Theta function

Summary

The Jacobi theta function is $\vartheta(z, \tau) = \sum_{n \in \Z} e^{\pi i n^2 \tau + 2 \pi i n z}$ for $z \in \C$ and $\tau$ in the upper half-plane. It is periodic in $z$ with period $1$ and quasi-periodic with period $\tau$. Theta functions are the building blocks of elliptic functions and of functions on abelian varieties. In the variable $\tau$, the theta series of quadratic forms are modular forms that count representations of numbers.