orthogonal functions with respect to weighted summation

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31: 22.15 Inverse Functions

§22.15 Inverse Functions

… ►The inverse Jacobian elliptic functions can be defined in an analogous manner to the inverse trigonometric functions (§4.23). …are denoted respectively by …The general solutions of (22.15.1), (22.15.2), (22.15.3) are, respectively, … ►

§22.15(ii) Representations as Elliptic Integrals

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32: 30.11 Radial Spheroidal Wave Functions

§30.11 Radial Spheroidal Wave Functions

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§30.11(i) Definitions

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Connection Formulas

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§30.11(ii) Graphics

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§30.11(iv) Wronskian

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33: 28.12 Definitions and Basic Properties

… ►As in §28.7 values of $q$ for which (28.2.16) has simple roots $\lambda$ are called normal values with respect to $\nu$. … ►

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

… ►The Floquet solution with respect to $\nu$ is denoted by ${\mathrm{me}}_{\nu }(z,q)$. …They have the following pseudoperiodic and orthogonality properties: … ► …

34: 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).

35: 1.16 Distributions

… ►A mapping $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is a linear functional if …$\mathrm{\Lambda }:𝒟(I)\to ℂ$ 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 $𝒟(I)$, … ►We denote a regular distribution by ${\mathrm{\Lambda }}_f$, or simply $f$, where $f$ is the function giving rise to the distribution. … ►where $H\left(x\right)$ is the Heaviside function defined in (1.16.13), and the derivatives are to be understood in the sense of distributions. … ►Friedman (1990) gives an overview of generalized functions and their relation to distributions. …

36: 16.17 Definition

§16.17 Definition

… ►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 $p=q\ge 1$.

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

37: 21.2 Definitions

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§21.2(i) Riemann Theta Functions

… ► $\theta \left(𝐳\right|𝛀)$ is also referred to as a theta function with $g$ components, a $g$-dimensional theta function or as a genus $g$ theta function. … ►

§21.2(ii) Riemann Theta Functions with Characteristics

… ►This function is referred to as a Riemann theta function with characteristics $\left[\begin{array}{c}𝜶\\ 𝜷\end{array}\right]$. … ►

§21.2(iii) Relation to Classical Theta Functions

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38: 22.2 Definitions

§22.2 Definitions

… ►As a function of $z$, with fixed $k$, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … ►The Jacobian functions are related in the following way. … ►In terms of Neville’s theta functions (§20.1) …

39: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

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§35.7(iii) Partial Differential Equations

►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 $f(𝟎)=1$. Subject to the conditions (a)–(c), the function $f(𝐓)={}_2{}^{}F_1^{}(a,b;c;𝐓)$ is the unique solution of each partial differential equation … ►Butler and Wood (2002) applies Laplace’s method (§2.3(iii)) to (35.7.5) to derive uniform asymptotic approximations for the functions …

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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Convergence Properties

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§35.8(ii) Relations to Other Functions

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Confluence

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