
<a name=r38_0450>

<title>ZETA</title></a>
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E"></p>
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<b>ZETA</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Zeta</em> operator returns Riemann's Zeta function, 
<P>
<P>
Zeta (z) := sum(1/(k**z),k,1,infinity) 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Zeta</em>(&lt;expression&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Zeta(2); 

    2
  pi  / 6 


on rounded; 

Zeta 1.01; 

  100.577943338

</tt></pre><p>Numerical computation for the Zeta function for arguments close to
 1 are 
tedious, because the series is converging very slowly. In this case a formula 
(e.g. found in Bender/Orzag: Advanced Mathematical Methods for 
Scientists and Engineers, McGraw-Hill) is used. 
<P>
<P>
No numerical approximation for complex arguments is done. 
<P>
<P>
<P>

<a name=r38_0451>

<title>Bernoulli Euler Zeta</title></a>
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E"></p>
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<b>Bernoulli Euler Zeta</b><menu>
<li><a href=r38_0400.html#r38_0446>BERNOULLI operator</a><P>
<li><a href=r38_0400.html#r38_0447>BERNOULLIP operator</a><P>
<li><a href=r38_0400.html#r38_0448>EULER operator</a><P>
<li><a href=r38_0400.html#r38_0449>EULERP operator</a><P>
<li><a href=r38_0450.html#r38_0450>ZETA operator</a><P>
</menu>
<a name=r38_0452>

<title>BESSELJ</title></a>
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E"></p>
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<b>BESSELJ</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>BesselJ</em> operator returns the Bessel function of the first kind. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>BesselJ</em>(&lt;order&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
BesselJ(1/2,pi); 

  0 


on rounded; 

BesselJ(0,1); 

  0.765197686558  

</tt></pre><p>
<a name=r38_0453>

<title>BESSELY</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>BESSELY</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>BesselY</em> operator returns the Bessel function of the second kind. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>BesselY</em>(&lt;order&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
BesselY (1/2,pi); 

  - sqrt(2) / pi 


on rounded; 

BesselY (1,3); 

  0.324674424792

</tt></pre><p>The operator <em>BesselY</em> is also called Weber's function. 
<P>
<P>
<P>

<a name=r38_0454>

<title>HANKEL1</title></a>
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E"></p>
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<b>HANKEL1</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Hankel1</em> operator returns the Hankel function of the first kind. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Hankel1</em>(&lt;order&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
on complex; 

Hankel1 (1/2,pi); 

  - i * sqrt(2) / pi 


Hankel1 (1,pi); 

  besselj(1,pi) + i*bessely(1,pi)

</tt></pre><p>The operator <em>Hankel1</em> is also called Bessel function of th
e third kind. 
There is currently no numeric evaluation of Hankel functions. 
<P>
<P>
<P>

<a name=r38_0455>

<title>HANKEL2</title></a>
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E"></p>
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<b>HANKEL2</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Hankel2</em> operator returns the Hankel function of the second kind. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Hankel2</em>(&lt;order&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
on complex; 

Hankel2 (1/2,pi); 

  - i * sqrt(2) / pi 


Hankel2 (1,pi); 

  besselj(1,pi) - i*bessely(1,pi)

</tt></pre><p>The operator <em>Hankel2</em> is also called Bessel function of th
e third kind. 
There is currently no numeric evaluation of Hankel functions. 
<P>
<P>
<P>

<a name=r38_0456>

<title>BESSELI</title></a>
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E"></p>
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<b>BESSELI</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>BesselI</em> operator returns the modified Bessel function I. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>BesselI</em>(&lt;order&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
on rounded; 

Besseli (1,1); 

  0.565159103992

</tt></pre><p>The knowledge about the operator <em>BesselI</em> is currently fai
rly limited. 
<P>
<P>
<P>

<a name=r38_0457>

<title>BESSELK</title></a>
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E"></p>
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<b>BESSELK</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>BesselK</em> operator returns the modified Bessel function K. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>BesselK</em>(&lt;order&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
df(besselk(0,x),x); 

