Theta Functions

A contour plot in the complex plane of the real part of the elliptic modular function phi(z). The unit circle is a natural boundary for this function, blocking any analytic continuation to a larger domain.

Jacobi and Abel first studied theta functions in 1827. They are named EllipticTheta[n, q, z] in Mathematica. Their series definition is:

The elliptic modular function phi(z) is defined by:

Here are a pair of formulas from the myriads in this field: