Neville Theta Functions

In mathematics, the Neville theta functions, named after Eric Harold Neville, are defined as follows:

: $\theta_c(z,m)=\frac \,\, \sum _^\infty (q(m))^ \cos \left(\frac \right)$

: $\theta_d(z,m)=\frac\,\,\left( 1+2\,\sum _^\infty (q(m))^ \cos \left( \frac \right) \right)$

: $\theta_n(z, m) =\frac \,\,\left( 1+2\sum _^\infty (-1)^k (q(m))^ \cos \left(\frac \right) \right)$

: $\theta_s(z, m)=\frac\,\, \sum_^\infty (-1)^k (q(m))^ \sin\left(\frac \right)$

where: ''K(m)'' is the complete elliptic integral of the first kind, $K'(m)=K(1-m)$, and $q(m)=e^$ is the elliptic nome.

Note that the functions ''θ_p(z,m)'' are sometimes defined in terms of the nome ''q(m)'' and written ''θ_p(z,q)'' (e.g. NIST). The functions may also be written in terms of the ''τ'' parameter ''θ_p(z, τ)'' where $q=e^$.

Relationship to other functions

The Neville theta functions may be expressed in terms of the Jacobi theta functions

:$\theta_s(z, \tau)=\theta_3^2(0, \tau)\theta_1(z', \tau)/\theta'_1(0, \tau)$
:$\theta_c(z, \tau)=\theta_2(z', \tau)/\theta_2(0, \tau)$
:$\theta_n(z, \tau)=\theta_4(z', \tau)/\theta_4(0, \tau)$
:$\theta_d(z, \tau)=\theta_3(z', \tau)/\theta_3(0, \tau)$

where $z'=z/\theta_3^2(0, \tau)$.

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

:$\operatorname(u,m)=\frac.$

Examples

*$\theta_c(2.5, 0.3)\approx -0.65900466676738154967$
*$\theta_d(2.5, 0.3)\approx 0.95182196661267561994$
*$\theta_n(2.5, 0.3)\approx 1.0526693354651613637$
*$\theta_s(2.5, 0.3)\approx 0.82086879524530400536$

Symmetry

*$\theta_c(z,m)=\theta_c(-z,m)$
*$\theta_d(z,m)=\theta_d(-z,m)$
*$\theta_n(z,m)=\theta_n(-z,m)$
*$\theta_s(z,m)=-\theta_s(-z,m)$

Complex 3D plots

Implementation

NetvilleThetaC ,m NevilleThetaD ,m NevilleThetaN ,m and NevilleThetaS ,mare built-in functions of Mathematica.

Notes

References

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*{{MathWorld , urlname=NevilleThetaFunctions , title=Neville Theta Functions

Special functions
Theta functions
Elliptic functions
Analytic functions