  - besselk(1,x)

</tt></pre><p>There is currently no numeric support for the operator <em>BesselK
</em>. 
<P>
<P>
<P>

<a name=r38_0458>

<title>StruveH</title></a>
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E"></p>
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<b>STRUVEH</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>StruveH</em> operator returns Struve's H function. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>StruveH</em>(&lt;order&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
struveh(-3/2,x); 

  - besselj(3/2,x) / i

</tt></pre><p>
<a name=r38_0459>

<title>StruveL</title></a>
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E"></p>
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<b>STRUVEL</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>StruveL</em> operator returns the modified Struve L function . 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>StruveL</em>(&lt;order&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
struvel(-3/2,x); 

  besseli(3/2,x)

</tt></pre><p>
<a name=r38_0460>

<title>KummerM</title></a>
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E"></p>
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<b>KUMMERM</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>KummerM</em> operator returns Kummer's M function. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>KummerM</em>(&lt;parameter&gt;,&lt;parameter&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
kummerm(1,1,x); 

   x
  e  


on rounded; 

kummerm(1,3,1.3); 

  1.62046942914

</tt></pre><p>Kummer's M function is one of the Confluent Hypergeometric functio
ns. 
For reference see the 
<a href=r38_0500.html#r38_0529>hypergeometric</a> operator. 
<P>
<P>
<P>

<a name=r38_0461>

<title>KummerU</title></a>
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E"></p>
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<b>KUMMERU</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>KummerU</em> operator returns Kummer's U function. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>KummerU</em>(&lt;parameter&gt;,&lt;parameter&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
df(kummeru(1,1,x),x) 

  - kummeru(2,2,x)

</tt></pre><p>Kummer's U function is one of the Confluent Hypergeometric functio
ns. 
For reference see the 
<a href=r38_0500.html#r38_0529>hypergeometric</a> operator. 
<P>
<P>
<P>

<a name=r38_0462>

<title>WhittakerW</title></a>
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<b>WHITTAKERW</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>WhittakerW</em> operator returns Whittaker's W function. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>WhittakerW</em>(&lt;parameter&gt;,&lt;parameter&gt;,&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
WhittakerW(2,2,2); 

                    1
  4*sqrt(2)*kummeru(-,5,2)
                    2
  -------------------------
             e

</tt></pre><p>Whittaker's W function is one of the Confluent Hypergeometric func
tions. 
For reference see the 
<a href=r38_0500.html#r38_0529>hypergeometric</a> operator. 
<P>
<P>
<P>

<a name=r38_0463>

<title>Bessel Functions</title></a>
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E"></p>
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<b>Bessel Functions</b><menu>
<li><a href=r38_0450.html#r38_0452>BESSELJ operator</a><P>
<li><a href=r38_0450.html#r38_0453>BESSELY operator</a><P>
<li><a href=r38_0450.html#r38_0454>HANKEL1 operator</a><P>
<li><a href=r38_0450.html#r38_0455>HANKEL2 operator</a><P>
<li><a href=r38_0450.html#r38_0456>BESSELI operator</a><P>
<li><a href=r38_0450.html#r38_0457>BESSELK operator</a><P>
<li><a href=r38_0450.html#r38_0458>StruveH operator</a><P>
<li><a href=r38_0450.html#r38_0459>StruveL operator</a><P>
<li><a href=r38_0450.html#r38_0460>KummerM operator</a><P>
<li><a href=r38_0450.html#r38_0461>KummerU operator</a><P>
<li><a href=r38_0450.html#r38_0462>WhittakerW operator</a><P>
</menu>
<a name=r38_0464>

<title>Airy_Ai</title></a>
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<b>AIRY_AI</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Airy_Ai</em> operator returns the Airy Ai function for a given argument.
 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Airy_Ai</em>(&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
on complex;
on rounded;
Airy_Ai(0); 


