# differential equations of Jacobi $\vartheta $ functions

The theta functions the following partial differential equation^{}:

$$\frac{\pi i}{4}\frac{{\partial}^{2}{\vartheta}_{i}}{\partial {z}^{2}}+\frac{\partial {\vartheta}_{i}}{\partial \tau}=0$$ |

It is easy to check that each in the series which define the theta functions 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.

Title | differential equations of Jacobi $\vartheta $ functions |
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Canonical name | DifferentialEquationsOfJacobivarthetaFunctions |

Date of creation | 2013-03-22 14:41:19 |

Last modified on | 2013-03-22 14:41:19 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 10 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 35H30 |