Neville theta functions

Definition

Let $0 < k < 1$, $k' = \sqrt{1 - k^2}$, let $K = K(k)$ be the complete elliptic integral of the first kind, and let $q$ be the nome of $k$. For complex $\zeta$ let $$\vartheta_1(\zeta) = 2\sum_{n \ge 0} (-1)^n q^{(n + 1/2)^2}\sin((2n + 1)\zeta), \qquad \vartheta_2(\zeta) = 2\sum_{n \ge 0} q^{(n + 1/2)^2}\cos((2n + 1)\zeta),$$ $$\vartheta_3(\zeta) = 1 + 2\sum_{n \ge 1} q^{n^2}\cos(2n\zeta), \qquad \vartheta_4(\zeta) = 1 + 2\sum_{n \ge 1} (-1)^n q^{n^2}\cos(2n\zeta)$$ be the four Jacobi theta functions with nome $q$. With $\zeta = \pi u/(2K)$, the Neville theta functions of $u \in \C$ are $$\theta_s(u) = \sqrt{\frac{\pi}{2kk'K}}\,\vartheta_1(\zeta), \quad \theta_c(u) = \sqrt{\frac{\pi}{2kK}}\,\vartheta_2(\zeta), \quad \theta_d(u) = \sqrt{\frac{\pi}{2K}}\,\vartheta_3(\zeta), \quad \theta_n(u) = \sqrt{\frac{\pi}{2k'K}}\,\vartheta_4(\zeta).$$