multidimensional integrals

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1: 5.14 Multidimensional Integrals

§5.14 Multidimensional Integrals

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5.14.7

\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}\prod_{1\leq j<k\leq n}|e^{i\theta_{j% }}-e^{i\theta_{k}}|^{2b}\,\mathrm{d}\theta_{1}\cdots\,\mathrm{d}\theta_{n}=% \frac{\Gamma\left(1+bn\right)}{(\Gamma\left(1+b\right))^{n}},

\Re b>-1/n

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : 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

2: 2.5 Mellin Transform Methods

… ►The first reference also contains explicit expressions for the error terms, as do Soni (1980) and Carlson and Gustafson (1985). ►The Mellin transform method can also be extended to derive asymptotic expansions of multidimensional integrals having algebraic or logarithmic singularities, or both; see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 7), and McClure and Wong (1987). …

3: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions

{{}_{p}F_{q}}

and

{{}_{p+1}F_{p}}

of matrix argument. …These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case

m=1

. …

4: 3.5 Quadrature

… ►For a comprehensive survey of quadrature of highly oscillatory integrals, including multidimensional integrals, see Iserles et al. (2006). … ►For integrals in higher dimensions, Monte Carlo methods are another—often the only—alternative. …

5: Bibliography C

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J. N. L. Connor (1973) Evaluation of multidimensional canonical integrals in semiclassical collision theory. Molecular Phys. 26 (6), pp. 1371–1377.

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External Links:: Document
Referenced by:: §36.8.
Encodings:: BibTeX

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6: Software Index

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	Open Source	With Book		Commercial
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6 Exponential, Logarithmic, Sine, and Cosine Integrals
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11.16(iii) Integrals of …		✓	✓		✓	✓	a
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11.16(vi) Integrals of …		✓			✓	✓	a
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19 Elliptic Integrals
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21 Multidimensional Theta Functions
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7: 21.1 Special Notation

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$g, h$	positive integers.
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$\oint_{a}\omega$	line integral of the differential $\omega$ over the cycle $a$ .

… ►The function

\Theta(\boldsymbol{{\phi}}|\mathbf{B})=\theta\left(\boldsymbol{{\phi}}/(2\pi i% )\middle|\mathbf{B}/(2\pi i)\right)

is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).

8: 20.11 Generalizations and Analogs

… ►As in §20.11(ii), the modulus

k

of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in

q

-series via (20.9.1). … ►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. …