# orthogonal functions with respect to weighted summation

(0.005 seconds)

## 1—10 of 1004 matching pages

##### 1: 9.1 Special Notation
(For other notation see Notation for the Special Functions.)
 $k$ nonnegative integer, except in §9.9(iii). … derivatives with respect to argument.
The main functions treated in this chapter are the Airy functions $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, and the Scorer functions $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ (also known as inhomogeneous Airy functions). Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi}\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi}\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-\ifrac{1}{3}}\pi\mathrm{Ai}\left(-3^{-\ifrac{1}{3}}x\right)$ (Szegő (1967, §1.81)); $e_{0}(x)=\pi\mathrm{Hi}(-x)$, $\widetilde{e}_{0}(x)=-\pi\mathrm{Gi}(-x)$ (Tumarkin (1959)).
##### 3: 5.15 Polygamma Functions
###### §5.15 Polygamma Functions
The functions $\psi^{(n)}\left(z\right)$, $n=1,2,\dots$, are called the polygamma functions. In particular, $\psi'\left(z\right)$ is the trigamma function; $\psi''$, $\psi^{(3)}$, $\psi^{(4)}$ are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … For $B_{2k}$ see §24.2(i). …
##### 4: 20.2 Definitions and Periodic Properties
###### §20.2(i) Fourier Series
Corresponding expansions for $\theta_{j}'\left(z\middle|\tau\right)$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$.
###### §20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …
##### 5: 15.2 Definitions and Analytical Properties
###### §15.2(i) Gauss Series
The hypergeometric function $F\left(a,b;c;z\right)$ is defined by the Gauss series … …
###### §15.2(ii) Analytic Properties
Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. …
##### 6: 14.20 Conical (or Mehler) Functions
The interval $-1 is mapped one-to-one to the interval $0<\eta<\infty$, with the points $x=-1$ and $x=1$ corresponding to $\eta=\infty$ and $\eta=0$, respectively. … The interval $-1 is mapped one-to-one to the interval $-\infty<\rho<\infty$, with the points $x=-1$, $x=0$, and $x=1$ corresponding to $\rho=-\infty$, $\rho=0$, and $\rho=\infty$, respectively. …
###### §14.20(x) Zeros and Integrals
For integrals with respect to $\tau$ involving $\mathsf{P}_{-\frac{1}{2}+i\tau}\left(x\right)$, see Prudnikov et al. (1990, pp. 218–228).
where …
##### 8: 14.19 Toroidal (or Ring) Functions
###### §14.19(i) Introduction
This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates $(\eta,\theta,\phi)$, which are related to Cartesian coordinates $(x,y,z)$ by …
##### 9: 5.2 Definitions
###### Euler’s Integral
It is a meromorphic function with no zeros, and with simple poles of residue $(-1)^{n}/n!$ at $z=-n$. …
5.2.2 $\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right),$ $z\neq 0,-1,-2,\dots$.
5.2.3 $\gamma=\lim_{n\to\infty}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}-\ln n% \right)=0.57721\;56649\;01532\;86060\;\dots.$