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--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
)abbrev category ELEMFUN ElementaryFunctionCategory
++ Category for the elementary functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description: Category for the elementary functions;
ElementaryFunctionCategory(): Category == with
log : $ -> $ ++ log(x) returns the natural logarithm of x.
exp : $ -> $ ++ exp(x) returns %e to the power x.
**: ($, $) -> $ ++ x**y returns x to the power y.
add
if $ has Monoid then
x ** y == exp(y * log x)
)abbrev category TRIGCAT TrigonometricFunctionCategory
++ Category for the trigonometric functions
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description: Category for the trigonometric functions;
TrigonometricFunctionCategory(): Category == with
cos: $ -> $ ++ cos(x) returns the cosine of x.
cot: $ -> $ ++ cot(x) returns the cotangent of x.
csc: $ -> $ ++ csc(x) returns the cosecant of x.
sec: $ -> $ ++ sec(x) returns the secant of x.
sin: $ -> $ ++ sin(x) returns the sine of x.
tan: $ -> $ ++ tan(x) returns the tangent of x.
add
if $ has Ring then
csc x ==
(a := recip(sin x)) case "failed" => error "csc: no reciprocal"
a::$
sec x ==
(a := recip(cos x)) case "failed" => error "sec: no reciprocal"
a::$
tan x == sin x * sec x
cot x == cos x * csc x
)abbrev category ATRIG ArcTrigonometricFunctionCategory
++ Category for the inverse trigonometric functions
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description: Category for the inverse trigonometric functions;
ArcTrigonometricFunctionCategory(): Category == with
acos: $ -> $ ++ acos(x) returns the arc-cosine of x.
acot: $ -> $ ++ acot(x) returns the arc-cotangent of x.
acsc: $ -> $ ++ acsc(x) returns the arc-cosecant of x.
asec: $ -> $ ++ asec(x) returns the arc-secant of x.
asin: $ -> $ ++ asin(x) returns the arc-sine of x.
atan: $ -> $ ++ atan(x) returns the arc-tangent of x.
add
if $ has Ring then
asec(x) ==
(a := recip x) case "failed" => error "asec: no reciprocal"
acos(a::$)
acsc(x) ==
(a := recip x) case "failed" => error "acsc: no reciprocal"
asin(a::$)
)abbrev category HYPCAT HyperbolicFunctionCategory
++ Category for the hyperbolic trigonometric functions
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description: Category for the hyperbolic trigonometric functions;
HyperbolicFunctionCategory(): Category == with
cosh: $ -> $ ++ cosh(x) returns the hyperbolic cosine of x.
coth: $ -> $ ++ coth(x) returns the hyperbolic cotangent of x.
csch: $ -> $ ++ csch(x) returns the hyperbolic cosecant of x.
sech: $ -> $ ++ sech(x) returns the hyperbolic secant of x.
sinh: $ -> $ ++ sinh(x) returns the hyperbolic sine of x.
tanh: $ -> $ ++ tanh(x) returns the hyperbolic tangent of x.
add
if $ has Ring then
csch x ==
(a := recip(sinh x)) case "failed" => error "csch: no reciprocal"
a::$
sech x ==
(a := recip(cosh x)) case "failed" => error "sech: no reciprocal"
a::$
tanh x == sinh x * sech x
coth x == cosh x * csch x
if $ has ElementaryFunctionCategory then
cosh x ==
e := exp x
(e + recip(e)::$) * recip(2::$)::$
sinh(x):$ ==
e := exp x
(e - recip(e)::$) * recip(2::$)::$
)abbrev category AHYP ArcHyperbolicFunctionCategory
++ Category for the inverse hyperbolic trigonometric functions
++ Author: ???
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the inverse hyperbolic trigonometric functions;
ArcHyperbolicFunctionCategory(): Category == with
acosh: $ -> $ ++ acosh(x) returns the hyperbolic arc-cosine of x.
acoth: $ -> $ ++ acoth(x) returns the hyperbolic arc-cotangent of x.
acsch: $ -> $ ++ acsch(x) returns the hyperbolic arc-cosecant of x.
asech: $ -> $ ++ asech(x) returns the hyperbolic arc-secant of x.
asinh: $ -> $ ++ asinh(x) returns the hyperbolic arc-sine of x.
atanh: $ -> $ ++ atanh(x) returns the hyperbolic arc-tangent of x.
