View source for Neville theta functions

In mathematics, the '''Neville theta functions''', named after [[Biography:Eric Harold Neville|Eric Harold Neville]],<ref>Abramowitz and Stegun, pp. 578-579</ref> are defined as follows:<ref>Neville (1944)</ref><ref>[http://functions.wolfram.com/EllipticFunctions/NevilleThetaC/02/  The Mathematical Functions Site]</ref>
<ref>[http://functions.wolfram.com/EllipticFunctions/NevilleThetaD/02/ The Mathematical Functions Site]</ref>

: <math> \theta_c(z,m)=\frac {\sqrt{2\pi}\,q(m)^{1/4}}{m^{1/4}\sqrt {K(m)}}\,\, \sum _{k=0}^\infty (q(m))^{k(k+1)} \cos \left(\frac{( 2k+1) \pi z}{2 K(m)} \right) 
</math>

: <math> \theta_d(z,m)=\frac{\sqrt{2\pi}}{2\sqrt{K(m)}}\,\,\left( 1+2\,\sum _{k=1}^\infty (q(m))^{k^2} \cos \left( \frac {\pi zk}{K(m)} \right) \right)
</math>

: <math> \theta_n(z, m) =\frac {\sqrt {2\pi }}{2(1-m)^{1/4}\sqrt {K(m)}}\,\,\left( 1+2\sum _{k=1}^\infty (-1)^k (q(m))^{k^2} \cos \left(\frac{\pi zk}{K(m)} \right) \right)
</math>

: <math> \theta_s(z, m)=\frac{\sqrt {2\pi}\,q(m)^{1/4}}{m^{1/4}(1-m)^{1/4}\sqrt{K(m)}}\,\, \sum_{k=0}^\infty (-1)^k (q(m))^{k(k+1) } \sin\left(\frac { (2k+1) \pi z}{2K(m)} \right)
</math>

where: ''K(m)'' is the complete [[Elliptic integral|elliptic integral]] of the first kind, <math>K'(m)=K(1-m)</math>, and  <math>q(m)=e^{-\pi K'(m)/K(m)}</math> is the elliptic nome.

Note that the functions  ''&theta;<sub>p</sub>(z,m)'' are sometimes defined in terms of the nome ''q(m)'' and written ''&theta;<sub>p</sub>(z,q)'' (e.g. NIST<ref name="DLMF20"/>). The functions may also be written in terms of the ''&tau;'' parameter ''&theta;<sub>p</sub>(z|&tau;)'' where <math>q=e^{i\pi\tau}</math>.

==Relationship to other functions==

The Neville theta functions may be expressed in terms of the Jacobi theta functions<ref name="DLMF20">{{cite web|url=http://dlmf.nist.gov/20|title=NIST Digital Library of Mathematical Functions (Release 1.0.17)|editor-last=Olver|editor-first=F. W. J. |display-editors=etal |date=2017-12-22|publisher=National Institute of Standards and Technology|access-date=2018-02-26 }}</ref>

:<math>\theta_s(z|\tau)=\theta_3^2(0|\tau)\theta_1(z'|\tau)/\theta'_1(0|\tau)</math>
:<math>\theta_c(z|\tau)=\theta_2(z'|\tau)/\theta_2(0|\tau)</math>
:<math>\theta_n(z|\tau)=\theta_4(z'|\tau)/\theta_4(0|\tau)</math>
:<math>\theta_d(z|\tau)=\theta_3(z'|\tau)/\theta_3(0|\tau)</math>

where <math>z'=z/\theta_3^2(0|\tau)</math>.

The Neville theta functions are related to the [[Jacobi elliptic functions]]. If pq(u,m) is a Jacobi elliptic function (p and q are one of s,c,n,d), then

:<math>\operatorname{pq}(u,m)=\frac{\theta_p(u,m)}{\theta_q(u,m)}.</math>

==Examples==

*<math>\theta_c(2.5, 0.3)\approx -0.65900466676738154967 </math>
*<math> \theta_d(2.5, 0.3)\approx 0.95182196661267561994 </math>
*<math> \theta_n(2.5, 0.3)\approx 1.0526693354651613637 </math>
*<math> \theta_s(2.5, 0.3)\approx 0.82086879524530400536 </math>

==Symmetry==
*<math>\theta_c(z,m)=\theta_c(-z,m)</math>
*<math>\theta_d(z,m)=\theta_d(-z,m)</math>
*<math>\theta_n(z,m)=\theta_n(-z,m)</math>
*<math>\theta_s(z,m)=-\theta_s(-z,m)</math>

==Complex 3D plots==
{|
|[[File:NevilleThetaC Maple complex plot 01.png|200px]]
|[[File:NevilleThetaD Maple complex plot.png|200px]]
|[[File:NevilleThetaN Maple complex plot.png|200px]]
|[[File:NevilleThetaS Maple complex plot.png|200px]]
|}

==Implementation==
NetvilleThetaC[z,m], NevilleThetaD[z,m], NevilleThetaN[z,m], and NevilleThetaS[z,m] are built-in functions of [[Software:Mathematica|Mathematica]].<ref>{{cite web | url=http://functions.wolfram.com/EllipticFunctions/NevilleThetaD/02/ShowAll.html | title=Neville theta function: Primary definition }}</ref>

==Notes==
{{reflist}}

==References==
*{{AS ref}}
*{{cite book
 | last = Neville
 | first = E. H. (Eric Harold)
 | title =Jacobian Elliptic Functions
 | publisher =Oxford Clarendon Press
 | date =1944
 | url = https://archive.org/details/jacobianelliptic00neviuoft}}
*{{MathWorld | urlname=NevilleThetaFunctions | title=Neville Theta Functions}}

[[Category:Special functions]]
[[Category:Theta functions]]
[[Category:Elliptic functions]]
[[Category:Analytic functions]]

{{Sourceattribution|Neville theta functions}}