for 3F2 hypergeometric functions of matrix argument

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31: 1.10 Functions of a Complex Variable

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Phase (or Argument) Principle

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Analytic Functions

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§1.10(vi) Multivalued Functions

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§1.10(vii) Inverse Functions

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§1.10(xi) Generating Functions

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32: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

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§11.10(vi) Relations to Other Functions

… ► ►For $n=1,2,3,\mathrm{\dots }$, … ►

§11.10(viii) Expansions in Series of Products of Bessel Functions

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33: 25.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$k,m,n$	nonnegative integers.
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primes	on function symbols: derivatives with respect to argument.

►The main function treated in this chapter is the Riemann zeta function $\zeta \left(s\right)$. … ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

34: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

… ►Graphs of the principal values for real arguments are given in §4.29. This section also indicates conformal mappings, and surface plots for complex arguments. … ►

Other Inverse Functions

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§4.37(vi) Interrelations

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35: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

… ►Graphs of the principal values for real arguments are given in §4.15. This section also includes conformal mappings, and surface plots for complex arguments. … ►

Other Inverse Functions

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§4.23(viii) Gudermannian Function

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36: 12.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. ►The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, and $W(a,z)$. …An older notation, due to Whittaker (1902), for $U(a,z)$ is $D_{\nu }\left(z\right)$. …

37: 14.1 Special Notation

§14.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). …

38: 30.11 Radial Spheroidal Wave Functions

§30.11 Radial Spheroidal Wave Functions

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§30.11(i) Definitions

… ►In (30.11.3) $z\ne 0$ when $j=1$, and $|z|>1$ when $j=2,3,4$. ►

Connection Formulas

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§30.11(ii) Graphics

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39: 22.15 Inverse Functions

§22.15 Inverse Functions

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§22.15(i) Definitions

… ►Each of these inverse functions is multivalued. The principal values satisfy … ►

40: 16.17 Definition

§16.17 Definition

… ►Then the Meijer $G$-function is defined via the Mellin–Barnes integral representation: … ►When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer $G$-function. … ►Then ►

16.17.2 $$G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})=\sum _{k=1}^m{A}_{p,q,k}^{m,n}(z){}_p{}^{}F_{q-1}^{}(\genfrac{}{}{0pt}{}{1+b_k-a_1,\mathrm{\dots },1+b_k-a_p}{1+b_k-b_1,\mathrm{\dots }\ast \mathrm{\dots },1+b_k-b_q};{(-1)}^{p-m-n}z),$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $A_{p,q}^{m,n}(z)$: coefficient

Permalink:

http://dlmf.nist.gov/16.17.E2

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See also:

Annotations for §16.17 and Ch.16

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