3F2 functions of matrix argument

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31: 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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32: 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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33: 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 $D_{\nu }\left(z\right)$. …

34: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

… ► … ►Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … ►

Polynomials

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§16.2(v) Behavior with Respect to Parameters

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35: 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). …

36: 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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37: 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 … ►

38: 28.12 Definitions and Basic Properties

§28.12 Definitions and Basic Properties

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§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization … ► … ►

39: 1.16 Distributions

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§1.16(i) Test Functions

… ► … ► … ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►

§1.16(iv) Heaviside Function

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40: 16.17 Definition

§16.17 Definition

… ►Then the Meijer $G$-function is defined via the Mellin–Barnes integral representation: … ►

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Figure 16.17.1: s-plane. Path $L$ for the integral representation (16.17.1) of the Meijer $G$-function.

►When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer $G$-function. … ►Then …