About the Project
NIST

orthogonal functions with respect to weighted summation

AdvancedHelp

(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).

primes

derivatives with respect to argument.

The main functions treated in this chapter are the Airy functions Ai ( z ) and Bi ( z ) , and the Scorer functions Gi ( z ) and Hi ( z ) (also known as inhomogeneous Airy functions). Other notations that have been used are as follows: Ai ( - x ) and Bi ( - x ) for Ai ( x ) and Bi ( x ) (Jeffreys (1928), later changed to Ai ( x ) and Bi ( x ) ); U ( x ) = π Bi ( x ) , V ( x ) = π Ai ( x ) (Fock (1945)); A ( x ) = 3 - 1 / 3 π Ai ( - 3 - 1 / 3 x ) (Szegő (1967, §1.81)); e 0 ( x ) = π Hi ( - x ) , e ~ 0 ( x ) = - π Gi ( - x ) (Tumarkin (1959)).
2: 23.15 Definitions
§23.15 Definitions
§23.15(i) General Modular Functions
Elliptic Modular Function
Dedekind’s Eta Function (or Dedekind Modular Function)
3: 5.15 Polygamma Functions
§5.15 Polygamma Functions
The functions ψ ( n ) ( z ) , n = 1 , 2 , , are called the polygamma functions. In particular, ψ ( z ) is the trigamma function; ψ ′′ , ψ ( 3 ) , ψ ( 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 2 k see §24.2(i). …
4: 20.2 Definitions and Periodic Properties
§20.2(i) Fourier Series
Corresponding expansions for θ j ( z | τ ) , 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: …
§20.2(iv) z -Zeros
5: 15.2 Definitions and Analytical Properties
§15.2(i) Gauss Series
The hypergeometric function F ( a , b ; c ; z ) 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 < x < 1 is mapped one-to-one to the interval 0 < η < , with the points x = - 1 and x = 1 corresponding to η = and η = 0 , respectively. … The interval - 1 < x < 1 is mapped one-to-one to the interval - < ρ < , with the points x = - 1 , x = 0 , and x = 1 corresponding to ρ = - , ρ = 0 , and ρ = , respectively. …
§14.20(x) Zeros and Integrals
For integrals with respect to τ involving P - 1 2 + i τ ( x ) , see Prudnikov et al. (1990, pp. 218–228).
7: 9.12 Scorer Functions
§9.12 Scorer Functions
where …
§9.12(ii) Graphs
Functions and Derivatives
8: 14.19 Toroidal (or Ring) Functions
§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 ( η , θ , ϕ ) , which are related to Cartesian coordinates ( x , y , z ) by …
§14.19(iv) Sums
§14.19(v) Whipple’s Formula for Toroidal Functions
9: 5.2 Definitions
§5.2(i) Gamma and Psi Functions
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 ψ ( z ) = Γ ( z ) / Γ ( z ) , z 0 , - 1 , - 2 , .
5.2.3 γ = lim n ( 1 + 1 2 + 1 3 + + 1 n - ln n ) = 0.57721 56649 01532 86060 .
10: 11.9 Lommel Functions
§11.9 Lommel Functions
Reflection Formulas
§11.9(ii) Expansions in Series of Bessel Functions