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

The specialization of the Jacobi triple product to the Jacobi theta function isSince $\tau \in \mathfrak{h}$, one has $|q|<1$, so every displayed factor is nonzero. MoreoverThe 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 $\mathfrak{h}$.