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Theta function

Jacobi theta function

The <Jacobi triple product> gives
$$
\theta(\tau)=\prod_{n\geq1}(1-q^{2n})(1+q^{2n-1})^2,
\qquad q=e^{\pi i\tau}.
$$
Every factor is nonzero for $|q|<1$, and the product converges to a nonzero limit, so $\theta$ has no zero in the upper half-plane.

Nonvanishing of the Jacobi theta function

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 3 / d / Solution

Nonvanishing of the Jacobi theta function

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Nonvanishing of the Jacobi theta function

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