi%29%29" class="RktValLink" data-pltdoc="x">psi</a></span><span class="hspace"> </span><span class="RktVal">0</span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span>.1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">10.154163959143848</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi0%29%29" class="RktValLink" data-pltdoc="x">psi0</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span>.1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">10.154163959143848</span></p></td></tr></table></blockquote></div></p><p>From spot checks with <span class="RktVar">m</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3e%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">></a></span><span class="stt"> </span><span class="RktVal">0</span>, error appears to be as with <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._psi0%29%29" class="RktValLink" data-pltdoc="x">psi0</a></span>: very low except
near negative roots. Near negative roots, relative error is apparently unbounded, but absolute error
is low.</p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><table cellspacing="0" cellpadding="0" class="together"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._erf))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erf%29%29" class="RktValDef RktValLink" data-pltdoc="x">erf</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></td></tr><tr><td><table cellspacing="0" cellpadding="0" class="together"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._erfc))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="RktValDef RktValLink" data-pltdoc="x">erfc</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></td></tr></table></blockquote></div><div class="SIntrapara">Compute the <a href="http://en.wikipedia.org/wiki/Error_function">error function and
complementary error function</a>, respectively. The only exact cases are <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erf%29%29" class="RktValLink" data-pltdoc="x">erf</a></span><span class="stt"> </span><span class="RktVal">0</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">0</span>
and <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="RktValLink" data-pltdoc="x">erfc</a></span><span class="stt"> </span><span class="RktVal">0</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">1</span>.</div></p><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=pairs.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._list%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">list</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erf%29%29" class="RktValLink" data-pltdoc="x">erf</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-2</span></span><span class="hspace"> </span><span class="RktVal">2</span><span class="hspace"> </span><span class="RktPn">#:label</span><span class="hspace"> </span><span class="RktVal">"erf(x)"</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="RktValLink" data-pltdoc="x">erfc</a></span><span class="hspace"> </span><span class="RktPn">#:color</span><span class="hspace"> </span><span class="RktVal">2</span><span class="hspace"> </span><span class="RktPn">#:label</span><span class="hspace"> </span><span class="RktVal">"erfc(x)"</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_12.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erf%29%29" class="RktValLink" data-pltdoc="x">erf</a></span><span class="hspace"> </span><span class="RktVal">0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erf%29%29" class="RktValLink" data-pltdoc="x">erf</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">0.8427007929497148</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="RktValLink" data-pltdoc="x">erfc</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">0.8427007929497148</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erf%29%29" class="RktValLink" data-pltdoc="x">erf</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-0.8427007929497149</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="RktValLink" data-pltdoc="x">erfc</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-0.8427007929497148</span></p></td></tr></table></blockquote></div></p><p>Mathematically, erfc(<span style="font-style: italic">x</span>) = 1 - erf(<span style="font-style: italic">x</span>), but having separate implementations
can help maintain accuracy. To compute an expression containing erf, use <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erf%29%29" class="RktValLink" data-pltdoc="x">erf</a></span> for
<span class="RktVar">x</span> near <span class="RktVal">0.0</span>. For positive <span class="RktVar">x</span> away from <span class="RktVal">0.0</span>,
manipulate <span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="stt"> </span><span class="RktVal">1.0</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="RktValLink" data-pltdoc="x">erfc</a></span><span class="stt"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span> and its surrounding expressions to avoid the subtraction:</p><p><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVal">5.2</span><span class="RktPn">)</span></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="bigfloat.html#%28def._%28%28lib._math%2Fbigfloat..rkt%29._bf-precision%29%29" class="RktValLink" data-pltdoc="x">bf-precision</a></span><span class="hspace"> </span><span class="RktVal">128</span><span class="RktPn">)</span></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">log-erf-x</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="bigfloat.html#%28def._%28%28lib._math%2Fbigfloat..rkt%29._bigfloat-~3erational%29%29" class="RktValLink" data-pltdoc="x">bigfloat->rational</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="bigfloat.html#%28def._%28%28lib._math%2Fbigfloat..rkt%29._bflog%29%29" class="RktValLink" data-pltdoc="x">bflog</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="bigfloat.html#%28def._%28%28lib._math%2Fbigfloat..rkt%29._bferf%29%29" class="RktValLink" data-pltdoc="x">bferf</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="bigfloat.html#%28def._%28%28lib._math%2Fbigfloat..rkt%29._bf%29%29" class="RktValLink" data-pltdoc="x">bf</a></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="flonum.html#%28def._%28%28lib._math%2Fflonum..rkt%29._flulp-error%29%29" class="RktValLink" data-pltdoc="x">flulp-error</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erf%29%29" class="RktValLink" data-pltdoc="x">erf</a></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktSym">log-erf-x</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">873944876280.6095</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="flonum.html#%28def._%28%28lib._math%2Fflonum..rkt%29._flulp-error%29%29" class="RktValLink" data-pltdoc="x">flulp-error</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="hspace"> </span><span class="RktVal">1.0</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="RktValLink" data-pltdoc="x">erfc</a></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktSym">log-erf-x</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">873944876280.6095</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="flonum.html#%28def._%28%28lib._math%2Fflonum..rkt%29._flulp-error%29%29" class="RktValLink" data-pltdoc="x">flulp-error</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="flonum.html#%28def._%28%28lib._math%2Fflonum..rkt%29._fllog1p%29%29" class="RktValLink" data-pltdoc="x">fllog1p</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="RktValLink" data-pltdoc="x">erfc</a></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktSym">log-erf-x</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">1.609486456125461</span></p></td></tr></table></blockquote></div><div class="SIntrapara">For negative <span class="RktVar">x</span> away from <span class="RktVal">0.0</span>, do the same with <span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="RktValLink" data-pltdoc="x">erfc</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="stt"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktVal">1.0</span><span class="RktPn">)</span>.</div></p><p>For <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erf%29%29" class="RktValLink" data-pltdoc="x">erf</a></span>, error is no greater than 2 <a href="flonum.html#%28tech._ulp%29" class="techoutside" data-pltdoc="x"><span class="techinside">ulps</span></a> everywhere that has been tested, and
is almost always no greater than 1. For <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._erfc%29%29" class="RktValLink" data-pltdoc="x">erfc</a></span>, observed error is no greater than 4 ulps,
and is usually no greater than 2.</p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><table cellspacing="0" cellpadding="0" class="together"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._lambert))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValDef RktValLink" data-pltdoc="x">lambert</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._.U%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">U</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Zero%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Zero</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span><span class="RktPn">)</span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></td></tr><tr><td><table cellspacing="0" cellpadding="0" class="together"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._lambert-))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert-%29%29" class="RktValDef RktValLink" data-pltdoc="x">lambert-</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></td></tr></table></blockquote></div><div class="SIntrapara">Compute the <a href="http://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>,
