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21: Bibliography H

… ►

Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

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External Links:

MathReview (V. L. Makarov)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

…

22: 11.1 Special Notation

… ►Unless indicated otherwise, primes denote derivatives with respect to the argument. For the functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, $I_{\nu }\left(z\right)$, and $K_{\nu }\left(z\right)$ see §§10.2(ii), 10.25(ii). …

23: 35.9 Applications

… ►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument ${}_p{}^{}F_q^{}$, with $p\le 2$ and $q\le 1$. …

24: 17.4 Basic Hypergeometric Functions

… ►

17.4.5 $${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)=\sum _{m,n\ge 0}\frac{{(a;q)}_{m+n}{(b;q)}_m{(b^{\prime };q)}_n{x}^m{y}^n}{{(q;q)}_m{(q;q)}_n{(c;q)}_{m+n}},$$

ⓘ

Defines:

${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$: first $q$-Appell function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.4.E5

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.4(iii), §17.4 and Ch.17

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17.4.6 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\sum _{m,n\ge 0}\frac{{(a;q)}_{m+n}{(b;q)}_m{(b^{\prime };q)}_n{x}^m{y}^n}{{(q,c;q)}_m{(q,c^{\prime };q)}_n},$$

ⓘ

Defines:

${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$: second $q$-Appell function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1, Erratum (V1.1.3) for Equation (17.4.6)

Permalink:

http://dlmf.nist.gov/17.4.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.3):

The explicit usage of ${(q,c;q)}_m{(q,c^{\prime };q)}_n$ in the denominator of the right-hand side was used.

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(2)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.4(iii), §17.4 and Ch.17

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17.4.7 $${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)=\sum _{m,n\ge 0}\frac{{(a,b;q)}_m{(a^{\prime },b^{\prime };q)}_n{x}^m{y}^n}{{(q;q)}_m{(q;q)}_n{(c;q)}_{m+n}},$$

ⓘ

Defines:

${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$: third $q$-Appell function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.4.E7

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(3)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.4(iii), §17.4 and Ch.17

►

17.4.8 $${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)=\sum _{m,n\ge 0}\frac{{(a,b;q)}_{m+n}{x}^m{y}^n}{{(q,c;q)}_m{(q,c^{\prime };q)}_n}.$$

ⓘ

Defines:

${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$: fourth $q$-Appell function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $y$: real variable

Referenced by:

Erratum (V1.0.10) for Section 17.1

Permalink:

http://dlmf.nist.gov/17.4.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.0.10):

The notation for ${\mathrm{\Phi }}^{(4)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument.

See also:

Annotations for §17.4(iii), §17.4 and Ch.17

…

25: Bibliography B

… ►

Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.

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External Links:

ISSN 0002-9939, FullText, MathReview (R. K. Raina), ZentralBlatt

Referenced by:

§16.4(ii), §16.8(ii).

Encodings:

BibTeX

… ►

Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

ⓘ

Notes:

Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.

External Links:

ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt

Referenced by:

§19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).

Encodings:

BibTeX

…

26: 34.7 Basic Properties: $\mathit{9}j$ Symbol

… ►Odd permutations of columns or rows introduce a phase factor ${(-1)}^R$, where $R$ is the sum of all arguments of the $\mathit{9}j$ symbol. …

27: 35.5 Bessel Functions of Matrix Argument

… ►

35.5.1 $$A_{\nu }\left(𝟎\right)=\frac{1}{{\mathrm{\Gamma }}_m\left(\nu +\frac{1}{2}(m+1)\right)},$$ $\nu \in ℂ$.

ⓘ

Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

►

35.5.2 $$A_{\nu }\left(𝐓\right)=A_{\nu }\left(𝟎\right)\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\sum _{|\kappa |=k}\frac{1}{{\left[\nu +\frac{1}{2}(m+1)\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $\nu \in ℂ$, $𝐓\in 𝓢$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $k$: nonnegative integer, $m$: positive integer, $𝟎$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

►

35.5.3 $$B_{\nu }\left(𝐓\right)={\int }_{𝛀}\mathrm{etr}\left(-(𝐓𝐗+{𝐗}^{-1})\right){\left|𝐗\right|}^{\nu -\frac{1}{2}(m+1)}d𝐗,$$ $\nu \in ℂ$, $𝐓\in 𝛀$.

