theta functions

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1: 20.2 Definitions and Periodic Properties

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§20.2(i) Fourier Series

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§20.2(ii) Periodicity and Quasi-Periodicity

… ►The theta functions are quasi-periodic on the lattice: … ►

§20.2(iii) Translation of the Argument by Half-Periods

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§20.2(iv) $z$-Zeros

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2: 21.2 Definitions

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§21.2(i) Riemann Theta Functions

… ► $\theta \left(𝐳\right|𝛀)$ 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. … ►

§21.2(ii) Riemann Theta Functions with Characteristics

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§21.2(iii) Relation to Classical Theta Functions

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3: 21 Multidimensional Theta Functions

Chapter 21 Multidimensional Theta Functions

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4: 21.9 Integrable Equations

§21.9 Integrable Equations

… ►Typical examples of such equations are the Korteweg–de Vries equation … ► … ►

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Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3). … Magnify

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5: 21.10 Methods of Computation

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§21.10(i) General Riemann Theta Functions

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§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface

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Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

6: 20.12 Mathematical Applications

§20.12 Mathematical Applications

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§20.12(i) Number Theory

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§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 $$\vartheta \left(x\right)=1+2\sum _{m=1}^{\mathrm{\infty }}{x}^{m^2},$$ .

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Defines:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation

Symbols:

${\theta }_j(z,q)$: theta function, $m$: positive integer and $x$: real number

A&S Ref:

16.27.3 (with $z=0$, $q=x$)

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

►(In §20.2(i), $\vartheta \left(x\right)$ is denoted by ${\theta }_3(0,x)$.) … ►

27.13.6 $${(\vartheta \left(x\right))}^2=1+4\sum _{n=1}^{\mathrm{\infty }}\left({\delta }_1(n)-{\delta }_3(n)\right)x^n,$$

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Symbols:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation, $n$: positive integer, $x$: real number and ${\delta }_j(n)$: number of divisors

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §27.13(iv), §27.13 and Ch.27

… ►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: 20.11 Generalizations and Analogs

§20.11 Generalizations and Analogs

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§20.11(ii) Ramanujan’s Theta Function and $q$-Series

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§20.11(iv) Theta Functions with Characteristics

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§20.11(v) Permutation Symmetry

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10: 21.3 Symmetry and Quasi-Periodicity

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§21.3(i) Riemann Theta Functions

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§21.3(ii) Riemann Theta Functions with Characteristics

… ► …For Riemann theta functions with half-period characteristics, …