  0.355028053888          


Airy_Ai(3.45 + 17.97i); 

  - 5.5561528511e+9 - 8.80397899932e+9*i  

</tt></pre><p>
<a name=r38_0465>

<title>Airy_Bi</title></a>
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E"></p>
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<b>AIRY_BI</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Airy_Bi</em> operator returns the Airy Bi function for a given 
argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Airy_Bi</em>(&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Airy_Bi(0); 

  0.614926627446          


Airy_Bi(3.45 + 17.97i); 

  8.80397899932e+9 - 5.5561528511e+9*i   

</tt></pre><p>
<a name=r38_0466>

<title>Airy_Aiprime</title></a>
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E"></p>
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<b>AIRY_AIPRIME</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Airy_Aiprime</em> operator returns the Airy Aiprime function for a 
given argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Airy_Aiprime</em>(&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Airy_Aiprime(0); 

  - 0.258819403793           


Airy_Aiprime(3.45+17.97i);

  - 3.83386421824e+19 + 2.16608828136e+19*i 

</tt></pre><p>
<a name=r38_0467>

<title>Airy_Biprime</title></a>
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E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>AIRY_BIPRIME</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Airy_Biprime</em> operator returns the Airy Biprime function for a 
given argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Airy_Biprime</em>(&lt;argument&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Airy_Biprime(0); 


Airy_Biprime(3.45 + 17.97i); 

  3.84251916792e+19 - 2.18006297399e+19*i

</tt></pre><p>
<a name=r38_0468>

<title>Airy Functions</title></a>
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E"></p>
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<b>Airy Functions</b><menu>
<li><a href=r38_0450.html#r38_0464>Airy_Ai operator</a><P>
<li><a href=r38_0450.html#r38_0465>Airy_Bi operator</a><P>
<li><a href=r38_0450.html#r38_0466>Airy_Aiprime operator</a><P>
<li><a href=r38_0450.html#r38_0467>Airy_Biprime operator</a><P>
</menu>
<a name=r38_0469>

<title>JacobiSN</title></a>
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E"></p>
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<b>JACOBISN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobisn</em> operator returns the Jacobi Elliptic function sn. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobisn</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobisn(0.672, 0.36) 

  0.609519691792 


Jacobisn(1,0.9) 

  0.770085724907881 

</tt></pre><p>
<a name=r38_0470>

<title>JacobiCN</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBICN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobicn</em> operator returns the Jacobi Elliptic function cn. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobicn</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobicn(7.2, 0.6) 

  0.837288298482018  


Jacobicn(0.11, 19) 

  0.994403862690043 - 1.6219006985556e-16*i  

</tt></pre><p>
<a name=r38_0471>

<title>JacobiDN</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBIDN</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobidn</em> operator returns the Jacobi Elliptic function dn. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobidn</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobidn(15, 0.683) 

  0.640574162024592 


Jacobidn(0,0) 

  1 

</tt></pre><p>
<a name=r38_0472>

<title>JacobiCD</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBICD</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobicd</em> operator returns the Jacobi Elliptic function cd. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobicd</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobicd(1, 0.34) 

  0.657683337805273 


Jacobicd(0.8,0.8) 

  0.925587311582301 

</tt></pre><p>
<a name=r38_0473>

<title>JacobiSD</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBISD</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobisd</em> operator returns the Jacobi Elliptic function sd. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobisd</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobisd(12, 0.4) 

  0.357189729437272    


Jacobisd(0.35,1) 

  - 1.17713873203043  

</tt></pre><p>
<a name=r38_0474>

<title>JacobiND</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBIND</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobind</em> operator returns the Jacobi Elliptic function nd. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobind</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobind(0.2, 17) 

  1.46553203037507 + 0.0000000000334032759313703*i 


Jacobind(30, 0.001) 

