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11: 7.7 Integral Representations
§7.7(ii) Auxiliary Functions
7.7.10 f ( z ) = 1 π 2 0 e - π z 2 t / 2 t ( t 2 + 1 ) d t , | ph z | 1 4 π ,
7.7.11 g ( z ) = 1 π 2 0 t e - π z 2 t / 2 t 2 + 1 d t , | ph z | 1 4 π ,
Mellin–Barnes Integrals
12: 6.18 Methods of Computation
§6.18(ii) Auxiliary Functions
13: 6.2 Definitions and Interrelations
§6.2(iii) Auxiliary Functions
6.2.17 f ( z ) = Ci ( z ) sin z - si ( z ) cos z ,
6.2.18 g ( z ) = - Ci ( z ) cos z - si ( z ) sin z .
14: 6.12 Asymptotic Expansions
6.12.3 f ( z ) 1 z ( 1 - 2 ! z 2 + 4 ! z 4 - 6 ! z 6 + ) ,
6.12.4 g ( z ) 1 z 2 ( 1 - 3 ! z 2 + 5 ! z 4 - 7 ! z 6 + ) ,
6.12.5 f ( z ) = 1 z m = 0 n - 1 ( - 1 ) m ( 2 m ) ! z 2 m + R n ( f ) ( z ) ,
15: 8.21 Generalized Sine and Cosine Integrals
§8.21(vii) Auxiliary Functions
8.21.18 f ( a , z ) = si ( a , z ) cos z - ci ( a , z ) sin z ,
8.21.20 si ( a , z ) = f ( a , z ) cos z + g ( a , z ) sin z ,
8.21.21 ci ( a , z ) = - f ( a , z ) sin z + g ( a , z ) cos z .
§8.21(viii) Asymptotic Expansions
16: 6.7 Integral Representations
§6.7(iii) Auxiliary Functions
6.7.13 f ( z ) = 0 sin t t + z d t = 0 e - z t t 2 + 1 d t ,
6.7.15 f ( z ) = 2 0 K 0 ( 2 z t ) cos t d t ,
6.7.16 g ( z ) = 2 0 K 0 ( 2 z t ) sin t d t .
17: 8.24 Physical Applications
With more general values of p , E p ( x ) supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).
18: 12.19 Tables
  • Miller (1955) includes W ( a , x ) , W ( a , - x ) , and reduced derivatives for a = - 10 ( 1 ) 10 , x = 0 ( .1 ) 10 , 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

  • Fox (1960) includes modulus and phase functions for W ( a , x ) and W ( a , - x ) , and several auxiliary functions for x - 1 = 0 ( .005 ) 0.1 , a = - 10 ( 1 ) 10 , 8S.

  • 19: 6.19 Tables
    §6.19(ii) Real Variables
    20: 7.12 Asymptotic Expansions
    §7.12(ii) Fresnel Integrals
    7.12.2 f ( z ) 1 π z m = 0 ( - 1 ) m ( 1 2 ) 2 m ( π z 2 / 2 ) 2 m ,
    7.12.3 g ( z ) 1 π z m = 0 ( - 1 ) m ( 1 2 ) 2 m + 1 ( π z 2 / 2 ) 2 m + 1 ,
    7.12.4 f ( z ) = 1 π z m = 0 n - 1 ( - 1 ) m ( 1 2 ) 2 m ( π z 2 / 2 ) 2 m + R n ( f ) ( z ) ,
    7.12.5 g ( z ) = 1 π z m = 0 n - 1 ( - 1 ) m ( 1 2 ) 2 m + 1 ( π z 2 / 2 ) 2 m + 1 , + R n ( g ) ( z ) ,