Neville Theta Function

The functions

$$\vartheta_s(u) = {H(u)\over H'(0)}$$ (1)

$$\vartheta_d(u) = {\Theta(u+K)\over\Theta(k)}$$ (2)

$$\vartheta_c(u) = {H(u)\over H(K)}$$ (3)

$$\vartheta_n(u) = {\Theta(u)\over\Theta(0)},$$ (4)

where $H$ and $\Theta$ are the Jacobi Theta Functions and $K(u)$ is the complete Elliptic Integral of the First Kind.

See also Jacobi Theta Function, Theta Function