Heat equation for theta functions with characteristics ($4\pi i{\partial }_{\tau }\theta ={\partial }_z^2\theta$)

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.