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 $\mathfrak h$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-137/3/d/solution