/usr/share/doc/axiom-doc/hypertex/calseries5.xhtml

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  <div align="center"><img align="middle" src="doctitle.png"/></div>
  <hr/>
  <div align="center">Converting to Power Series</div>
  <hr/>
The <a href="db.xhtml?ExpressionToUnivariatePowerSeries">
ExpressionToUnivariatePowerSeries</a> package provides operations for
computing series expansions of functions. 

Evaluate this to compute the Taylor expansion of sin x about x=0. The first
argument, sin(x), specifies the function whose series expansion is to be
computed and the second argument, x=0, specifies that the series is to be
expanded in powers of (x-0), that is, in powers of x.
<ul>
 <li>
  <input type="submit" id="p1" class="subbut" 
    onclick="makeRequest('p1');"
    value="taylor(sin(x),x=0)" />
  <div id="ansp1"><div></div></div>
 </li>
</ul>
Here is the Taylor expansion of sin x about x=%pi/6:
<ul>
 <li>
  <input type="submit" id="p2" class="subbut" 
    onclick="makeRequest('p2');"
    value="taylor(sin(x),x=%pi/6)" />
  <div id="ansp2"><div></div></div>
 </li>
</ul>
The function to be expanded into a series may have variables other than the
series variable. For example, we may expand tan(x*y) as a Taylor series in x.
<ul>
 <li>
  <input type="submit" id="p3" class="subbut" 
    onclick="makeRequest('p3');"
    value="taylor(tan(x*y),x=0)" />
  <div id="ansp3"><div></div></div>
 </li>
</ul>
or as a Taylor series in y.
<ul>
 <li>
  <input type="submit" id="p4" class="subbut" 
    onclick="makeRequest('p4');"
    value="taylor(tan(x*y),y=0)" />
  <div id="ansp4"><div></div></div>
 </li>
</ul>
A more interesting function it (t*%e^(x*t))/(%e^t-1).
When we expand this function as a Taylor series in t the nth order
coefficient is the nth Bernoulli polynomial divided by n!.
<ul>
 <li>
  <input type="submit" id="p5" class="subbut" 
    onclick="makeRequest('p5');"
    value="bern:=taylor(t*exp(x*t)/(exp(t)-1),t=0)" />
  <div id="ansp5"><div></div></div>
 </li>
</ul>
Therefore, this and the next expression produce the same result.
<ul>
 <li>
  <input type="submit" id="p6" class="subbut" 
    onclick="handleFree(['p5','p6']);"
    value="factorial(6)*coefficient(bern,6)" />
  <div id="ansp6"><div></div></div>
 </li>
 <li>
  <input type="submit" id="p7" class="subbut" 
    onclick="handleFree(['p6','p7']);"
    value="bernoulliB(6,x)" />
  <div id="ansp7"><div></div></div>
 </li>
</ul>
Technically, a series with terms of negative degree is not considered to
be a Taylor series, but rather a Laurent series. If you try to compute a
Taylor series expansion of x/log(x) at x=1 via taylor(x/log(x),x=1) you 
get an error message. The reason is that the function has a pole at x=1,
meaning that its series expansion about this point has terms of negative
degree. A series with finitely many terms of negative degree is called a
Laurent series. You get the desired series expansion by issuing this.
<ul>
 <li>
  <input type="submit" id="p8" class="subbut" 
    onclick="makeRequest('p8');"
    value="laurent(x/log(x),x=1)" />
  <div id="ansp8"><div></div></div>
 </li>
</ul>
Similarly, a series with terms of fractional degree is neither a Taylor
series nor a Laurent series. Such a series is called a Puiseux series. The
expression laurent(sqrt(sec(x)),x=3*%pi/2) results in an error message 
because the series expansion about this point has terms of fractional degree.
However, this command produces what you want.
<ul>
 <li>
  <input type="submit" id="p9" class="subbut" 
    onclick="makeRequest('p9');"
    value="puiseux(sqrt(sec(x)),x=3*%pi/2)" />
  <div id="ansp9"><div></div></div>
 </li>
</ul>
Finally, consider the case of functions that do not have Puiseux expansions
about certain points. An example of this is x^x about x=0. puiseux(x^x,x=0)
produces an error message because of the type of singularity of the 
function at x=0. The general function <a href="dbopseries.xhtml">series</a>
can be used in this case. Notice that the series returned is not, strictly
speaking, a power series because of the log(x) in the expansion.
<ul>
 <li>
  <input type="submit" id="p10" class="subbut" 
    onclick="makeRequest('p10');"
    value="series(x^x,x=0)" />
  <div id="ansp10"><div></div></div>
 </li>
</ul>
<hr/>
The operation <a href="dbopseries.xhtml">series</a> returns the most general
type of infinite series. The user who is not interested in distinguishing
between various types of infinite series may wish to use this operation
exclusively.
<hr/>
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axiom-hypertex-data 20120501-8 / usr / share / doc / axiom-doc / hypertex / calseries5.xhtml
