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1: 7.24 Approximations
  • Cody et al. (1970) gives minimax rational approximations to Dawson’s integral F ( x ) (maximum relative precision 20S–22S).

  • 2: 7.8 Inequalities
    7.8.5 x 2 2 x 2 + 1 x 2 ( 2 x 2 + 5 ) 4 x 4 + 12 x 2 + 3 x 𝖬 ( x ) < 2 x 4 + 9 x 2 + 4 4 x 4 + 20 x 2 + 15 < x 2 + 1 2 x 2 + 3 , x 0 .
    7.8.6 0 x e a t 2 d t < 1 3 a x ( 2 e a x 2 + a x 2 2 ) , a , x > 0 .
    7.8.7 sinh x 2 x < e x 2 F ( x ) = 0 x e t 2 d t < e x 2 1 x , x > 0 .
    The function F ( x ) / 1 e 2 x 2 is strictly decreasing for x > 0 . For these and similar results for Dawson’s integral F ( x ) see Janssen (2021). …
    3: 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.

  • Abramowitz and Stegun (1964, Table 27.6) includes the Goodwin–Staton integral G ( x ) , x = 1 ( .1 ) 3 ( .5 ) 8 , 4D; also G ( x ) + ln x , x = 0 ( .05 ) 1 , 4D.

  • 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.

  • Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of erf z , x [ 0 , 5 ] , y = 0.5 ( .5 ) 3 , 7D and 8D, respectively; the real and imaginary parts of x e ± i t 2 d t , ( 1 / π ) e i ( x 2 + ( π / 4 ) ) x e ± i t 2 d t , x = 0 ( .5 ) 20 ( 1 ) 25 , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

  • Zhang and Jin (1996, p. 642) includes the first 10 zeros of erf z , 9D; the first 25 distinct zeros of C ( z ) and S ( z ) , 8S.

  • 4: Software Index
    5: Bibliography C
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • W. J. Cody, K. A. Paciorek, and H. C. Thacher (1970) Chebyshev approximations for Dawson’s integral. Math. Comp. 24 (109), pp. 171–178.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • D. Cvijović and J. Klinowski (1999) Integrals involving complete elliptic integrals. J. Comput. Appl. Math. 106 (1), pp. 169–175.