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1: 23.2 Definitions and Periodic Properties
§23.2(i) Lattices
§23.2(ii) Weierstrass Elliptic Functions
§23.2(iii) Periodicity
2: 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)).
3: 23.15 Definitions
§23.15 Definitions
In §§23.1523.19, k and k ( ) denote the Jacobi modulus and complementary modulus, respectively, and q = e i π τ ( τ > 0 ) denotes the nome; compare §§20.1 and 22.1. … 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. …
4: 5.12 Beta Function
§5.12 Beta Function
In (5.12.1)–(5.12.4) it is assumed a > 0 and b > 0 . … In (5.12.8) the fractional powers have their principal values when w > 0 and z > 0 , and are continued via continuity. … When a , b …where the contour starts from an arbitrary point P in the interval ( 0 , 1 ) , circles 1 and then 0 in the positive sense, circles 1 and then 0 in the negative sense, and returns to P . …
5: 5.2 Definitions
§5.2(i) Gamma and Psi Functions
Euler’s Integral
When z 0 , Γ ( z ) is defined by analytic continuation. It is a meromorphic function with no zeros, and with simple poles of residue ( 1 ) n / n ! at z = n . …
( a ) 0 = 1 ,
6: 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 , …
7: 15.2 Definitions and Analytical Properties
§15.2(i) Gauss Series
  • (a)

    Converges absolutely when ( c a b ) > 0 .

  • As a multivalued function of z , 𝐅 ( a , b ; c ; z ) is analytic everywhere except for possible branch points at z = 0 , 1 , and . The same properties hold for F ( a , b ; c ; z ) , except that as a function of c , F ( a , b ; c ; z ) in general has poles at c = 0 , 1 , 2 , . … For example, when a = m , m = 0 , 1 , 2 , , and c 0 , 1 , 2 , , F ( a , b ; c ; z ) is a polynomial: …
    8: 9.12 Scorer Functions
    §9.12 Scorer Functions
    Gi ( x ) is a numerically satisfactory companion to the complementary functions Ai ( x ) and Bi ( x ) on the interval 0 x < . Hi ( x ) is a numerically satisfactory companion to Ai ( x ) and Bi ( x ) on the interval < x 0 . …
    Functions and Derivatives
    9: 14.20 Conical (or Mehler) Functions
    For μ > 0 and x 1 , … uniformly for θ ( 0 , π δ ] , where I and K are the modified Bessel functions10.25(ii)) and δ is an arbitrary constant such that 0 < δ < π . …
    §14.20(viii) Asymptotic Approximations: Large τ , 0 μ A τ
    §14.20(ix) Asymptotic Approximations: Large μ , 0 τ A μ
    10: 16.13 Appell Functions
    §16.13 Appell Functions
    The following four functions of two real or complex variables x and y cannot be expressed as a product of two F 1 2 functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1):
    16.13.1 F 1 ( α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , max ( | x | , | y | ) < 1 ,
    16.13.4 F 4 ( α , β ; γ , γ ; x , y ) = m , n = 0 ( α ) m + n ( β ) m + n ( γ ) m ( γ ) n m ! n ! x m y n , | x | + | y | < 1 .