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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 ⁢ ℓ ⁡ ( 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
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§23.15(i) General Modular Functions
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Elliptic Modular Function
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Dedekind’s Eta Function (or Dedekind Modular Function)
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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
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§5.2(i) Gamma and Psi Functions
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Euler’s Integral
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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
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§14.19(i) Introduction
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§14.19(ii) Hypergeometric Representations
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§14.19(iv) Sums
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§14.19(v) Whipple’s Formula for Toroidal Functions
8: 11.9 Lommel Functions
§11.9 Lommel Functions
► ►
Reflection Formulas
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§11.9(ii) Expansions in Series of Bessel Functions
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9: 20.2 Definitions and Periodic Properties
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§20.2(i) Fourier Series
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§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
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§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 ,
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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 .
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