ⓘ

Defines:

$B_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (second kind)

Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices and $m$: positive integer

Referenced by:

§35.9

Permalink:

http://dlmf.nist.gov/35.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

… ►

35.5.7 $${\int }_{𝛀}{A}_{{\nu }_1}\left(𝐓𝐗\right)B_{-{\nu }_2}\left(𝐒𝐗\right){\left|𝐗\right|}^{{\nu }_1}d𝐗=\frac{1}{A_{{\nu }_1+{\nu }_2}\left(𝟎\right)}{\left|𝐒\right|}^{{\nu }_2}{\left|𝐓+𝐒\right|}^{-({\nu }_1+{\nu }_2+\frac{1}{2}(m+1))},$$ $\mathrm{\Re }\left({\nu }_1+{\nu }_2\right)>-1$; $𝐒,𝐓\in 𝛀$.

ⓘ

Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $B_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (second kind), $\in$: element of, $\int$: integral, $\mathrm{\Re }$: real part, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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35.5.8 $${\int }_{𝐎(m)}\mathrm{etr}\left(𝐒𝐇\right)\mathrm{d}𝐇=\frac{A_{-1/2}\left(-\frac{1}{4}𝐒{𝐒}^{\mathrm{T}}\right)}{A_{-1/2}\left(𝟎\right)},$$ $𝐒$ arbitrary.

ⓘ

Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐒$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.10, §35.9

Permalink:

http://dlmf.nist.gov/35.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

…

28: 19.9 Inequalities

… ►

19.9.11 $$\varphi \le F(\varphi ,k)\le \min (\varphi /\mathrm{\Delta },{\mathrm{gd}}^{-1}\left(\varphi \right)),$$

ⓘ

Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Permalink:

http://dlmf.nist.gov/19.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.13 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,0)\le \mathrm{\Pi }(\varphi ,{\alpha }^2,k)\le \min (\mathrm{\Pi }(\varphi ,{\alpha }^2,0)/\mathrm{\Delta },\mathrm{\Pi }(\varphi ,{\alpha }^2,1)).$$

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Symbols:

$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $\mathrm{\Delta }$

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

… ►

19.9.14

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Referenced by:

§19.9(ii), §19.9(ii)

Permalink:

http://dlmf.nist.gov/19.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

►

19.9.15

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Referenced by:

§19.9(ii), §19.9(ii)

Permalink:

http://dlmf.nist.gov/19.9.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.17 $$L\le F(\varphi ,k)\le \sqrt{UL}\le \frac{1}{2}(U+L)\le U,$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $L$: lower limit and $U$: upper limit

Permalink:

http://dlmf.nist.gov/19.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

…

29: 2.11 Remainder Terms; Stokes Phenomenon

… ►However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive …

30: 19.14 Reduction of General Elliptic Integrals

… ►

19.14.1 $${\int }_1^x\frac{dt}{\sqrt{t^3-1}}=3^{-1/4}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{\sqrt{3}+1-x}{\sqrt{3}-1+x}}$, $k^2={\displaystyle \frac{2-\sqrt{3}}{4}}$.

ⓘ

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i)

Permalink:

http://dlmf.nist.gov/19.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

►

19.14.2 $${\int }_x^1\frac{dt}{\sqrt{1-t^3}}=3^{-1/4}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{\sqrt{3}-1+x}{\sqrt{3}+1-x}}$, $k^2={\displaystyle \frac{2+\sqrt{3}}{4}}$.

ⓘ

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\varphi$: real or complex argument and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.3 $${\int }_0^x\frac{dt}{\sqrt{1+t^4}}=\frac{\mathrm{sign}\left(x\right)}{2}F(\varphi ,k),$$ $\cos \varphi ={\displaystyle \frac{1-x^2}{1+x^2}}$, $k^2={\displaystyle \frac{1}{2}}$.

ⓘ

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\int$: integral, $\mathrm{sign}x$: sign of, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i)

Permalink:

http://dlmf.nist.gov/19.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.7 $${\sin }^2\varphi =\frac{(\gamma -\alpha )x^2}{a_1{a}_2+\gamma x^2}.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

… ►

19.14.8 $${\sin }^2\varphi =\frac{\gamma -\alpha }{b_1{b}_2{y}^2+\gamma }.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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