  1.00048958438  

</tt></pre><p>
<a name=r38_0475>

<title>JacobiDC</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBIDC</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobidc</em> operator returns the Jacobi Elliptic function dc. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobidc</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobidc(0.003,1) 

  1 


Jacobidc(2, 0.75) 

  6.43472885111  

</tt></pre><p>
<a name=r38_0476>

<title>JacobiNC</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBINC</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobinc</em> operator returns the Jacobi Elliptic function nc. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobinc</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobinc(1,0) 

  1.85081571768093 


Jacobinc(56, 0.4387) 

  39.304842663512  

</tt></pre><p>
<a name=r38_0477>

<title>JacobiSC</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBISC</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobisc</em> operator returns the Jacobi Elliptic function sc. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobisc</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobisc(9, 0.88) 

  - 1.16417697982095  


Jacobisc(0.34, 7) 

  0.305851938390775 - 9.8768100944891e-12*i 

</tt></pre><p>
<a name=r38_0478>

<title>JacobiNS</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBINS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobins</em> operator returns the Jacobi Elliptic function ns. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobins</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobins(3, 0.9) 

  1.00945801599785 


Jacobins(0.887, 15) 

  0.683578280513975 - 0.85023411082469*i 

</tt></pre><p>
<a name=r38_0479>

<title>JacobiDS</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBIDS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobisn</em> operator returns the Jacobi Elliptic function ds. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobids</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobids(98,0.223) 

  - 1.061253961477 


Jacobids(0.36,0.6) 

  2.76693172243692 

</tt></pre><p>
<a name=r38_0480>

<title>JacobiCS</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBICS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Jacobics</em> operator returns the Jacobi Elliptic function cs. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Jacobics</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Jacobics(0, 0.767) 

  infinity   


Jacobics(1.43, 0) 

  0.141734127352112 

</tt></pre><p>
<a name=r38_0481>

<title>JacobiAMPLITUDE</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBIAMPLITUDE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>JacobiAmplitude</em> operator returns the amplitude of u. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>JacobiAmplitude</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
JacobiAmplitude(7.239, 0.427) 

  0.0520978301448978 


JacobiAmplitude(0,0.1) 

  0 

</tt></pre><p>Amplitude u = asin(<em>Jacobisn(u,m)</em>) 
<P>
<P>
<P>

<a name=r38_0482>

<title>AGM_FUNCTION</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>AGM_FUNCTION</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>AGM_function</em> operator returns a list of (N, AGM, 
 list of aNtoa0, list of bNtob0, list of cNtoc0) where a0, b0 and c0 
are the initial values; N is the index number of the last term 
used to generate the AGM. AGM is the Arithmetic Geometric Mean. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>AGM_function</em>(&lt;integer&gt;,&lt;integer&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
AGM_function(1,1,1) 

  1,1,1,1,1,1,0,1  


AGM_function(1, 0.1, 1.3) 

  {6,
   2.27985615996629, 
   {2.27985615996629, 2.27985615996629,
    2.2798561599706, 2.2798624278857, 
    2.28742283656583, 2.55, 1},
   {2.27985615996629, 2.27985615996629,
    2.27985615996198, 2.2798498920555, 
    2.27230201920557, 2.02484567313166, 4.1},
   {0, 4.30803136219904e-12, 0.0000062679151007581,
    0.00756040868012758, 0.262577163434171, - 1.55, 5.9}}

</tt></pre><p>The other Jacobi functions use this function with initial values 
a0=1, b0=sqrt(1-m), c0=sqrt(m). 
<P>
<P>
<P>

<a name=r38_0483>

<title>LANDENTRANS</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>LANDENTRANS</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>landentrans</em> operator generates the descending landen 
transformation of the given imput values, returning a list of these 
values; initial to final in each case. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>landentrans</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
landentrans(0,0.1) 

  {{0,0,0,0,0},{0.1,0.0025041751943776, 


 

  0.00000156772498954046,6.1444078 9914461e-13,0}}  

</tt></pre><p>The first list ascends in value, and the second descends in value.
 
