# theta functions

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##### 1: 20.2 Definitions and Periodic Properties
###### §20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …
##### 2: 21.2 Definitions
###### §21.2(i) Riemann ThetaFunctions
$\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ is also referred to as a theta function with $g$ components, a $g$-dimensional theta function or as a genus $g$ theta function. … Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. …
##### 4: 21.9 Integrable Equations
###### §21.9 Integrable Equations
Typical examples of such equations are the Korteweg–de Vries equation … Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3). … Magnify
##### 5: 21.10 Methods of Computation
###### §21.10(ii) Riemann ThetaFunctions Associated with a Riemann Surface
• Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

• ##### 6: 20.12 Mathematical Applications
###### §20.12(ii) Uniformization and Embedding of Complex Tori
Thus theta functions “uniformize” the complex torus. …
##### 7: 21.8 Abelian Functions
###### §21.8 Abelian Functions
For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …
##### 8: 27.13 Functions
Jacobi (1829) notes that $r_{2}\left(n\right)$ is the coefficient of $x^{n}$ in the square of the theta function $\vartheta\left(x\right)$:
27.13.4 $\vartheta\left(x\right)=1+2\sum_{m=1}^{\infty}x^{m^{2}},$ $|x|<1$.
(In §20.2(i), $\vartheta\left(x\right)$ is denoted by $\theta_{3}\left(0,x\right)$.) … Mordell (1917) notes that $r_{k}\left(n\right)$ is the coefficient of $x^{n}$ in the power-series expansion of the $k$th power of the series for $\vartheta\left(x\right)$. …
##### 9: 20.8 Watson’s Expansions
###### §20.8 Watson’s Expansions
20.8.1 $\frac{\theta_{2}\left(0,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q% \right)}{\theta_{2}\left(z,q\right)}=2\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q% ^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}.$