integral identities

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21: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

35.7.5

{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left% (a,c-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}-\mathbf{X}\right|}^{% c-a-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{T}\mathbf{X}\right|}^{-b}\,% \mathrm{d}{\mathbf{X}},

\Re\left(a\right),\Re\left(c-a\right)>\frac{1}{2}(m-1)

\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, $b$ : complex variable, $<$ : less-than for matrices, $c$ : complex variable, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Referenced by:: §35.7(iv)
Permalink:: http://dlmf.nist.gov/35.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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22: 22.8 Addition Theorems

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22.8.22

z_{1}+z_{2}+z_{3}+z_{4}=2K\left(k\right).

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Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(iii), §22.8 and Ch.22

… ►For these and related identities see Copson (1935, pp. 415–416). ►If sums/differences of the

z_{j}

’s are rational multiples of

K\left(k\right)

, then further relations follow. … ►

22.8.24

z_{1}-z_{2}=z_{2}-z_{3}=\tfrac{2}{3}K\left(k\right),

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Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E24
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(iii), §22.8 and Ch.22

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22.8.26

z_{1}-z_{2}=z_{2}-z_{3}=z_{3}-z_{4}=\tfrac{1}{2}K\left(k\right),

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Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(iii), §22.8 and Ch.22

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National Bureau of Standards (1958) tabulates $A_{0}(x)\equiv\pi\operatorname{Hi}\left(-x\right)$ and $-A_{0}^{\prime}(x)\equiv\pi\operatorname{Hi}'\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$ ; $\int_{0}^{x}A_{0}(t)\,\mathrm{d}t$ for $x=0.5,1(1)11$ . Precision is 8D.

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24: 7.12 Asymptotic Expansions

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§7.12(ii) Fresnel Integrals

►The asymptotic expansions of

C\left(z\right)

and

S\left(z\right)

are given by (7.5.3), (7.5.4), and … ►They are bounded by

|\csc\left(4\operatorname{ph}z\right)|

times the first neglected terms when

\frac{1}{8}\pi\leq|\operatorname{ph}z|<\frac{1}{4}\pi

. … ►

§7.12(iii) Goodwin–Staton Integral

►See Olver (1997b, p. 115) for an expansion of

G\left(z\right)

with bounds for the remainder for real and complex values of

z

25: 23.6 Relations to Other Functions

… ►In (23.6.27)–(23.6.29) the modulus

k

is given and

K=K\left(k\right)

{K^{\prime}}=K\left(k^{\prime}\right)

are the corresponding complete elliptic integrals (§19.2(ii)). … ►

§23.6(iv) Elliptic Integrals

… ►For (23.6.30)–(23.6.35) and further identities see Lawden (1989, §6.12). … ►For relations to symmetric elliptic integrals see §19.25(vi). … ►

26: Bibliography K

… ►

J. Kamimoto (1998) On an integral of Hardy and Littlewood. Kyushu J. Math. 52 (1), pp. 249–263.

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External Links:: ISSN 1340-6116, Document, MathReview (Fritz Haslinger), ZentralBlatt
Referenced by:: §9.13(ii).
Encodings:: BibTeX

… ►

E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §19.12.
Encodings:: BibTeX

… ►

A. Khare, A. Lakshminarayan, and U. Sukhatme (2003) Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.

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External Links:: ISSN 0022-2488, Document, MathReview (Zhi Guo Liu), ZentralBlatt
Referenced by:: §22.9(iv), §22.9(v).
Encodings:: BibTeX

►

A. Khare and U. Sukhatme (2002) Cyclic identities involving Jacobi elliptic functions. J. Math. Phys. 43 (7), pp. 3798–3806.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §22.9(i), §22.9(ii), §22.9(iii), §22.9(iv), §22.9(v).
Encodings:: BibTeX

… ►

A. N. Kirillov (1995) Dilogarithm identities. Progr. Theoret. Phys. Suppl. (118), pp. 61–142.

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External Links:: ISSN 0375-9687, FullText, MathReview (J. Browkin), ZentralBlatt
Referenced by:: §25.12(i).
Encodings:: BibTeX

…

27: 1.13 Differential Equations

… ►Then the following relation is known as Abel’s identity … ►

1.13.13

w(z)=W(z)\exp\left(-\tfrac{1}{2}\int f(z)\,\mathrm{d}z\right)

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $z$ : variable, $w(z)$ : solution, $f(z)$ : analytic coefficient and $W(\xi)$ : solution
Permalink:: http://dlmf.nist.gov/1.13.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

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1.13.16

\eta=\int\exp\left(-\int f(z)\,\mathrm{d}z\right)\,\mathrm{d}z.

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Defines:: $\eta$ : change of variable (locally)
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $z$ : variable and $f(z)$ : analytic coefficient
Permalink:: http://dlmf.nist.gov/1.13.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

… ►

Cayley’s Identity

… ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues,

\lambda

; (ii) the corresponding (real) eigenfunctions,

u(x)

and

w(t)

, have the same number of zeros, also called nodes, for

t\in(0,c)

as for

x\in(a,b)

; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

28: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►In this section we will only consider the special case

w(x)=1

, so

\,\mathrm{d}\alpha(x)=\,\mathrm{d}x

; in which case

L^{2}\left(X\right)\equiv L^{2}\left(X,\,\mathrm{d}x\right)

. … ►where the integral kernel is given by … ►Other choices of boundary conditions, identical for

f(x)

and

g(x)

, and which also lead to the vanishing of the boundary terms in (1.18.26), each lead to a distinct self adjoint extension of

T

. … ► It is to be noted that if any of the

\lambda\in\boldsymbol{\sigma}

have degenerate sub-spaces, that is subspaces of orthogonal eigenfunctions with identical eigenvalues, that in the expansions below all such distinct eigenfunctions are to be included. … ►What then is the condition on

q(x)

to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if

q(x)\equiv 0

then there is only a continuous spectrum. …

29: 2.10 Sums and Sequences

… ►This identity can be used to find asymptotic approximations for large

n

when the factor

v_{j}

changes slowly with

j

, and

u_{j}

is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). …

30: Bibliography S

… ►

I. Shavitt and M. Karplus (1965) Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.

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External Links:: Document, MathReview (R. K. Nesbet)
Referenced by:: §8.24(i).
Encodings:: BibTeX

… ►

B. L. Shea (1988) Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.

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Notes:: Double-precision Fortran.
External Links:: FullText, Code
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

L. Shen (1998) On an identity of Ramanujan based on the hypergeometric series

{}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{1}{2};x)

. J. Number Theory 69 (2), pp. 125–134.

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External Links:: ISSN 0022-314X, Document, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §20.11(iii).
Encodings:: BibTeX

… ►

R. Sitaramachandrarao and B. Davis (1986) Some identities involving the Riemann zeta function. II. Indian J. Pure Appl. Math. 17 (10), pp. 1175–1186.

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External Links:: ISSN 0019-5588, MathReview (Jean-Marie De Koninck), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

… ►

B. D. Sleeman (1969) Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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External Links:: Document, MathReview (J. D. DePree), ZentralBlatt
Referenced by:: §31.10(ii).
Encodings:: BibTeX

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