<P>
<P>
<P>

<a name=r38_0484>

<title>EllipticF</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>ELLIPTICF</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>EllipticF</em> operator returns the Elliptic Integral of the 
First Kind. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>EllitpicF</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
EllipticF(0.3, 8.222) 

  0.3 


EllipticF(7.396, 0.1) 

  7.58123216114307 

</tt></pre><p>The Complete Elliptic Integral of the First Kind can be found by 
putting the first argument to pi/2 or by using <em>EllipticK</em> 
and the second argument. 
<P>
<P>
<P>

<a name=r38_0485>

<title>EllipticK</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>ELLIPTICK</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>EllipticK</em> operator returns the Elliptic value K. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>EllipticK</em>(&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
EllipticK(0.2) 

  1.65962359861053   


EllipticK(4.3) 

  0.808442364282734 - 1.05562492399206*i  


EllipticK(0.000481) 

  1.57098526617635    

</tt></pre><p>The <em>EllipticK</em> function is the Complete Elliptic Integral 
of 
the First Kind. 
<P>
<P>
<P>

<a name=r38_0486>

<title>EllipticKprime</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>ELLIPTICKPRIME</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>EllipticK'</em> operator returns the Elliptic value K(m). 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>EllipticKprime</em>(&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
EllipticKprime(0.2) 

  2.25720532682085 


EllipticKprime(4.3) 

  1.05562492399206 


EllipticKprime(0.000481) 

  5.206621921966   

</tt></pre><p>The <em>EllipticKprime</em> function is the Complete Elliptic Inte
gral of 
the First Kind of (1-m). 
<P>
<P>
<P>

<a name=r38_0487>

<title>EllipticE</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>ELLIPTICE</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>EllipticE</em> operator used with two arguments 
returns the Elliptic Integral of the Second Kind. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>EllipticE</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
EllipticE(1.2,0.22) 

  1.15094019180949 


EllipticE(0,4.35) 

  0                


EllipticE(9,0.00719) 

  8.98312465929145  

</tt></pre><p>The Complete Elliptic Integral of the Second Kind can be obtained 
by 
using just the second argument, or by using pi/2 as the first argument. 
<P>
<P>
<P>
The <em>EllipticE</em> operator used with one argument 
returns the Elliptic value E. 
 <P> <H3> 
syntax: </H3>
<P>
<P>
<em>EllipticE</em>(&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
EllipticE(0.22) 

  1.48046637439519  


EllipticE(pi/2, 0.22) 

  1.48046637439519  

</tt></pre><p>
<a name=r38_0488>

<title>EllipticTHETA</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>ELLIPTICTHETA</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>EllipticTheta</em> operator returns one of the four Theta 
functions. It cannot except any number other than 1,2,3 or 4 as 
its first argument. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>EllipticTheta</em>(&lt;integer&gt;,&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
EllipticTheta(1, 1.4, 0.72) 

  0.91634775373  


EllipticTheta(2, 3.9, 6.1 ) 

  -48.0202736969 + 20.9881034377 i 


EllipticTheta(3, 0.67, 0.2) 

  1.0083077448   


EllipticTheta(4, 8, 0.75) 

  0.894963369304 


EllipticTheta(5, 1, 0.1) 

  ***** In EllipticTheta(a,u,m); a = 1,2,3 or 4.   

</tt></pre><p>Theta functions are important because every one of the Jacobian 
Elliptic functions can be expressed as the ratio of two theta functions. 
<P>
<P>
<P>

<a name=r38_0489>

<title>JacobiZETA</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>JACOBIZETA</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>JacobiZeta</em> operator returns the Jacobian function Zeta. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>JacobiZeta</em>(&lt;expression&gt;,&lt;integer&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
JacobiZeta(3.2, 0.8) 

  - 0.254536403439 


JacobiZeta(0.2, 1.6) 