or the inverse of <span class="RktVar">x</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252A%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">*</a></span><span class="stt"> </span><span class="RktSym">y</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._exp%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exp</a></span><span class="stt"> </span><span class="RktSym">y</span><span class="RktPn">)</span><span class="RktPn">)</span>.</div></p><p>This function has two real branches. The <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span> variant computes the upper branch,
and is defined for <span class="RktVar">x</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3e%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">>=</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._exp%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exp</a></span><span class="stt"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="RktPn">)</span><span class="RktPn">)</span>. The <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert-%29%29" class="RktValLink" data-pltdoc="x">lambert-</a></span> variant computes the
lower branch, and is defined for <span style="font-style: italic">negative</span> <span class="RktVar">x</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3e%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">>=</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._exp%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exp</a></span><span class="stt"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="RktPn">)</span><span class="RktPn">)</span>.
The only exact case is <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="stt"> </span><span class="RktVal">0</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">0</span>.</p><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=pairs.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._list%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">list</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._exp%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exp</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert-%29%29" class="RktValLink" data-pltdoc="x">lambert-</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._exp%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exp</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktSym"><a href="flonum.html#%28def._%28%28lib._math%2Fflonum..rkt%29._-min..0%29%29" class="RktValLink" data-pltdoc="x"><span class="nobreak">-m</span>in.0</a></span><span class="hspace"> </span><span class="RktPn">#:color</span><span class="hspace"> </span><span class="RktVal">2</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:y-min</span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-4</span></span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_13.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="hspace"> </span><span class="RktVal">0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._exp%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exp</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-1.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span>/2</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">lambert: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: Real >= (- (exp -1))</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: -1/2</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert-%29%29" class="RktValLink" data-pltdoc="x">lambert-</a></span><span class="hspace"> </span><span class="RktVal">0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">lambert-: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: Negative-Real >= (- (exp -1))</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: 0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">y0</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-0</span>.1</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">y1</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert-%29%29" class="RktValLink" data-pltdoc="x">lambert-</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-0</span>.1</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="stt">> </span><span class="RktSym">y0</span></td></tr><tr><td><p><span class="RktRes">-0.11183255915896297</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktSym">y1</span></td></tr><tr><td><p><span class="RktRes">-3.577152063957297</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252A%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">*</a></span><span class="hspace"> </span><span class="RktSym">y0</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._exp%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exp</a></span><span class="hspace"> </span><span class="RktSym">y0</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-0.1</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252A%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">*</a></span><span class="hspace"> </span><span class="RktSym">y1</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._exp%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exp</a></span><span class="hspace"> </span><span class="RktSym">y1</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-0.10000000000000002</span></p></td></tr></table></blockquote></div></p><p>The Lambert W function often appears in solutions to equations that contain
<span style="font-style: italic">n</span> log(<span style="font-style: italic">n</span>), such as those that describe the running time of divide-and-conquer
algorithms.</p><p><div class="SIntrapara">For example, suppose we have a sort that takes <span class="RktSym">t</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252A%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">*</a></span><span class="stt"> </span><span class="RktSym">c</span><span class="stt"> </span><span class="RktSym">n</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="stt"> </span><span class="RktSym">n</span><span class="RktPn">)</span><span class="RktPn">)</span> time, and we measure
the time it takes to sort an <span class="RktSym">n</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">10000</span>-element list at <span class="RktSym">t</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">0.245</span> ms. Solving for
<span class="RktSym">c</span>, we get
</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">n</span><span class="hspace"> </span><span class="RktVal">10000</span><span class="RktPn">)</span></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">t</span><span class="hspace"> </span><span class="RktVal">0.245</span><span class="RktPn">)</span></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">c</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="hspace"> </span><span class="RktSym">t</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252A%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">*</a></span><span class="hspace"> </span><span class="RktSym">n</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="hspace"> </span><span class="RktSym">n</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="stt">> </span><span class="RktSym">c</span></td></tr><tr><td><p><span class="RktRes">2.6600537016574172e-06</span></p></td></tr></table></blockquote></div><div class="SIntrapara">Now we would like to know how many elements we can sort in 100ms. We solve for <span class="RktSym">n</span> and
use the solution to define a function <span class="RktSym">time->sort-size</span>:
</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym">time->sort-size</span><span class="hspace"> </span><span class="RktSym">t</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528lib._racket%252Fmath..rkt%2529._exact-floor%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exact-floor</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._exp%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">exp</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._lambert%29%29" class="RktValLink" data-pltdoc="x">lambert</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="hspace"> </span><span class="RktSym">t</span><span class="hspace"> </span><span class="RktSym">c</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym">time->sort-size</span><span class="hspace"> </span><span class="RktVal">100</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">2548516</span></p></td></tr></table></blockquote></div><div class="SIntrapara">Testing the solution, we get
</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">lst2</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=pairs.html%23%2528def._%2528%2528lib._racket%252Fprivate%252Flist..rkt%2529._build-list%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">build-list</a></span><span class="hspace"> </span><span class="RktVal">2548516</span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=values.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._values%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">values</a></span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=time.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fmore-scheme..rkt%2529._time%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">time</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=pairs.html%23%2528def._%2528%2528lib._racket%252Fprivate%252Flist..rkt%2529._sort%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">sort</a></span><span class="hspace"> </span><span class="RktSym">lst2</span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3c%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><</a></span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktOut">cpu time: 80 real time: 93 gc time: 0</span></p></td></tr><tr><td><p></p></td></tr></table></blockquote></div></p><p>For both branches, error is no more than 2 <a href="flonum.html#%28tech._ulp%29" class="techoutside" data-pltdoc="x"><span class="techinside">ulps</span></a> everywhere tested.</p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._zeta))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValDef RktValLink" data-pltdoc="x">zeta</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></blockquote></div><div class="SIntrapara">Computes the <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>.
