About the Project
NIST

Dawson integral

AdvancedHelp

(0.001 seconds)

1—10 of 21 matching pages

1: 7.2 Definitions
erf z , erfc z , and w ( z ) are entire functions of z , as is F ( z ) in the next subsection. …
§7.2(ii) Dawson’s Integral
7.2.5 F ( z ) = e - z 2 0 z e t 2 d t .
2: 7.24 Approximations
  • Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

  • 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).

  • Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for ( 1 + 2 x ) e x 2 erfc x on ( 0 , ) (22D).

  • 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.

  • 3: 7.3 Graphics
    See accompanying text
    Figure 7.3.2: Dawson’s integral F ( x ) , - 3.5 x 3.5 . Magnify
    4: 7.4 Symmetry
    7.4.4 F ( - z ) = - F ( z ) .
    5: 7.1 Special Notation
    Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the error function erf z ; the complementary error functions erfc z and w ( z ) ; Dawson’s integral F ( z ) ; the Fresnel integrals ( z ) , C ( z ) , and S ( z ) ; the Goodwin–Staton integral G ( z ) ; the repeated integrals of the complementary error function i n erfc ( z ) ; the Voigt functions U ( x , t ) and V ( x , t ) . Alternative notations are Q ( z ) = 1 2 erfc ( z / 2 ) , P ( z ) = Φ ( z ) = 1 2 erfc ( - z / 2 ) , Erf z = 1 2 π erf z , Erfi z = e z 2 F ( z ) , C 1 ( z ) = C ( 2 / π z ) , S 1 ( z ) = S ( 2 / π z ) , C 2 ( z ) = C ( 2 z / π ) , S 2 ( z ) = S ( 2 z / π ) . …
    6: 7.5 Interrelations
    §7.5 Interrelations
    7.5.1 F ( z ) = 1 2 i π ( e - z 2 - w ( z ) ) = - 1 2 i π e - z 2 erf ( i z ) .
    7.5.13 G ( x ) = π F ( x ) - 1 2 e - x 2 Ei ( x 2 ) , x > 0 .
    7: 7.23 Tables
  • Abramowitz and Stegun (1964, Chapter 7) includes erf x , ( 2 / π ) e - x 2 , x [ 0 , 2 ] , 10D; ( 2 / π ) e - x 2 , x [ 2 , 10 ] , 8S; x e x 2 erfc x , x - 2 [ 0 , 0.25 ] , 7D; 2 n Γ ( 1 2 n + 1 ) i n erfc ( x ) , n = 1 ( 1 ) 6 , 10 , 11 , x [ 0 , 5 ] , 6S; F ( x ) , x [ 0 , 2 ] , 10D; x F ( x ) , x - 2 [ 0 , 0.25 ] , 9D; C ( x ) , S ( x ) , x [ 0 , 5 ] , 7D; f ( x ) , g ( x ) , x [ 0 , 1 ] , x - 1 [ 0 , 1 ] , 15D.

  • Zhang and Jin (1996, pp. 637, 639) includes ( 2 / π ) e - x 2 , erf x , x = 0 ( .02 ) 1 ( .04 ) 3 , 8D; C ( x ) , S ( x ) , x = 0 ( .2 ) 10 ( 2 ) 100 ( 100 ) 500 , 8D.

  • 8: 8.4 Special Values
    For erf ( z ) , erfc ( z ) , and F ( z ) , see §§7.2(i), 7.2(ii). …
    8.4.3 γ * ( 1 2 , - z 2 ) = 2 e z 2 z π F ( z ) .
    9: 7.16 Generalized Error Functions
    §7.16 Generalized Error Functions
    Generalizations of the error function and Dawson’s integral are 0 x e - t p d t and 0 x e t p d t . …
    10: 7.14 Integrals
    7.14.1 0 e 2 i a t erfc ( b t ) d t = 1 a π F ( a b ) + i 2 a ( 1 - e - ( a / b ) 2 ) , a , | ph b | < 1 4 π .