Jacobi Theta Functions

These are the definitions of the Jacobi theta functions as singly-infinite sums, taken from Wolfram MathWorld .

\[\vartheta_1(z,q) = 2q^{1/4} \sum_{n=0}^{\infty} (-1)^n q^{n(n+1)} \sin[(2n+1)z]\]

\[\vartheta_2(z,q) = 2q^{1/4} \sum_{n=0}^{\infty} q^{n(n+1)} \cos[(2n+1)z]\]

\[\vartheta_3(z,q) = 1 + 2 \sum_{n=1}^{\infty} q^{n^2} \cos(2nz)\]

\[\vartheta_4(z,q) = 1 + 2 \sum_{n=1}^{\infty} (-1)^n q^{n^2} \cos(2nz)\]