If <span class="RktVar">x</span> is a nonpositive exact integer, <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValLink" data-pltdoc="x">zeta</a></span><span class="stt"> </span><span class="RktVar">x</span><span class="RktPn">)</span> is exact.</div></p><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValLink" data-pltdoc="x">zeta</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-2</span></span><span class="hspace"> </span><span class="RktVal">10</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">#:y-min</span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-4</span></span><span class="hspace"> </span><span class="RktPn">#:y-max</span><span class="hspace"> </span><span class="RktVal">4</span><span class="RktPn">)</span></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_14.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValLink" data-pltdoc="x">zeta</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span>4</span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-2</span></span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_15.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValLink" data-pltdoc="x">zeta</a></span><span class="hspace"> </span><span class="RktVal">0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-1/2</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValLink" data-pltdoc="x">zeta</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">zeta: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: Real, not One</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: 1</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValLink" data-pltdoc="x">zeta</a></span><span class="hspace"> </span><span class="RktVal">1.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-inf.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValLink" data-pltdoc="x">zeta</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">-1/12</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">num</span><span class="hspace"> </span><span class="RktVal">1000000</span><span class="RktPn">)</span></td></tr><tr><td><p></p></td></tr><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=define.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._define%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">define</a></span><span class="hspace"> </span><span class="RktSym">num-coprime</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=for.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._for%252Fsum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">for/sum</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktPn">[</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=stx-patterns.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fstxcase-scheme..rkt%2529.__%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">_</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=sequences.html%23%2528def._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._in-range%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">in-range</a></span><span class="hspace"> </span><span class="RktSym">num</span><span class="RktPn">)</span><span class="RktPn">]</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=if.html%23%2528form._%2528%2528quote._%7E23%7E25kernel%2529._if%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">if</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="number-theory.html#%28def._%28%28lib._math%2Fnumber-theory..rkt%29._coprime~3f%29%29" class="RktValLink" data-pltdoc="x">coprime?</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="base.html#%28def._%28%28lib._math%2Fbase..rkt%29._random-bits%29%29" class="RktValLink" data-pltdoc="x">random-bits</a></span><span class="hspace"> </span><span class="RktVal">16</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="base.html#%28def._%28%28lib._math%2Fbase..rkt%29._random-bits%29%29" class="RktValLink" data-pltdoc="x">random-bits</a></span><span class="hspace"> </span><span class="RktVal">16</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVal">0</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="flonum.html#%28def._%28%28lib._math%2Fflonum..rkt%29._fl%29%29" class="RktValLink" data-pltdoc="x">fl</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="hspace"> </span><span class="RktSym">num-coprime</span><span class="hspace"> </span><span class="RktSym">num</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">0.607901</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValLink" data-pltdoc="x">zeta</a></span><span class="hspace"> </span><span class="RktVal">2</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">0.6079271018540264</span></p></td></tr></table></blockquote></div></p><p>When <span class="RktSym">s</span> is an odd, negative exact integer, <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValLink" data-pltdoc="x">zeta</a></span><span class="stt"> </span><span class="RktSym">s</span><span class="RktPn">)</span> computes
<span class="RktPn">(</span><span class="RktSym">bernoulli</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="stt"> </span><span class="RktVal">1</span><span class="stt"> </span><span class="RktSym">s</span><span class="RktPn">)</span><span class="RktPn">)</span>, which can be rather slow.</p><p>Maximum observed error is 6 <a href="flonum.html#%28tech._ulp%29" class="techoutside" data-pltdoc="x"><span class="techinside">ulps</span></a>, but is usually 3 or less.</p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._eta))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._eta%29%29" class="RktValDef RktValLink" data-pltdoc="x">eta</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></blockquote></div><div class="SIntrapara">Computes the <a href="http://en.wikipedia.org/wiki/Dirichlet_eta_function">Dirichlet eta function</a>.
