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Homedifferential equations of Jacobi $\vartheta$ functions

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# differential equations of Jacobi $\vartheta$ functions

The theta functions satisfy the following partial differential equation:

${\pi i\over 4}{\partial^{2}\vartheta_{i}\over\partial z^{2}}+{\partial% \vartheta_{i}\over\partial\tau}=0$ |

It is easy to check that each term in the series which define the theta functions satisfies this differential equation. Furthermore, by the Weierstrass M-test, the series obtained by differentiating the series which define the theta functions term-by-term converge absolutely, and hence one may compute derivatives of the theta functions by taking derivatives of the series term-by-term.

Students of mathematical physics will recognize this equation as a one-dimensional diffusion equation. Furthermore, as may be seen by examining the series defining the theta functions, the theta functions approach periodic delta distributions in the limit $\tau\to 0$. Hence, the theta functions are the Green’s functions of the one-dimensional diffusion equation with periodic boundary conditions.

## Mathematics Subject Classification

35H30*no label found*

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