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1: 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)).
2: 31.1 Special Notation
(For other notation see Notation for the Special Functions.)
x , y real variables.
The main functions treated in this chapter are H ( a , q ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) Hf m ( a , q m ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) Hf m ν ( a , q m ; α , β , γ , δ ; z ) , and the polynomial Hp n , m ( a , q n , m ; - n , β , γ , δ ; z ) . …Sometimes the parameters are suppressed.
3: 23.15 Definitions
§23.15 Definitions
§23.15(i) General Modular Functions
Elliptic Modular Function
Dedekind’s Eta Function (or Dedekind Modular Function)
4: 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). …
5: 5.2 Definitions
§5.2(i) Gamma and Psi Functions
Euler’s Integral
5.2.1 Γ ( z ) = 0 e - t t z - 1 d t , z > 0 .
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 , .
6: 9.12 Scorer Functions
§9.12 Scorer Functions
where …
§9.12(ii) Graphs
Functions and Derivatives
7: 14.19 Toroidal (or Ring) Functions
§14.19 Toroidal (or Ring) Functions
§14.19(i) Introduction
§14.19(ii) Hypergeometric Representations
§14.19(iv) Sums
§14.19(v) Whipple’s Formula for Toroidal Functions
8: 11.9 Lommel Functions
§11.9 Lommel Functions
Reflection Formulas
§11.9(ii) Expansions in Series of Bessel Functions
9: 20.2 Definitions and Periodic Properties
§20.2(i) Fourier Series
§20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …
§20.2(iii) Translation of the Argument by Half-Periods
§20.2(iv) z -Zeros
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 .