  0.171766095970451 - 0.0717028569800147*i  

</tt></pre><p>The Jacobian function Zeta is related to the Jacobian function The
ta. 
But it is significantly different from Riemann's Zeta Function 
<a href=r38_0450.html#r38_0450>Zeta</a>. 
<P>
<P>
<P>

<a name=r38_0490>

<title>Jacobi's Elliptic Functions and Elliptic Integrals</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>
<b>Jacobi's Elliptic Functions and Elliptic Integrals</b><menu>
<li><a href=r38_0450.html#r38_0469>JacobiSN operator</a><P>
<li><a href=r38_0450.html#r38_0470>JacobiCN operator</a><P>
<li><a href=r38_0450.html#r38_0471>JacobiDN operator</a><P>
<li><a href=r38_0450.html#r38_0472>JacobiCD operator</a><P>
<li><a href=r38_0450.html#r38_0473>JacobiSD operator</a><P>
<li><a href=r38_0450.html#r38_0474>JacobiND operator</a><P>
<li><a href=r38_0450.html#r38_0475>JacobiDC operator</a><P>
<li><a href=r38_0450.html#r38_0476>JacobiNC operator</a><P>
<li><a href=r38_0450.html#r38_0477>JacobiSC operator</a><P>
<li><a href=r38_0450.html#r38_0478>JacobiNS operator</a><P>
<li><a href=r38_0450.html#r38_0479>JacobiDS operator</a><P>
<li><a href=r38_0450.html#r38_0480>JacobiCS operator</a><P>
<li><a href=r38_0450.html#r38_0481>JacobiAMPLITUDE operator</a><P>
<li><a href=r38_0450.html#r38_0482>AGM_FUNCTION operator</a><P>
<li><a href=r38_0450.html#r38_0483>LANDENTRANS operator</a><P>
<li><a href=r38_0450.html#r38_0484>EllipticF operator</a><P>
<li><a href=r38_0450.html#r38_0485>EllipticK operator</a><P>
<li><a href=r38_0450.html#r38_0486>EllipticKprime operator</a><P>
<li><a href=r38_0450.html#r38_0487>EllipticE operator</a><P>
<li><a href=r38_0450.html#r38_0488>EllipticTHETA operator</a><P>
<li><a href=r38_0450.html#r38_0489>JacobiZETA operator</a><P>
</menu>
<a name=r38_0491>

<title>POCHHAMMER</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>POCHHAMMER</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
<P>
<P>
The <em>Pochhammer</em> operator implements the Pochhammer notation 
(shifted factorial). 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Pochhammer</em>(&lt;expression&gt;,&lt;expression&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
pochhammer(17,4); 

  116280 



pochhammer(1/2,z); 

    factorial(2*z)
  --------------------
    2*z
  (2   *factorial(z))

</tt></pre><p>A number of complex rules for <em>Pochhammer</em> are inactive, be
cause they 
cause a huge system load in algebraic mode. If one wants to use more rules 
for the simplification of Pochhammer's notation, one can do: 
<P>
<P>
let special!*pochhammer!*rules; 
<P>
<P>
<P>
<P>

<a name=r38_0492>

<title>GAMMA</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>GAMMA</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Gamma</em> operator returns the Gamma function. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Gamma</em>(&lt;expression&gt;) 
<P>
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
gamma(10); 

  362880    


gamma(1/2); 

  sqrt(pi)

</tt></pre><p>
<a name=r38_0493>

<title>BETA</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>BETA</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Beta</em> operator returns the Beta function defined by 
<P>
<P>
Beta (z,w) := defint(t**(z-1)* (1 - t)**(w-1),t,0,1) . 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Beta</em>(&lt;expression&gt;,&lt;expression&gt;) 
<P>
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Beta(2,2); 

  1 / 6 


Beta(x,y); 

  gamma(x)*gamma(y) / gamma(x + y)

</tt></pre><p>The operator <em>Beta</em> is simplified towards the 
<a href=r38_0450.html#r38_0492>GAMMA</a> operator. 
<P>
<P>
<P>

