{{for|other θ functions|Theta function (disambiguation)}}
In mathematics, the '''Neville theta functions''', named after 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 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 ''θ<sub>p</sub>(z,m)'' are sometimes defined in terms of the nome ''q(m)'' and written ''θ<sub>p</sub>(z,q)'' (e.g. NIST<ref name="DLMF20"/>). The functions may also be written in terms of the ''τ'' parameter ''θ<sub>p</sub>(z|τ)'' 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== {| |200px |200px |200px |200px |}
==Notes== {{reflist}}
==References== *{{AS ref}} *{{cite book | last = Neville | first = E. H. (Eric Harold) | author-link =Eric Harold Neville | 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