About the Project

generators

AdvancedHelp

(0.002 seconds)

11—20 of 394 matching pages

11: 24.16 Generalizations
§24.16 Generalizations
For = 0 , 1 , 2 , , Bernoulli and Euler polynomials of order are defined respectively by … B n ( x ) is a polynomial in x of degree n . …
§24.16(ii) Character Analogs
§24.16(iii) Other Generalizations
12: 8.24 Physical Applications
§8.24 Physical Applications
§8.24(iii) Generalized Exponential Integral
With more general values of p , E p ( x ) supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).
13: 4.44 Other Applications
§4.44 Other Applications
For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). For an application of the Lambert W -function to generalized Gaussian noise see Chapeau-Blondeau and Monir (2002). …
14: 16.24 Physical Applications
§16.24 Physical Applications
§16.24(i) Random Walks
Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. …
§16.24(iii) 3 j , 6 j , and 9 j Symbols
15: 4.12 Generalized Logarithms and Exponentials
§4.12 Generalized Logarithms and Exponentials
A generalized exponential function ϕ ( x ) satisfies the equations …Its inverse ψ ( x ) is called a generalized logarithm. It, too, is strictly increasing when 0 x 1 , and … For analytic generalized logarithms, see Kneser (1950).
16: 14.29 Generalizations
§14.29 Generalizations
14.29.1 ( 1 z 2 ) d 2 w d z 2 2 z d w d z + ( ν ( ν + 1 ) μ 1 2 2 ( 1 z ) μ 2 2 2 ( 1 + z ) ) w = 0
are called Generalized Associated Legendre Functions. … …
17: 19.35 Other Applications
§19.35(i) Mathematical
Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute π to high precision (Borwein and Borwein (1987, p. 26)). …
18: 12.15 Generalized Parabolic Cylinder Functions
§12.15 Generalized Parabolic Cylinder Functions
can be viewed as a generalization of (12.2.4). …
19: 9.13 Generalized Airy Functions
§9.13 Generalized Airy Functions
§9.13(i) Generalizations from the Differential Equation
§9.13(ii) Generalizations from Integral Representations
20: 16.23 Mathematical Applications
§16.23 Mathematical Applications
These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. …
§16.23(ii) Random Graphs
§16.23(iv) Combinatorics and Number Theory