Editing Theta function (section)

{{Short description|Special functions of several complex variables}}
{{for|other θ functions|Theta function (disambiguation)}}

[[File:Cplot of Jacobi theta 1.svg|400px|thumb|upright=1.2|Jacobi's theta function {{math|''θ''<sub>1</sub>}} with nome {{math|''q'' {{=}} ''e''<sup>''i''π''τ''</sup> {{=}} 0.1''e''<sup>0.1''i''π</sup>}}:
<math>\begin{align}
\theta_1(z,q) &= 2 q^\frac14 \sum_{n=0}^\infty (-1)^n q^{n(n+1)} \sin(2n+1)z \\
&= \sum_{n=-\infty}^\infty (-1)^{n-\frac12} q^{\left(n+\frac12\right)^2} e^{(2n+1)i z} 
.\end{align}</math>]]

In [[mathematics]], '''theta functions''' are [[special function]]s of [[several complex variables]]. They show up in many topics, including [[Abelian variety|Abelian varieties]], [[moduli space]]s, [[quadratic form]]s, and [[soliton]]s. Theta functions are parametrized by points in a [[tube domain]] inside a complex [[Lagrangian Grassmannian]],<ref name="Tyurin2002">{{cite arXiv|last1=Tyurin|first1=Andrey N.|title=Quantization, Classical and Quantum Field Theory and Theta-Functions|eprint=math/0210466v1|date=30 October 2002}}</ref> namely the [[Siegel upper half space]].

The most common form of theta function is that occurring in the theory of [[elliptic function]]s. With respect to one of the complex variables (conventionally called {{mvar|z}}), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a [[quasiperiodic function]]. In the abstract theory this quasiperiodicity comes from the [[Group cohomology|cohomology class]] of a [[Complex torus|line bundle on a complex torus]], a condition of [[descent (category theory)|descent]].

One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".<ref>{{Cite book |last=Chang |first=Der-Chen |title=Heat Kernels for Elliptic and Sub-elliptic Operators |publisher=Birkhäuser |year=2011 |pages=7}}</ref>

Throughout this article, <math>(e^{\pi i\tau})^{\alpha}</math> should be interpreted as <math>e^{\alpha \pi i\tau}</math> (in order to resolve issues of choice of [[Branch point#Branch cuts|branch]]).<ref group="note">See e.g. https://dlmf.nist.gov/20.1. Note that this is, in general, not equivalent to the usual interpretation <math>(e^z)^\alpha=e^{\alpha \operatorname{Log} e^z}</math> when <math>z</math> is outside the strip <math>-\pi<\operatorname{Im}z\le\pi</math>. Here, <math>\operatorname{Log}</math> denotes the principal branch of the [[Complex logarithm#Properties|complex logarithm]].</ref>