Weierstrass%0Aelliptic functions

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31: 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 … ►

32: 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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33: 28.12 Definitions and Basic Properties

… ►The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict $\widehat{\nu }\ne 0,1$; equivalently $\nu \ne n$. … ►As a function of $\nu$ with fixed $q$ ($\ne 0$), ${\lambda }_{\nu }\left(q\right)$ is discontinuous at $\nu =\pm 1,\pm 2,\mathrm{\dots }$. … ►

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

… ►For $q=0$, … ►However, these functions are not the limiting values of ${\mathrm{me}}_{\pm \nu }(z,q)$ as $\nu \to n$ $(\ne 0)$. …

34: 1.16 Distributions

… ►The closure of the set of points where $\varphi \ne 0$ is called the support of $\varphi$. … ►A sequence $\{{\varphi }_n\}$ of test functions converges to a test function $\varphi$ if the support of every ${\varphi }_n$ is contained in a fixed compact set $K$ and as $n\to \mathrm{\infty }$ the sequence $\{{\varphi }_n^{(k)}\}$ converges uniformly on $K$ to ${\varphi }^{(k)}$ for $k=0,1,2,\mathrm{\dots }$. … ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ 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 $𝒟(I)$, … ►Suppose $f(x)$ is infinitely differentiable except at $x_0$, where left and right derivatives of all orders exist, and … ►For $\alpha >0$, …

35: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

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§35.5(i) Definitions

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§35.5(ii) Properties

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§35.5(iii) Asymptotic Approximations

►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).

36: 16.17 Definition

§16.17 Definition

… ►Assume also that $m$ and $n$ are integers such that $0\le m\le q$ and $0\le n\le p$, and none of $a_k-b_j$ is a positive integer when $1\le k\le n$ and $1\le j\le m$. Then the Meijer $G$-function is defined via the Mellin–Barnes integral representation: … ►

$L$ is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all $z$ ($\ne 0$) if , and for if $p=q\ge 1$.

… ►Then …

37: 22.2 Definitions

§22.2 Definitions

… ►For $k\in [0,1]$, all functions are real for $z\in ℝ$. … ►The Jacobian functions are related in the following way. … ►In terms of Neville’s theta functions (§20.1) …

38: 21.2 Definitions

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§21.2(i) Riemann Theta Functions

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§21.2(ii) Riemann Theta Functions with Characteristics

… ►This function is referred to as a Riemann theta function with characteristics $\left[\begin{array}{c}𝜶\\ 𝜷\end{array}\right]$. …Characteristics whose elements are either $0$ or $\frac{1}{2}$ are called half-period characteristics. … ►

§21.2(iii) Relation to Classical Theta Functions

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39: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

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§35.7(i) Definition

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Jacobi Form

… ►Let $f:𝛀\to ℂ$ (a) be orthogonally invariant, so that $f(𝐓)$ is a symmetric function of $t_1,\mathrm{\dots },t_m$, the eigenvalues of the matrix argument $𝐓\in 𝛀$; (b) be analytic in $t_1,\mathrm{\dots },t_m$ in a neighborhood of $𝐓=𝟎$; (c) satisfy $f(𝟎)=1$. … ►Systems of partial differential equations for the ${}_0{}^{}F_1^{}$ (defined in §35.8) and ${}_1{}^{}F_1^{}$ functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …

40: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(i) Definition

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§35.8(ii) Relations to Other Functions

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Value at $𝐓=𝟎$

… ►A similar result for the ${}_0{}^{}F_1^{}$ function of matrix argument is given in Faraut and Korányi (1994, p. 346). …