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11: 19.13 Integrals of Elliptic Integrals

§19.13 Integrals of Elliptic Integrals

… ►For definite and indefinite integrals of complete elliptic integrals see Byrd and Friedman (1971, pp. 610–612, 615), Prudnikov et al. (1990, §§1.11, 2.16), Glasser (1976), Bushell (1987), and Cvijović and Klinowski (1999). ►For definite and indefinite integrals of incomplete elliptic integrals see Byrd and Friedman (1971, pp. 613, 616), Prudnikov et al. (1990, §§1.10.2, 2.15.2), and Cvijović and Klinowski (1994). … ►

§19.13(iii) Laplace Transforms

►For direct and inverse Laplace transforms for the complete elliptic integrals

K\left(k\right)

E\left(k\right)

, and

D\left(k\right)

see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.

12: 25.12 Polylogarithms

… ►The right-hand side is called Clausen’s integral. … ►

Integral Representation

… ►

§25.12(iii) Fermi–Dirac and Bose–Einstein Integrals

►The Fermi–Dirac and Bose–Einstein integrals are defined by … ►In terms of polylogarithms …

13: 10.71 Integrals

§10.71 Integrals

►

§10.71(i) Indefinite Integrals

… ►

§10.71(ii) Definite Integrals

… ►

§10.71(iii) Compendia

… ►

14: 4.40 Integrals

§4.40 Integrals

… ►

§4.40(ii) Indefinite Integrals

►

4.40.1

\int\sinh x\,\mathrm{d}x=\cosh x,

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral and $x$ : real variable
A&S Ref:: 4.5.77
Permalink:: http://dlmf.nist.gov/4.40.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.40(ii), §4.40 and Ch.4

… ►

§4.40(iii) Definite Integrals

… ►Extensive compendia of indefinite and definite integrals of hyperbolic functions include Apelblat (1983, pp. 96–109), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 139–160), Gröbner and Hofreiter (1950, pp. 160–167), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.4, 1.8, 2.4, 2.8).

15: 14.17 Integrals

§14.17 Integrals

►

§14.17(i) Indefinite Integrals

… ►

§14.17(ii) Barnes’ Integral

… ►

§14.17(iii) Orthogonality Properties

… ►

§14.17(iv) Definite Integrals of Products

…

16: 9.11 Products

… ►

§9.11(iii) Integral Representations

… ►For further integral representations see Reid (1995, 1997a, 1997b). ►

§9.11(iv) Indefinite Integrals

… ►

§9.11(v) Definite Integrals

… ►For further definite integrals see Prudnikov et al. (1990, §1.8.2), Laurenzi (1993), Reid (1995, 1997a, 1997b), and Vallée and Soares (2010, Chapters 3, 4).

17: 4.26 Integrals

§4.26 Integrals

… ►

§4.26(ii) Indefinite Integrals

… ►

§4.26(iii) Definite Integrals

… ►

§4.26(iv) Inverse Trigonometric Functions

… ►Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).

18: 22.14 Integrals

§22.14 Integrals

►

§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions

… ►

§22.14(iii) Other Indefinite Integrals

… ► ►

§22.14(iv) Definite Integrals

…

19: 11.7 Integrals and Sums

§11.7 Integrals and Sums

►

§11.7(i) Indefinite Integrals

… ►

§11.7(ii) Definite Integrals

… ► … ►

§11.7(iv) Integrals with Respect to Order

…

20: 19.31 Probability Distributions

§19.31 Probability Distributions

►

R_{G}\left(x,y,z\right)

and

R_{F}\left(x,y,z\right)

occur as the expectation values, relative to a normal probability distribution in

{\mathbb{R}}^{2}

{\mathbb{R}}^{3}

, of the square root or reciprocal square root of a quadratic form. More generally, let

\mathbf{A}

(

=[a_{r,s}]

) and

\mathbf{B}

(

=[b_{r,s}]

) be real positive-definite matrices with

n

rows and

n

columns, and let

\lambda_{1},\dots,\lambda_{n}

be the eigenvalues of

\mathbf{A}\mathbf{B}^{-1}

. … ►

19.31.2

\int_{{\mathbb{R}}^{n}}(\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x})^{\mu}\exp% \left(-\mathbf{x}^{\mathrm{T}}\mathbf{B}\mathbf{x}\right)\,\mathrm{d}x_{1}% \cdots\,\mathrm{d}x_{n}=\frac{\pi^{n/2}\Gamma\left(\mu+\tfrac{1}{2}n\right)}{% \sqrt{\det\mathbf{B}}\Gamma\left(\tfrac{1}{2}n\right)}R_{\mu}\left(\tfrac{1}{2% },\dots,\tfrac{1}{2};\lambda_{1},\dots,\lambda_{n}\right),

\mu>-\tfrac{1}{2}n

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\mathbb{R}$ : real line, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $n$ : nonnegative integer and $\lambda_{n}$ : eigenvalues
Referenced by:: §19.31
Permalink:: http://dlmf.nist.gov/19.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.31 and Ch.19

►§19.16(iii) shows that for

n=3

the incomplete cases of

R_{F}

and

R_{G}

occur when

\mu=-1/2

and

\mu=1/2

, respectively, while their complete cases occur when

n=2

. …