)abbrev category TRANFUN TranscendentalFunctionCategory
++ Category for the transcendental elementary functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description: Category for the transcendental elementary functions;
TranscendentalFunctionCategory(): Category ==
Join(TrigonometricFunctionCategory,ArcTrigonometricFunctionCategory,
HyperbolicFunctionCategory,ArcHyperbolicFunctionCategory,
ElementaryFunctionCategory) with
pi : () -> $ ++ pi() returns the constant pi.
add
if $ has Ring then
pi() == 2*asin(1)
acsch x ==
(a := recip x) case "failed" => error "acsch: no reciprocal"
asinh(a::$)
asech x ==
(a := recip x) case "failed" => error "asech: no reciprocal"
acosh(a::$)
acoth x ==
(a := recip x) case "failed" => error "acoth: no reciprocal"
atanh(a::$)
if $ has Field and $ has sqrt: $ -> $ then
asin x == atan(x/sqrt(1-x**2))
acos x == pi()/2::$ - asin x
acot x == pi()/2::$ - atan x
asinh x == log(x + sqrt(x**2 + 1))
acosh x == 2*log(sqrt((x+1)/2::$) + sqrt((x-1)/2::$))
atanh x == (log(1+x)-log(1-x))/2::$
)abbrev category PRIMCAT PrimitiveFunctionCategory
++ Category for the integral functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description: Category for the functions defined by integrals;
PrimitiveFunctionCategory(): Category == with
integral: ($, Symbol) -> $
++ integral(f, x) returns the formal integral of f dx.
integral: ($, SegmentBinding $) -> $
++ integral(f, x = a..b) returns the formal definite integral
++ of f dx for x between \spad{a} and b.
)abbrev category LFCAT LiouvillianFunctionCategory
++ Category for the transcendental Liouvillian functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description: Category for the transcendental Liouvillian functions;
LiouvillianFunctionCategory(): Category ==
Join(PrimitiveFunctionCategory, TranscendentalFunctionCategory) with
Ei : $ -> $
++ Ei(x) returns the exponential integral of x, i.e.
++ the integral of \spad{exp(x)/x dx}.
Si : $ -> $
++ Si(x) returns the sine integral of x, i.e.
++ the integral of \spad{sin(x) / x dx}.
Ci : $ -> $
++ Ci(x) returns the cosine integral of x, i.e.
++ the integral of \spad{cos(x) / x dx}.
li : $ -> $
++ li(x) returns the logarithmic integral of x, i.e.
++ the integral of \spad{dx / log(x)}.
dilog : $ -> $
++ dilog(x) returns the dilogarithm of x, i.e.
++ the integral of \spad{log(x) / (1 - x) dx}.
erf : $ -> $
++ erf(x) returns the error function of x, i.e.
++ \spad{2 / sqrt(%pi)} times the integral of \spad{exp(-x**2) dx}.
)abbrev category CFCAT CombinatorialFunctionCategory
++ Category for the usual combinatorial functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 14 May 1991
++ Description: Category for the usual combinatorial functions;
CombinatorialFunctionCategory(): Category == with
binomial : ($, $) -> $
++ binomial(n,r) returns the \spad{(n,r)} binomial coefficient
++ (often denoted in the literature by \spad{C(n,r)}).
++ Note: \spad{C(n,r) = n!/(r!(n-r)!)} where \spad{n >= r >= 0}.
factorial : $ -> $
++ factorial(n) computes the factorial of n
++ (denoted in the literature by \spad{n!})
++ Note: \spad{n! = n (n-1)! when n > 0}; also, \spad{0! = 1}.
permutation: ($, $) -> $
++ permutation(n, m) returns the number of
++ permutations of n objects taken m at a time.
++ Note: \spad{permutation(n,m) = n!/(n-m)!}.
)abbrev category SPFCAT SpecialFunctionCategory
++ Category for the other special functions
++ Author: Manuel Bronstein
++ Date Created: ???
++ Date Last Updated: 11 May 1993
++ Description: Category for the other special functions;
SpecialFunctionCategory(): Category == with
abs : $ -> $
++ abs(x) returns the absolute value of x.
Gamma: $ -> $
++ Gamma(x) is the Euler Gamma function.
Beta: ($,$)->$
++ Beta(x,y) is \spad{Gamma(x) * Gamma(y)/Gamma(x+y)}.
digamma: $ -> $
++ digamma(x) is the logarithmic derivative of \spad{Gamma(x)}
++ (often written \spad{psi(x)} in the literature).
polygamma: ($, $) -> $
++ polygamma(k,x) is the \spad{k-th} derivative of \spad{digamma(x)},
++ (often written \spad{psi(k,x)} in the literature).
Gamma: ($, $) -> $
++ Gamma(a,x) is the incomplete Gamma function.
besselJ: ($,$) -> $
++ besselJ(v,z) is the Bessel function of the first kind.
besselY: ($,$) -> $
++ besselY(v,z) is the Bessel function of the second kind.
besselI: ($,$) -> $
++ besselI(v,z) is the modified Bessel function of the first kind.
besselK: ($,$) -> $
++ besselK(v,z) is the modified Bessel function of the second kind.
airyAi: $ -> $
++ airyAi(x) is the Airy function \spad{Ai(x)}.
airyBi: $ -> $
++ airyBi(x) is the Airy function \spad{Bi(x)}.
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