# Neville theta functions

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

$\theta _{c}(z,m)={\frac {{\sqrt {2\pi }}\,q(m)^{1/4}}{m^{1/4}{\sqrt {K(m)}}}}\,\,\sum _{k=0}^{\infty }(q(m))^{k(k+1)}\cos \left({\frac {(2k+1)\pi z}{2K(m)}}\right)$ $\theta _{d}(z,m)={\frac {\sqrt {2\pi }}{2{\sqrt {K(m)}}}}\,\,\left(1+2\,\sum _{k=1}^{\infty }(q(m))^{k^{2}}\cos \left({\frac {\pi zk}{K(m)}}\right)\right)$ $\theta _{n}(z,m)={\frac {\sqrt {2\pi }}{2(1-m)^{1/4}{\sqrt {K(m)}}}}\,\,\left(1+2\sum _{k=1}^{\infty }(-1)^{k}(q(m))^{k^{2}}\cos \left({\frac {\pi zk}{K(m)}}\right)\right)$ $\theta _{s}(z,m)={\frac {{\sqrt {2\pi }}\,q(m)^{1/4}}{m^{1/4}(1-m)^{1/4}{\sqrt {K(m)}}}}\,\,\sum _{k=0}^{\infty }(-1)^{k}(q(m))^{k(k+1)}\sin \left({\frac {(2k+1)\pi z}{2K(m)}}\right)$ where: K(m) is the complete elliptic integral of the first kind, K'(m)=K(1-m), and $q(m)=e^{-\pi K'(m)/K(m)}$ is the elliptic nome.

Note that the functions θp(z,m) are sometimes defined in terms of the nome q(m) and written θp(z,q) (e.g NIST). The functions may also be written in terms of the τ parameter θp(z|τ) where $q=e^{i\pi \tau }$ .

## Relationship to other functions

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

$\theta _{s}(z|\tau )=\theta _{23}(0|\tau )\theta _{1}(z'|\tau )/\theta '_{1}(0|\tau )$ $\theta _{c}(z|\tau )=\theta _{2}(z'|\tau )/\theta _{2}(0|\tau )$ $\theta _{n}(z|\tau )=\theta _{4}(z'|\tau )/\theta _{4}(0|\tau )$ $\theta _{d}(z|\tau )=\theta _{3}(z'|\tau )/\theta _{3}(0|\tau )$ where $z'=z/\theta _{3}(0|\tau )^{2}$ .

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

$\operatorname {pq} (u,m)={\frac {\theta _{p}(u,m)}{\theta _{q}(u,m)}}$ ## Examples

Substitute z = 2.5, m = 0.3 into the above definitions of Neville theta functions (using Maple) once obtain the following (consistent with results from wolfram math).

• $\theta _{c}(2.5,0.3)=-0.65900466676738154967$ • $\theta _{d}(2.5,0.3)=0.95182196661267561994$ • $\theta _{n}(2.5,0.3)=1.0526693354651613637$ • $\theta _{s}(2.5,0.3)=0.82086879524530400536$ ## Symmetry

• $\theta _{c}(z,m)=\theta _{c}(-z,m)$ • $\theta _{d}(z,m)=\theta _{d}(-z,m)$ • $\theta _{n}(z,m)=\theta _{n}(-z,m)$ • $\theta _{s}(z,m)=-\theta _{s}(-z,m)$ ## Implementation

NetvilleThetaC[z,m], NevilleThetaD[z,m], NevilleThetaN[z,m], and NevilleThetaS[z,m] are built-in functions of Mathematica No such functions in Maple.

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