If <span class="RktVar">x</span> is a nonpositive exact integer, <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._eta%29%29" class="RktValLink" data-pltdoc="x">eta</a></span><span class="stt"> </span><span class="RktVar">x</span><span class="RktPn">)</span> is exact.</div></p><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._eta%29%29" class="RktValLink" data-pltdoc="x">eta</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span>0</span><span class="hspace"> </span><span class="RktVal">6</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_16.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._eta%29%29" class="RktValLink" data-pltdoc="x">eta</a></span><span class="hspace"> </span><span class="RktVal">0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">1/2</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._eta%29%29" class="RktValLink" data-pltdoc="x">eta</a></span><span class="hspace"> </span><span class="RktVal"><span class="nobreak">-1</span></span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">1/4</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._eta%29%29" class="RktValLink" data-pltdoc="x">eta</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">0.6931471805599453</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="hspace"> </span><span class="RktVal">2</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">0.6931471805599453</span></p></td></tr></table></blockquote></div></p><p>When <span class="RktSym">s</span> is an odd, negative exact integer, <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._eta%29%29" class="RktValLink" data-pltdoc="x">eta</a></span><span class="stt"> </span><span class="RktSym">s</span><span class="RktPn">)</span> computes
<span class="RktPn">(</span><span class="RktSym">bernoulli</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._-%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><span class="nobreak">-</span></a></span><span class="stt"> </span><span class="RktVal">1</span><span class="stt"> </span><span class="RktSym">s</span><span class="RktPn">)</span><span class="RktPn">)</span>, which can be rather slow.</p><p>Maximum observed error is 11 <a href="flonum.html#%28tech._ulp%29" class="techoutside" data-pltdoc="x"><span class="techinside">ulps</span></a>, but is usually 4 or less.</p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._hurwitz-zeta))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._hurwitz-zeta%29%29" class="RktValDef RktValLink" data-pltdoc="x">hurwitz-zeta</a></span></span><span class="hspace"> </span><span class="RktVar">s</span><span class="hspace"> </span><span class="RktVar">q</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">s</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">q</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></blockquote></div><div class="SIntrapara">Computes the <a href="http://en.wikipedia.org/wiki/Hurwitz_zeta_function">Hurwitz
zeta function</a> for <span class="RktVar">s</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3e%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">></a></span><span class="stt"> </span><span class="RktVal">1</span> and <span class="RktVar">q</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3e%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">></a></span><span class="stt"> </span><span class="RktVal">0</span>. When <span class="RktVar">s</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">1.0</span> or <span class="RktVar">q</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">0.0</span>,
<span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._hurwitz-zeta%29%29" class="RktValLink" data-pltdoc="x">hurwitz-zeta</a></span><span class="stt"> </span><span class="RktVar">s</span><span class="stt"> </span><span class="RktVar">q</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">+inf.0</span>.</div></p><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=pairs.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._list%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">list</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._zeta%29%29" class="RktValLink" data-pltdoc="x">zeta</a></span><span class="hspace"> </span><span class="RktVal">1.5</span><span class="hspace"> </span><span class="RktVal">5</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer2d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._function%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">function</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=lambda.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._%7Ece%7Ebb%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">λ</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktVar">s</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._hurwitz-zeta%29%29" class="RktValLink" data-pltdoc="x">hurwitz-zeta</a></span><span class="hspace"> </span><span class="RktVar">s</span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:color</span><span class="hspace"> </span><span class="RktVal">2</span><span class="hspace"> </span><span class="RktPn">#:style</span><span class="hspace"> </span><span class="RktVal">'</span><span class="RktVal">long-dash</span><span class="hspace"> </span><span class="RktPn">#:width</span><span class="hspace"> </span><span class="RktVal">2</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_17.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._hurwitz-zeta%29%29" class="RktValLink" data-pltdoc="x">hurwitz-zeta</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVal">1</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">hurwitz-zeta: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: Real > 1</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: 1</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">argument position: 1st</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">other arguments...:</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">1</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._hurwitz-zeta%29%29" class="RktValLink" data-pltdoc="x">hurwitz-zeta</a></span><span class="hspace"> </span><span class="RktVal">1.0</span><span class="hspace"> </span><span class="RktVal">1.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">+inf.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._hurwitz-zeta%29%29" class="RktValLink" data-pltdoc="x">hurwitz-zeta</a></span><span class="hspace"> </span><span class="RktVal">2</span><span class="hspace"> </span><span class="RktVal">1/4</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">17.197329154507113</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252B%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">+</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528lib._racket%252Fmath..rkt%2529._sqr%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">sqr</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528lib._racket%252Fmath..rkt%2529._pi%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">pi</a></span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252A%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">*</a></span><span class="hspace"> </span><span class="RktVal">8</span><span class="hspace"> </span><span class="RktSym"><a href="base.html#%28def._%28%28lib._math%2Fbase..rkt%29._catalan..0%29%29" class="RktValLink" data-pltdoc="x">catalan.0</a></span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">17.19732915450711</span></p></td></tr></table></blockquote></div></p><p>While <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._hurwitz-zeta%29%29" class="RktValLink" data-pltdoc="x">hurwitz-zeta</a></span> currently raises an exception for <span class="RktVar">s</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3c%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><</a></span><span class="stt"> </span><span class="RktVal">1</span>, it may in the future
return real values.</p><p>Maximum observed error is 6 <a href="flonum.html#%28tech._ulp%29" class="techoutside" data-pltdoc="x"><span class="techinside">ulps</span></a>, but is usually 2 or less.</p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._beta))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta%29%29" class="RktValDef RktValLink" data-pltdoc="x">beta</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVar">y</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._.U%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">U</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Exact-.Rational%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Exact-Rational</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span><span class="RktPn">)</span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">y</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></blockquote></div><div class="SIntrapara">Computes the <a href="http://en.wikipedia.org/wiki/Beta_function">beta function</a> for positive
