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functions f(ϵ,ℓ;r),h(ϵ,ℓ;r)

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1: 7.4 Symmetry
f ( i z ) = ( 1 / 2 ) e 1 4 π i - 1 2 π i z 2 - i f ( z ) ,
f ( - z ) = 2 cos ( 1 4 π + 1 2 π z 2 ) - f ( z ) ,
2: 7.5 Interrelations
7.5.6 e ± 1 2 π i z 2 ( g ( z ) ± i f ( z ) ) = 1 2 ( 1 ± i ) - ( C ( z ) ± i S ( z ) ) .
7.5.10 g ( z ) ± i f ( z ) = 1 2 ( 1 ± i ) e ζ 2 erfc ζ .
7.5.11 | ( x ) | 2 = f 2 ( x ) + g 2 ( x ) , x 0 ,
7.5.12 | ( x ) | 2 = 2 + f 2 ( - x ) + g 2 ( - x ) - 2 2 cos ( 1 4 π + 1 2 π x 2 ) f ( - x ) - 2 2 cos ( 1 4 π - 1 2 π x 2 ) g ( - x ) , x 0 .
3: 7.10 Derivatives
7.10.1 d n + 1 erf z d z n + 1 = ( - 1 ) n 2 π H n ( z ) e - z 2 , n = 0 , 1 , 2 , .
d f ( z ) d z = - π z g ( z ) ,
d g ( z ) d z = π z f ( z ) - 1 .
4: 8.22 Mathematical Applications
The so-called terminant function F p ( z ) , defined by
8.22.1 F p ( z ) = Γ ( p ) 2 π z 1 - p E p ( z ) = Γ ( p ) 2 π Γ ( 1 - p , z ) ,
5: 7.24 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for erf x , erfc x and the auxiliary functions f ( x ) and g ( x ) .

  • Cody et al. (1970) gives minimax rational approximations to Dawson’s integral F ( x ) (maximum relative precision 20S–22S).

  • Luke (1969b, pp. 323–324) covers 1 2 π erf x and e x 2 F ( x ) for - 3 x 3 (the Chebyshev coefficients are given to 20D); π x e x 2 erfc x and 2 x F ( x ) for x 3 (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

  • Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions f ( x ) and g ( x ) for x 3 (15D).

  • Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for F ( z ) , erf z , erfc z , C ( z ) , and S ( z ) ; approximate errors are given for a selection of z -values.

  • 6: 7.25 Software
    §7.25(iv) C ( x ) , S ( x ) , f ( x ) , g ( x ) , x
    7: 7.7 Integral Representations
    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 ,
    7.7.13 f ( z ) = ( 2 π ) - 3 / 2 2 π i c - i c + i ζ - s Γ ( s ) Γ ( s + 1 2 ) Γ ( s + 3 4 ) Γ ( 1 4 - s ) d s ,
    7.7.15 0 e - a t cos ( t 2 ) d t = π 2 f ( a 2 π ) , a > 0 ,
    8: 2.11 Remainder Terms; Stokes Phenomenon
    2.11.11 F n + p ( z ) = e - z 2 π 0 e - z t t n + p - 1 1 + t d t = Γ ( n + p ) 2 π E n + p ( z ) z n + p - 1 .
    Owing to the factor e - ρ , that is, e - | z | in (2.11.13), F n + p ( z ) is uniformly exponentially small compared with E p ( z ) . … However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the F -functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the F -functions. In this context the F -functions are called terminants, a name introduced by Dingle (1973). …
    9: 7.2 Definitions
    erf z , erfc z , and w ( z ) are entire functions of z , as is F ( z ) in the next subsection. … ( z ) , C ( z ) , and S ( z ) are entire functions of z , as are f ( z ) and g ( z ) in the next subsection. …
    7.2.10 f ( z ) = ( 1 2 - S ( z ) ) cos ( 1 2 π z 2 ) - ( 1 2 - C ( z ) ) sin ( 1 2 π z 2 ) ,
    10: 9 Airy and Related Functions
    Chapter 9 Airy and Related Functions