Theta function

A theta function is formed by summing an exponential quadratic expression over a Euclidean lattice. A basic one-dimensional example, in period-one spatial normalization, is^[ref]

Gaussian decay gives local uniform convergence, so it is an entire function of $z$ and a holomorphic function of $\tau$. Its value at $z=0$ is the Jacobi theta function in the $Q=e^{\pi i\tau }$ convention:The theta function with characteristics shifts the summation lattice and its phase; evaluating at $z=0$ gives a theta constant.

These sums translate geometric counting into analytic functions. The coefficient of $Q^n$ in $\Theta (0,\tau {)}^d$ counts ordered integer $d$-tuples with sum of squares $n$. For instance, the coefficient of $Q$ in the square is $4$, from $(\pm 1,0)$ and $(0,\pm 1)$. Poisson summation relates the sum over a Euclidean lattice to the sum over its dual lattice, producing the lattice theta functional equation. Termwise differentiation also gives the heat equation for theta functions with characteristics. These connections explain the role of theta functions in modular forms, Euclidean lattice geometry, and diffusion.