Neville theta functions

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

$θ_{c} (z, m) = \frac{\sqrt{2 π} q (m)^{1 / 4}}{m^{1 / 4} \sqrt{K (m)}} \sum_{k = 0}^{\infty} (q (m))^{k (k + 1)} \cos (\frac{(2 k + 1) π z}{2 K (m)})$

$θ_{d} (z, m) = \frac{\sqrt{2 π}}{2 \sqrt{K (m)}} (1 + 2 \sum_{k = 1}^{\infty} (q (m))^{k^{2}} \cos (\frac{π z k}{K (m)}))$

$θ_{n} (z, m) = \frac{\sqrt{2 π}}{2 (1 - m)^{1 / 4} \sqrt{K (m)}} (1 + 2 \sum_{k = 1}^{\infty} (- 1)^{k} (q (m))^{k^{2}} \cos (\frac{π z k}{K (m)}))$

$θ_{s} (z, m) = \frac{\sqrt{2 π} q (m)^{1 / 4}}{m^{1 / 4} (1 - m)^{1 / 4} \sqrt{K (m)}} \sum_{k = 0}^{\infty} (- 1)^{k} (q (m))^{k (k + 1)} \sin (\frac{(2 k + 1) π z}{2 K (m)})$

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

Relationship to other functions

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

$θ_{s} (z | τ) = θ_{3}^{2} (0 | τ) θ_{1} (z^{'} | τ) / θ'_{1} (0 | τ)$ $θ_{c} (z | τ) = θ_{2} (z^{'} | τ) / θ_{2} (0 | τ)$ $θ_{n} (z | τ) = θ_{4} (z^{'} | τ) / θ_{4} (0 | τ)$ $θ_{d} (z | τ) = θ_{3} (z^{'} | τ) / θ_{3} (0 | τ)$