<a name=r38_0450>
<title>ZETA</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>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>(<expression>)
<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>
<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>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>
<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>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>(<order>,<argument>)
<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>(<order>,<argument>)
<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>
<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>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>(<order>,<argument>)
<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>
<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>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>(<order>,<argument>)
<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>
<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>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>(<order>,<argument>)
<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>
<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>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>(<order>,<argument>)
<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>
<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>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>(<order>,<argument>)
<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>
<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>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>(<order>,<argument>)
<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>
<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>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>(<parameter>,<parameter>,<argument>)
<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>
<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>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>(<parameter>,<parameter>,<argument>)
<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>
<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>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>(<parameter>,<parameter>,<argument>)
<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>
<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>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>
<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>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>(<argument>)
<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>
<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>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>(<argument>)
<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>
<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>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>(<argument>)
<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>
<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>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>(<argument>)
<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>
<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>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>
<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>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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<integer>,<integer>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<integer>)
<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>(<integer>)
<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>(<expression>,<integer>)
<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>(<integer>)
<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>(<integer>,<expression>,<integer>)
<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>(<expression>,<integer>)
<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>(<expression>,<expression>)
<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>(<expression>)
<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>(<expression>,<expression>)
<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>(<expression>)
<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>
<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>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>(<integer>,<expression>)
<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>
<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 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>
</menu>
<a name=r38_0497>
<title>DILOG_extended</title></a>
<p align="centre"><img src="redlogo.gif" width=621 height=60 border=0 alt="REDUC
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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>.
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<P>
dilog(x) := - defint(log(t)/(t - 1),t,1,x);
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<P> <H3>
syntax: </H3>
<em>Dilog</em>(<order>,<expression>)
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<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.
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<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>(<z>)
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<P>
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<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.
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<P>
<P>
<a name=r38_0499>
<title>Miscellaneous Functions</title></a>
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<b>Miscellaneous Functions</b><menu>
<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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