Neville theta functions

For other θ functions, see Theta function (disambiguation).

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

${\theta }_c(z,m)=\frac{\sqrt{2\pi }q(m{)}^{1/4}}{m^{1/4}\sqrt{K(m)}}\sum _{k=0}^{\mathrm{\infty }}(q(m){)}^{k(k+1)}\cos \left(\frac{(2k+1)\pi z}{2K(m)}\right)$

${\theta }_d(z,m)=\frac{\sqrt{2\pi }}{2\sqrt{K(m)}}(1+2\sum _{k=1}^{\mathrm{\infty }}(q(m){)}^{k^2}\cos \left(\frac{\pi zk}{K(m)}\right))$

${\theta }_n(z,m)=\frac{\sqrt{2\pi }}{2(1-m{)}^{1/4}\sqrt{K(m)}}(1+2\sum _{k=1}^{\mathrm{\infty }}(-1{)}^k(q(m){)}^{k^2}\cos \left(\frac{\pi zk}{K(m)}\right))$

${\theta }_s(z,m)=\frac{\sqrt{2\pi }q(m{)}^{1/4}}{m^{1/4}(1-m{)}^{1/4}\sqrt{K(m)}}\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k(q(m){)}^{k(k+1)}\sin \left(\frac{(2k+1)\pi z}{2K(m)}\right)$

where: K(m) is the complete elliptic integral of the first kind, $K^{\prime }(m)=K(1-m)$, and $q(m)=e^{-\pi K^{\prime }(m)/K(m)}$ 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 $q=e^{i\pi \tau }$.

Relationship to other functions

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

${\theta }_s(z|\tau )={\theta }_3^2(0|\tau ){\theta }_1(z^{\prime }|\tau )/\theta {\text{'}}_1(0|\tau )$

${\theta }_c(z|\tau )={\theta }_2(z^{\prime }|\tau )/{\theta }_2(0|\tau )$

${\theta }_n(z|\tau )={\theta }_4(z^{\prime }|\tau )/{\theta }_4(0|\tau )$

${\theta }_d(z|\tau )={\theta }_3(z^{\prime }|\tau )/{\theta }_3(0|\tau )$

where $z^{\prime }=z/{\theta }_3^2(0|\tau )$.

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

$\mathrm{pq}(u,m)=\frac{{\theta }_p(u,m)}{{\theta }_q(u,m)}.$

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