= Solution
The specialization of the <Jacobi triple product> to the <Jacobi theta function> is
$$
\theta(\tau)=\prod_{n\geq1}(1-q^{2n})(1+q^{2n-1})^2,
\qquad q=e^{\pi i\tau}.
$$
Since $\tau\in\mathfrak h$, one has $|q|<1$, so every displayed factor is nonzero. Moreover
$$
\sum_{n\geq1}\bigl(|q|^{2n}+2|q|^{2n-1}\bigr)<\infty.
$$
The standard convergence criterion for <infinite products> therefore shows that the product converges to a nonzero value. This proves the <nonvanishing of the Jacobi theta function> throughout $\mathfrak h$.
Back to article page
