= Heat equation for theta functions with characteristics
{title2=$4\pi i\,\partial_\tau\theta=\partial_z^2\theta$}
Normal convergence allows termwise differentiation of the defining <theta function with characteristics>. A tau derivative multiplies a summand by $\pi i(n+\alpha)^2$, while its second spatial derivative multiplies it by $-4\pi^2(n+\alpha)^2$. This proves the displayed <heat equation>, linking its lattice-sum construction with diffusion and analytic continuation.
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