real <span class="RktVar">x</span> and <span class="RktVar">y</span>. Like <span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252A%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">*</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="stt"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="stt"> </span><span class="RktVar">y</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252B%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">+</a></span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktVar">y</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="RktPn">)</span>,
but more accurate.</div></p><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot3d</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer3d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._contour-intervals3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">contour-intervals3d</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta%29%29" class="RktValLink" data-pltdoc="x">beta</a></span><span class="hspace"> </span><span class="RktVal">0.25</span><span class="hspace"> </span><span class="RktVal">2</span><span class="hspace"> </span><span class="RktVal">0.25</span><span class="hspace"> </span><span class="RktVal">2</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">#:angle</span><span class="hspace"> </span><span class="RktVal">250</span><span class="RktPn">)</span></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_18.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta%29%29" class="RktValLink" data-pltdoc="x">beta</a></span><span class="hspace"> </span><span class="RktVal">0</span><span class="hspace"> </span><span class="RktVal">0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">beta: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: positive Real</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: 0</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">argument position: 1st</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">other arguments...:</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta%29%29" class="RktValLink" data-pltdoc="x">beta</a></span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktVal">5</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">1/5</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta%29%29" class="RktValLink" data-pltdoc="x">beta</a></span><span class="hspace"> </span><span class="RktVal">1.0</span><span class="hspace"> </span><span class="RktVal">5.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">0.2</span></p></td></tr></table></blockquote></div></p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._log-beta))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-beta%29%29" class="RktValDef RktValLink" data-pltdoc="x">log-beta</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVar">y</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types-extra..rkt%2529._.U%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">U</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Zero%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Zero</a></span><span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span><span class="RktPn">)</span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">y</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr></table></blockquote></div><div class="SIntrapara">Like <span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta%29%29" class="RktValLink" data-pltdoc="x">beta</a></span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktVar">y</span><span class="RktPn">)</span><span class="RktPn">)</span>, but more accurate and without unnecessary overflow.
The only exact case is <span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-beta%29%29" class="RktValLink" data-pltdoc="x">log-beta</a></span><span class="stt"> </span><span class="RktVal">1</span><span class="stt"> </span><span class="RktVal">1</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">0</span>.</div></p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._gamma-inc))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValDef RktValLink" data-pltdoc="x">gamma-inc</a></span></span><span class="hspace"> </span><span class="RktVar">k</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>[<span class="RktVar">upper?</span><span class="hspace"> </span><span class="RktVar">regularized?</span>]<span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">k</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span><span class="hspace"> </span>=<span class="hspace"> </span><span class="RktVal">#f</span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span><span class="hspace"> </span>=<span class="hspace"> </span><span class="RktVal">#f</span></td></tr></table></blockquote></div><div class="SIntrapara">Computes the <a href="http://en.wikipedia.org/wiki/Incomplete_gamma_function">incomplete
gamma integral</a> for <span class="RktVar">k</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3e%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">></a></span><span class="stt"> </span><span class="RktVal">0</span> and <span class="RktVar">x</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3e%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">>=</a></span><span class="stt"> </span><span class="RktVal">0</span>. When <span class="RktVar">upper?</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">#f</span>, it integrates
from zero to <span class="RktVar">x</span>; otherwise it integrates from <span class="RktVar">x</span> to infinity.</div></p><p>If you are doing statistical work, you should probably use <span class="RktSym"><a href="Real_Distribution_Families.html#%28def._%28%28lib._math%2Fdistributions..rkt%29._gamma-dist%29%29" class="RktValLink" data-pltdoc="x">gamma-dist</a></span> instead, which
is defined in terms of <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span> and is more flexible (e.g. it allows negative <span class="RktVar">x</span>).</p><p><div class="SIntrapara">The following identities should hold:
</div><div class="SIntrapara"><ul><li><p><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="stt"> </span><span class="RktVal">0</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">0</span></p></li><li><p><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="stt"> </span><span class="RktVal">+inf.0</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="RktPn">)</span></p></li><li><p><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252B%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">+</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktVal">#f</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktVal">#t</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="RktPn">)</span> (approximately)</p></li><li><p><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktVar">upper?</span><span class="stt"> </span><span class="RktVal">#t</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktVar">upper?</span><span class="stt"> </span><span class="RktVal">#f</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="RktPn">)</span><span class="RktPn">)</span> (approximately)</p></li><li><p><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="stt"> </span><span class="RktVal">+inf.0</span><span class="stt"> </span><span class="RktVal">#t</span><span class="stt"> </span><span class="RktVal">#t</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">1.0</span></p></li><li><p><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252B%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">+</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktVal">#f</span><span class="stt"> </span><span class="RktVal">#t</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktVal">#t</span><span class="stt"> </span><span class="RktVal">#t</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">1.0</span> (approximately)</p></li></ul></div><div class="SIntrapara"><p><div class="SIntrapara">Examples:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=pairs.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._list%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">list</a></span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot3d</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer3d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._contour-intervals3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">contour-intervals3d</a></span><span class="hspace"> </span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="hspace"> </span><span class="RktVal">0.1</span><span class="hspace"> </span><span class="RktVal">4.5</span><span class="hspace"> </span><span class="RktVal">0</span><span class="hspace"> </span><span class="RktVal">10</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:x-label</span><span class="hspace"> </span><span class="RktVal">"k"</span><span class="hspace"> </span><span class="RktPn">#:y-label</span><span class="hspace"> </span><span class="RktVal">"x"</span><span class="hspace"> </span><span class="RktPn">#:width</span><span class="hspace"> </span><span class="RktVal">210</span><span class="hspace"> </span><span class="RktPn">#:height</span><span class="hspace"> </span><span class="RktVal">210</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot3d</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer3d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._contour-intervals3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">contour-intervals3d</a></span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=lambda.