Neville theta functions

In mathematics, the Neville theta functions, named after Eric Harold Neville,¹ are defined as follows:^{23 4}

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

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

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

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

where: K(m) is the complete elliptic integral of the first kind, ${\displaystyle K'(m)=K(1-m)}$, and ${\displaystyle q(m)=e^{-\pi K'(m)/K(m)}}$ 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. NIST⁵). The functions may also be written in terms of the τ parameter θ_p(z|τ) where ${\displaystyle q=e^{i\pi \tau }}$.

The Neville theta functions may be expressed in terms of the Jacobi theta functions⁵

${\displaystyle \theta _{s}(z|\tau )=\theta _{3}^{2}(0|\tau )\theta _{1}(z'|\tau )/\theta '_{1}(0|\tau )}$

${\displaystyle \theta _{c}(z|\tau )=\theta _{2}(z'|\tau )/\theta _{2}(0|\tau )}$

${\displaystyle \theta _{n}(z|\tau )=\theta _{4}(z'|\tau )/\theta _{4}(0|\tau )}$

${\displaystyle \theta _{d}(z|\tau )=\theta _{3}(z'|\tau )/\theta _{3}(0|\tau )}$

where ${\displaystyle z'=z/\theta _{3}^{2}(0|\tau )}$.

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

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

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

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