A theta function is a holomorphic function formed by summing an exponential quadratic form over a lattice.
The Jacobi theta function
satisfies and .
For and , the Jacobi triple product is
The theta group is . It is generated by and and has index three in the modular group.
The Jacobi triple product gives
Every factor is nonzero for , and the product converges to a nonzero limit, so has no zero in the upper half-plane.

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The theta function is a special mathematical function often used in various areas of mathematics, including complex analysis, number theory, and mathematical physics. There are several different definitions of theta functions, but the most common ones arise in the context of elliptic functions and modular forms.