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._%7Ece%7Ebb%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">λ</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktVar">k</span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="hspace"> </span><span class="RktVar">k</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVal">#t</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal">0.1</span><span class="hspace"> </span><span class="RktVal">4.5</span><span class="hspace"> </span><span class="RktVal">0</span><span class="hspace"> </span><span class="RktVal">10</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:x-label</span><span class="hspace"> </span><span class="RktVal">"k"</span><span class="hspace"> </span><span class="RktPn">#:y-label</span><span class="hspace"> </span><span class="RktVal">"x"</span><span class="hspace"> </span><span class="RktPn">#:width</span><span class="hspace"> </span><span class="RktVal">210</span><span class="hspace"> </span><span class="RktPn">#:height</span><span class="hspace"> </span><span class="RktVal">210</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p><span class="RktRes">'(</span><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_19.png" alt="image" width="216" height="216"/><span class="RktRes"></span><span class="hspace"> </span><span class="RktRes"></span><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_20.png" alt="image" width="216" height="216"/><span class="RktRes">)</span></p></td></tr><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot3d</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer3d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._contour-intervals3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">contour-intervals3d</a></span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=lambda.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._%7Ece%7Ebb%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">λ</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktVar">k</span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="hspace"> </span><span class="RktVar">k</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVal">#f</span><span class="hspace"> </span><span class="RktVal">#t</span><span class="RktPn">)</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktVal">0.1</span><span class="hspace"> </span><span class="RktVal">20</span><span class="hspace"> </span><span class="RktVal">0</span><span class="hspace"> </span><span class="RktVal">20</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:x-label</span><span class="hspace"> </span><span class="RktVal">"k"</span><span class="hspace"> </span><span class="RktPn">#:y-label</span><span class="hspace"> </span><span class="RktVal">"x"</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_21.png" alt="image" width="406" height="406"/></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal">4.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">6.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252B%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">+</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="hspace"> </span><span class="RktVal">4.0</span><span class="hspace"> </span><span class="RktVal">0.5</span><span class="hspace"> </span><span class="RktVal">#f</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="hspace"> </span><span class="RktVal">4.0</span><span class="hspace"> </span><span class="RktVal">0.5</span><span class="hspace"> </span><span class="RktVal">#t</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">6.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="hspace"> </span><span class="RktVal">4.0</span><span class="hspace"> </span><span class="RktVal">+inf.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">6.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%252F%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">/</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="hspace"> </span><span class="RktVal">200.0</span><span class="hspace"> </span><span class="RktVal">50.0</span><span class="hspace"> </span><span class="RktVal">#f</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma%29%29" class="RktValLink" data-pltdoc="x">gamma</a></span><span class="hspace"> </span><span class="RktVal">200.0</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">+nan.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="hspace"> </span><span class="RktVal">200.0</span><span class="hspace"> </span><span class="RktVal">50.0</span><span class="hspace"> </span><span class="RktVal">#f</span><span class="hspace"> </span><span class="RktVal">#t</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">2.0247590148473565e-57</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="hspace"> </span><span class="RktVal">0</span><span class="hspace"> </span><span class="RktVal">5.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktErr">gamma-inc: contract violation</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">expected: Positive-Real</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">given: 0</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">argument position: 1st</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">other arguments...:</span></p></td></tr><tr><td><p><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr"></span><span class="hspace"> </span><span class="RktErr">5.0</span></p></td></tr><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="hspace"> </span><span class="RktVal">0.0</span><span class="hspace"> </span><span class="RktVal">5.0</span><span class="RktPn">)</span></td></tr><tr><td><p><span class="RktRes">+inf.0</span></p></td></tr></table></blockquote></div></p></div></p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._log-gamma-inc))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-gamma-inc%29%29" class="RktValDef RktValLink" data-pltdoc="x">log-gamma-inc</a></span></span><span class="hspace"> </span><span class="RktVar">k</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>[<span class="RktVar">upper?</span><span class="hspace"> </span><span class="RktVar">regularized?</span>]<span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">k</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span><span class="hspace"> </span>=<span class="hspace"> </span><span class="RktVal">#f</span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span><span class="hspace"> </span>=<span class="hspace"> </span><span class="RktVal">#f</span></td></tr></table></blockquote></div><div class="SIntrapara">Like <span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span><span class="stt"> </span><span class="RktVar">k</span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktVar">upper?</span><span class="stt"> </span><span class="RktVar">regularized?</span><span class="RktPn">)</span><span class="RktPn">)</span>, but more accurate and without unnecessary
overflow.</div></p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._beta-inc))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta-inc%29%29" class="RktValDef RktValLink" data-pltdoc="x">beta-inc</a></span></span><span class="hspace"> </span><span class="RktVar">a</span><span class="hspace"> </span><span class="RktVar">b</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>[<span class="RktVar">upper?</span><span class="hspace"> </span><span class="RktVar">regularized?</span>]<span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">a</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">b</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span><span class="hspace"> </span>=<span class="hspace"> </span><span class="RktVal">#f</span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span><span class="hspace"> </span>=<span class="hspace"> </span><span class="RktVal">#f</span></td></tr></table></blockquote></div><div class="SIntrapara">Computes the <a href="http://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function">incomplete beta integral</a> for <span class="RktVar">a</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3e%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">></a></span><span class="stt"> </span><span class="RktVal">0</span>, <span class="RktVar">b</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3e%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">></a></span><span class="stt"> </span><span class="RktVal">0</span> and <span class="RktVal">0</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3c%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><=</a></span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3c%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x"><=</a></span><span class="stt"> </span><span class="RktVal">1</span>.
