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1976, Journal of Research of the National Bureau of Standards,vol. 80B, pp. 291–311; 1981, op. cit., vol. 86, pp. 661–686.Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 byDover Publications, New York), Chapters 5 and 7.6.10 Dawson’s IntegralDawson’s Integral F (x) is defined byZ2F (x) = e−xx2et dt(6.10.1)0The function can also be related to the complex error function by√i π −z2e[1 − erfc(−iz)] .F (z) =2(6.10.2)A remarkable approximation for F (x), due to Rybicki [1], is21 X e−(z−nh)F (z) = lim √h→0πn(6.10.3)n oddWhat makes equation (6.10.3) unusual is that its accuracy increases exponentiallyas h gets small, so that quite moderate values of h (and correspondingly quite rapidconvergence of the series) give very accurate approximations.We will discuss the theory that leads to equation (6.10.3) later, in §13.11, asan interesting application of Fourier methods.
Here we simply implement a routinebased on the formula.It is first convenient to shift the summation index to center it approximately onthe maximum of the exponential term. Define n0 to be the even integer nearest tox/h, and x0 ≡ n0 h, x0 ≡ x − x0 , and n0 ≡ n − n0 , so that002N1 X e−(x −n h),F (x) ≈ √π n0 =−N n0 + n0n0 odd(6.10.4)Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
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