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1: 16.2 Definition and Analytic Properties
§16.2(i) Generalized Hypergeometric Series
Polynomials
Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …
§16.2(v) Behavior with Respect to Parameters
2: 1.16 Distributions
Λ : 𝒟 ( I ) 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 ϕ n ϕ in 𝒟 ( I ) , … More generally, for α : [ a , b ] [ , ] a nondecreasing function the corresponding Lebesgue–Stieltjes measure μ α (see §1.4(v)) can be considered as a distribution: … More generally, if α ( x ) is an infinitely differentiable function, then … For α > 0 , … Friedman (1990) gives an overview of generalized functions and their relation to distributions. …
3: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8(i) Definition
Convergence Properties
§35.8(iv) General Properties
Confluence
4: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
§8.19(i) Definition and Integral Representations
§8.19(ii) Graphics
§8.19(vi) Relation to Confluent Hypergeometric Function
§8.19(xi) Further Generalizations
5: 8.21 Generalized Sine and Cosine Integrals
§8.21 Generalized Sine and Cosine Integrals
Spherical-Bessel-Function Expansions
§8.21(vii) Auxiliary Functions
§8.21(viii) Asymptotic Expansions
6: 19.2 Definitions
§19.2(i) General Elliptic Integrals
Here a , b , p are real parameters, and k c and x are real or complex variables, with p 0 , k c 0 . If < p < 0 , then the integral in (19.2.11) is a Cauchy principal value. …
§19.2(iv) A Related Function: R C ( x , y )
where the Cauchy principal value is taken if y < 0 . …
7: 9.1 Special Notation
(For other notation see Notation for the Special Functions.)
k nonnegative integer, except in §9.9(iii).
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)).
8: 23.15 Definitions
§23.15 Definitions
§23.15(i) General Modular Functions
A modular function f ( τ ) is a function of τ that is meromorphic in the half-plane τ > 0 , and has the property that for all 𝒜 SL ( 2 , ) , or for all 𝒜 belonging to a subgroup of SL ( 2 , ) , …If, as a function of q , f ( τ ) is analytic at q = 0 , then f ( τ ) is called a modular form. If, in addition, f ( τ ) 0 as q 0 , then f ( τ ) is called a cusp form. …
9: 14.19 Toroidal (or Ring) Functions
§14.19(i) Introduction
When ν = n 1 2 , n = 0 , 1 , 2 , , μ , and x ( 1 , ) solutions of (14.2.2) are known as toroidal or ring functions. … With ξ > 0 , … With ξ > 0 , … With ξ > 0 , …
10: 14.20 Conical (or Mehler) Functions
Lastly, for the range 1 < x < , P 1 2 + i τ μ ( x ) is a real-valued solution of (14.20.1); in terms of Q 1 2 ± i τ μ ( x ) (which are complex-valued in general): …
§14.20(vi) Generalized Mehler–Fock Transformation
§14.20(viii) Asymptotic Approximations: Large τ , 0 μ A τ
§14.20(ix) Asymptotic Approximations: Large μ , 0 τ A μ