Theta function

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In mathematics, a theta function is an analytic function which satisfies a particular kind of functional equation. Theta functions occur as generating functions associated to quadratic forms. They are modular forms of half-integer weight.

Jacobi's theta function is a power series in \exp(\pi i \tau) (traditionally referred to as the nome)

\vartheta(\tau) = \sum_{n \in \mathbf{Z}} \mathrm{e}^{\pi i n^2 \tau} .\,

Jacobi's triple product identity

\prod_{m=1}^\infty(1-x^{2n})(1+x^{2n-1}z^2)(1+x^{2n-1}z^{-2}) = \sum_{m \in \mathbf{Z}} x^{m^2}z^{2m} .

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