<a name=r38_0494>

<title>PSI</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
E"></p>
<b><a href=r38_idx.html>INDEX</a></b><p><p>



<b>PSI</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The <em>Psi</em> operator returns the Psi (or DiGamma) function. 
<P>
<P>
Psi(x) := df(Gamma(z),z)/ Gamma (z) 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Gamma</em>(&lt;expression&gt;) 
<P>
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Psi(3); 

  (2*log(2) + psi(1/2) + psi(1) + 3)/2 


on rounded; 

- Psi(1); 

  0.577215664902

</tt></pre><p>Euler's constant can be found as - Psi(1). 
<P>
<P>
<P>

<a name=r38_0495>

<title>POLYGAMMA</title></a>
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<b>POLYGAMMA</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
The <em>Polygamma</em> operator returns the Polygamma function. 
<P>
<P>
Polygamma(n,x) := df(Psi(z),z,n); 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Polygamma</em>(&lt;integer&gt;,&lt;expression&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
 Polygamma(1,2); 

     2
  (pi   - 6) / 6


on rounded; 

Polygamma(1,2.35); 

  0.52849689109

</tt></pre><p>The Polygamma function is used for simplification of the 
<a href=r38_0450.html#r38_0450>ZETA</a> 
function for some arguments. 
<P>
<P>
<P>

<a name=r38_0496>

<title>Gamma and Related Functions</title></a>
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<b>Gamma and Related Functions</b><menu>
<li><a href=r38_0450.html#r38_0491>POCHHAMMER operator</a><P>
<li><a href=r38_0450.html#r38_0492>GAMMA operator</a><P>
<li><a href=r38_0450.html#r38_0493>BETA operator</a><P>
<li><a href=r38_0450.html#r38_0494>PSI operator</a><P>
<li><a href=r38_0450.html#r38_0495>POLYGAMMA operator</a><P>
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<a name=r38_0497>

<title>DILOG_extended</title></a>
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<b>DILOG EXTENDED</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
 <P>
<P>
The package <em>specfn</em> supplies an extended support for the 

<a href=r38_0050.html#r38_0078>dilog</a> operator which implements the <em>dilog
arithm function</em>. 
<P>
<P>
dilog(x) := - defint(log(t)/(t - 1),t,1,x); 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Dilog</em>(&lt;order&gt;,&lt;expression&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
defint(log(t)/(t - 1),t,1,x); 

  - dilog (x) 


dilog 2; 

      2
  - pi  /12 



on rounded; 

Dilog 20; 

  - 5.92783972438

</tt></pre><p>The operator <em>Dilog</em> is sometimes called Spence's Integral 
for n = 2. 
<P>
<P>
<P>

<a name=r38_0498>

<title>Lambert_W_function</title></a>
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<b>LAMBERT\_W FUNCTION</b> _ _ _  _ _ _  _ _ _  _ _ _ <b>operator</b><P>
<P>
 
Lambert's W function is the inverse of the function w * e**w. 
It is used in the 
<a href=r38_0150.html#r38_0179>solve</a> package for equations containing 
exponentials and logarithms. 
<P>
<P>
 <P> <H3> 
syntax: </H3>
<em>Lambert_W</em>(&lt;z&gt;) 
<P>
<P>
<P>
 <P> <H3> 
examples: </H3>
<p><pre><tt>
Lambert_W(-1/e); 

  -1 


solve(w + log(w),w); 

  w=lambert_w(1)


on rounded; 

Lambert_W(-0.05); 

  - 0.0527059835515

</tt></pre><p>The current implementation will compute the principal branch in 
rounded mode only. 
<P>
<P>
<P>

<a name=r38_0499>

<title>Miscellaneous Functions</title></a>
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<li><a href=r38_0450.html#r38_0497>DILOG extended operator</a><P>
<li><a href=r38_0450.html#r38_0498>Lambert\_W function operator</a><P>
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