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1: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
As in the case of the logarithm4.2(i)) there is a cut along the interval ( , 0 ] and the principal value is two-valued on ( , 0 ) . … The logarithmic integral is defined by … Si ( z ) is an odd entire function. …
§6.2(iii) Auxiliary Functions
2: 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 . …
5.2.3 γ = lim n ( 1 + 1 2 + 1 3 + + 1 n ln n ) = 0.57721 56649 01532 86060 .
3: 9.12 Scorer Functions
§9.12 Scorer Functions
where …
§9.12(ii) Graphs
Functions and Derivatives
4: 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
5: 14.20 Conical (or Mehler) Functions
§14.20 Conical (or Mehler) Functions
§14.20(i) Definitions and Wronskians
§14.20(ii) Graphics
§14.20(x) Zeros and Integrals
6: 5.12 Beta Function
§5.12 Beta Function
Euler’s Beta Integral
5.12.6 0 π ( sin t ) a 1 e i b t d t = π 2 a 1 e i π b / 2 a B ( 1 2 ( a + b + 1 ) , 1 2 ( a b + 1 ) ) , a > 0 .
Pochhammer’s Integral
5.12.12 P ( 1 + , 0 + , 1 , 0 ) t a 1 ( 1 t ) b 1 d t = 4 e π i ( a + b ) sin ( π a ) sin ( π b ) B ( a , b ) ,
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: 23.15 Definitions
§23.15 Definitions
§23.15(i) General Modular Functions
Elliptic Modular Function
Dedekind’s Eta Function (or Dedekind Modular Function)
9: 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)).
10: 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.