 Nonvanishing of the Jacobi theta function

ID: nonvanishing-of-the-jacobi-theta-function

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

Every factor is nonzero for

∣ q ∣ < 1

, and the product converges to a nonzero limit, so

θ

has no zero in the upper half-plane.

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