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<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 E"></p> <b><a href=r38_idx.html>INDEX</a></b><p><p> <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>(<order>,<expression>) <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> <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>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>) <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> <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>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> </menu>