When <span class="RktVar">upper?</span><span class="stt"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._%7E3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">=</a></span><span class="stt"> </span><span class="RktVal">#f</span>, it integrates from zero to <span class="RktVar">x</span>; otherwise, it integrates from
<span class="RktVar">x</span> to one.</div></p><p>If you are doing statistical work, you should probably use <span class="RktSym"><a href="Real_Distribution_Families.html#%28def._%28%28lib._math%2Fdistributions..rkt%29._beta-dist%29%29" class="RktValLink" data-pltdoc="x">beta-dist</a></span> instead, which
is defined in terms of <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta-inc%29%29" class="RktValLink" data-pltdoc="x">beta-inc</a></span> and is more flexible (e.g. it allows negative <span class="RktVar">x</span>).</p><p>Similar identities should hold as with <span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._gamma-inc%29%29" class="RktValLink" data-pltdoc="x">gamma-inc</a></span>.</p><p><div class="SIntrapara">Example:</div><div class="SIntrapara"><blockquote class="SCodeFlow"><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><table cellspacing="0" cellpadding="0" class="RktBlk"><tr><td><span class="stt">> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=plotting.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._plot3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">plot3d</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=plot&rel=renderer3d.html%23%2528def._%2528%2528lib._plot%252Fmain..rkt%2529._isosurfaces3d%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">isosurfaces3d</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=lambda.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._%7Ece%7Ebb%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">λ</a></span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktVar">a</span><span class="hspace"> </span><span class="RktVar">b</span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta-inc%29%29" class="RktValLink" data-pltdoc="x">beta-inc</a></span><span class="hspace"> </span><span class="RktVar">a</span><span class="hspace"> </span><span class="RktVar">b</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVal">#f</span><span class="hspace"> </span><span class="RktVal">#t</span><span class="RktPn">)</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktVal">0.1</span><span class="hspace"> </span><span class="RktVal">2.5</span><span class="hspace"> </span><span class="RktVal">0.1</span><span class="hspace"> </span><span class="RktVal">2.5</span><span class="hspace"> </span><span class="RktVal">0</span><span class="hspace"> </span><span class="RktVal">1</span><span class="hspace"> </span><span class="RktPn">#:label</span><span class="hspace"> </span><span class="RktVal">"beta(a,b,x)"</span><span class="RktPn">)</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:x-label</span><span class="hspace"> </span><span class="RktVal">"a"</span><span class="hspace"> </span><span class="RktPn">#:y-label</span><span class="hspace"> </span><span class="RktVal">"b"</span><span class="hspace"> </span><span class="RktPn">#:z-label</span><span class="hspace"> </span><span class="RktVal">"x"</span></td></tr><tr><td><span class="hspace"> </span><span class="hspace"> </span><span class="RktPn">#:angle</span><span class="hspace"> </span><span class="RktVal">20</span><span class="hspace"> </span><span class="RktPn">#:altitude</span><span class="hspace"> </span><span class="RktVal">20</span><span class="hspace"> </span><span class="RktPn">#:legend-anchor</span><span class="hspace"> </span><span class="RktVal">'</span><span class="RktVal">top</span><span class="RktPn">)</span></td></tr></table></td></tr><tr><td><p><img style="vertical-align: 0px; margin: -3px -3px -3px -3px;" src="pict_22.png" alt="image" width="406" height="406"/></p></td></tr></table></blockquote></div></p><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._log-beta-inc))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._log-beta-inc%29%29" class="RktValDef RktValLink" data-pltdoc="x">log-beta-inc</a></span></span><span class="hspace"> </span><span class="RktVar">a</span><span class="hspace"> </span><span class="RktVar">b</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>[<span class="RktVar">upper?</span><span class="hspace"> </span><span class="RktVar">regularized?</span>]<span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">a</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">b</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Real%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Real</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span><span class="hspace"> </span>=<span class="hspace"> </span><span class="RktVal">#f</span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span><span class="hspace"> </span>=<span class="hspace"> </span><span class="RktVal">#f</span></td></tr></table></blockquote></div><div class="SIntrapara">Like <span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=generic-numbers.html%23%2528def._%2528%2528quote._%7E23%7E25kernel%2529._log%2529%2529&version=6.7" class="RktValLink Sq" data-pltdoc="x">log</a></span><span class="stt"> </span><span class="RktPn">(</span><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._beta-inc%29%29" class="RktValLink" data-pltdoc="x">beta-inc</a></span><span class="stt"> </span><span class="RktVar">a</span><span class="stt"> </span><span class="RktVar">b</span><span class="stt"> </span><span class="RktVar">x</span><span class="stt"> </span><span class="RktVar">upper?</span><span class="stt"> </span><span class="RktVar">regularized?</span><span class="RktPn">)</span><span class="RktPn">)</span>, but more accurate and without unnecessary
overflow.</div></p><p>While most areas of this function have error less than <span class="RktVal">5e-15</span>, when <span class="RktVar">a</span> and <span class="RktVar">b</span>
have very dissimilar magnitudes (e.g. <span class="RktVal">1e-16</span> and <span class="RktVal">1e+16</span>), it exhibits
