multidimensional theta functions

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1: 21 Multidimensional Theta Functions

Chapter 21 Multidimensional Theta Functions

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2: Bernard Deconinck

… ►He is the coauthor of several Maple commands to work with Riemann surfaces and the command to compute multidimensional theta functions numerically. … ►

21 Multidimensional Theta Functions

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3: 21.1 Special Notation

… ►The function $\mathrm{\Theta }(\mathit{\varphi }|𝐁)=\theta \left(\mathit{\varphi }/(2\pi \mathrm{i})\right|𝐁/(2\pi \mathrm{i}))$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).

4: 20.2 Definitions and Periodic Properties

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§20.2(i) Fourier Series

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5: 20.11 Generalizations and Analogs

… ►Multidimensional theta functions with characteristics are defined in §21.2(ii) and their properties are described in §§21.3(ii), 21.5(ii), and 21.6. …

6: Software Index

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	Open Source													With Book						Commercial
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21 Multidimensional Theta Functions
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7: 5.14 Multidimensional Integrals

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5.14.7 $\mathrm{\Re }b>-1/n$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $j$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer and $b$: real or complex variable

Referenced by:

§5.14, §5.20

Permalink:

http://dlmf.nist.gov/5.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5

8: 3.5 Quadrature

… ►For a comprehensive survey of quadrature of highly oscillatory integrals, including multidimensional integrals, see Iserles et al. (2006). … ►

Example

… ►The steepest descent path is given by $\mathrm{\Im }\left(t-2\sqrt{t}\right)=0$, or in polar coordinates $t=r{\mathrm{e}}^{\mathrm{i}\theta }$ we have $r={\sec }^2\left(\frac{1}{2}\theta \right)$. … ►For integrals in higher dimensions, Monte Carlo methods are another—often the only—alternative. …