Editing Theta function (section)

==A solution to the heat equation==
The Jacobi theta function is the [[fundamental solution]] of the one-dimensional [[heat equation]] with spatially periodic boundary conditions.<ref>{{Cite journal|last=Ohyama|first=Yousuke|date=1995|title=Differential relations of theta functions|url=https://projecteuclid.org/euclid.ojm/1200786061|journal=Osaka Journal of Mathematics|language=EN|volume=32|issue=2|pages=431–450|issn=0030-6126}}</ref>  Taking {{math|''z'' {{=}} ''x''}} to be real and {{math|''τ'' {{=}} ''it''}} with {{mvar|t}} real and positive, we can write

:<math>\vartheta (x;it)=1+2\sum_{n=1}^\infty \exp\left(-\pi n^2 t\right) \cos(2\pi nx)</math>

which solves the heat equation

:<math>\frac{\partial}{\partial t} \vartheta(x;it)=\frac{1}{4\pi} \frac{\partial^2}{\partial x^2} \vartheta(x;it).</math>

This theta-function solution is 1-periodic in {{mvar|x}}, and as {{math|''t'' → 0}} it approaches the periodic [[Dirac delta function|delta function]], or [[Dirac comb]], in the sense of [[Distribution (mathematics)|distributions]]

:<math>\lim_{t\to 0} \vartheta(x;it)=\sum_{n=-\infty}^\infty \delta(x-n)</math>.

General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at {{math|''t'' {{=}} 0}} with the theta function.