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

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- Solution d

Solution

The specialization of the Jacobi triple product to the Jacobi theta function is

θ (τ) = \prod_{n \geq 1} (1 - q^{2 n}) (1 + q^{2 n - 1})^{2}, q = e^{πi τ} .

Since

τ \in h

, one has

∣ q ∣ < 1

, so every displayed factor is nonzero. Moreover

\sum_{n \geq 1} (∣ q ∣^{2 n} + 2∣ q ∣^{2 n - 1}) < \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

h

.

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