<a href="flonum.html#%28tech._catastrophic._cancellation%29" class="techoutside" data-pltdoc="x"><span class="techinside">catastrophic cancellation</span></a>. We are working on it.</p><h4 x-source-module="(lib "math/scribblings/math.scrbl")" x-source-pkg="math-doc" x-part-tag=""flonum-functions"">3.2<tt> </tt><a name="(part._flonum-functions)"></a>Flonum Functions</h4><p><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._flgamma))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flgamma%29%29" class="RktValDef RktValLink" data-pltdoc="x">flgamma</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._fllog-gamma))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllog-gamma%29%29" class="RktValDef RktValLink" data-pltdoc="x">fllog-gamma</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._flpsi0))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flpsi0%29%29" class="RktValDef RktValLink" data-pltdoc="x">flpsi0</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._flpsi))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flpsi%29%29" class="RktValDef RktValLink" data-pltdoc="x">flpsi</a></span></span><span class="hspace"> </span><span class="RktVar">m</span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">m</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Integer%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Integer</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._flerf))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flerf%29%29" class="RktValDef RktValLink" data-pltdoc="x">flerf</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._flerfc))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flerfc%29%29" class="RktValDef RktValLink" data-pltdoc="x">flerfc</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._fllambert))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllambert%29%29" class="RktValDef RktValLink" data-pltdoc="x">fllambert</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._fllambert-))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllambert-%29%29" class="RktValDef RktValLink" data-pltdoc="x">fllambert-</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._flzeta))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flzeta%29%29" class="RktValDef RktValLink" data-pltdoc="x">flzeta</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._fleta))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fleta%29%29" class="RktValDef RktValLink" data-pltdoc="x">fleta</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._flhurwitz-zeta))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flhurwitz-zeta%29%29" class="RktValDef RktValLink" data-pltdoc="x">flhurwitz-zeta</a></span></span><span class="hspace"> </span><span class="RktVar">s</span><span class="hspace"> </span><span class="RktVar">q</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">s</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">q</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._flbeta))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flbeta%29%29" class="RktValDef RktValLink" data-pltdoc="x">flbeta</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVar">y</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">y</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._fllog-beta))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllog-beta%29%29" class="RktValDef RktValLink" data-pltdoc="x">fllog-beta</a></span></span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVar">y</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">y</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._flgamma-inc))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flgamma-inc%29%29" class="RktValDef RktValLink" data-pltdoc="x">flgamma-inc</a></span></span><span class="hspace"> </span><span class="RktVar">k</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">k</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._fllog-gamma-inc))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllog-gamma-inc%29%29" class="RktValDef RktValLink" data-pltdoc="x">fllog-gamma-inc</a></span></span><span class="hspace"> </span><span class="RktVar">k</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">k</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._flbeta-inc))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._flbeta-inc%29%29" class="RktValDef RktValLink" data-pltdoc="x">flbeta-inc</a></span></span><span class="hspace"> </span><span class="RktVar">a</span><span class="hspace"> </span><span class="RktVar">b</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">a</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">b</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span></td></tr></table></blockquote></div><div class="SIntrapara"><blockquote class="SVInsetFlow"><table cellspacing="0" cellpadding="0" class="boxed RBoxed"><tr><td><blockquote class="SubFlow"><div class="RBackgroundLabel SIEHidden"><div class="RBackgroundLabelInner"><p>procedure</p></div></div><p class="RForeground"><span class="RktPn">(</span><a name="(def._((lib._math/special-functions..rkt)._fllog-beta-inc))"></a><span title="Provided from: math/special-functions | Package: math-lib"><span class="RktSym"><a href="#%28def._%28%28lib._math%2Fspecial-functions..rkt%29._fllog-beta-inc%29%29" class="RktValDef RktValLink" data-pltdoc="x">fllog-beta-inc</a></span></span><span class="hspace"> </span><span class="RktVar">a</span><span class="hspace"> </span><span class="RktVar">b</span><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="RktPn">)</span><span class="hspace"> </span>→<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></p></blockquote></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">a</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">b</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">x</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Flonum%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Flonum</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">upper?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span></td></tr><tr><td><span class="hspace"> </span><span class="RktVar">regularized?</span><span class="hspace"> </span>:<span class="hspace"> </span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=type-ref.html%23%2528form._%2528%2528lib._typed-racket%252Fbase-env%252Fbase-types..rkt%2529._.Any%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">Any</a></span></td></tr></table></blockquote></div><div class="SIntrapara">Flonum versions of the above functions. These return <span class="RktVal">+nan.0</span> instead of raising errors and do
not have optional arguments. They can be a little faster to apply because they check fewer special
cases.</div></p><div class="navsetbottom"><span class="navleft"><form class="searchform"><input class="searchbox" style="color: #888;" type="text" value="...search manuals..." title="Enter a search string to search the manuals" onkeypress="return DoSearchKey(event, this, "6.7", "../");" onfocus="this.style.color="black"; this.style.textAlign="left"; if (this.value == "...search manuals...") this.value="";" onblur="if (this.value.match(/^ *$/)) { this.style.color="#888"; this.style.textAlign="center"; this.value="...search manuals..."; }"/></form> <a href="../index.html" title="up to the documentation top" data-pltdoc="x" onclick="return GotoPLTRoot("6.7");">top</a></span><span class="navright"> <a href="flonum.html" title="backward to "2 Flonums"" data-pltdoc="x">← prev</a> <a href="index.html" title="up to "Math Library"" data-pltdoc="x">up</a> <a href="number-theory.html" title="forward to "4 Number Theory"" data-pltdoc="x">next →</a></span> </div></div></div><div id="contextindicator"> </div></body></html>
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