Numerical recipes

608 Chapter 13. Fourier and Spectral Applications
(13.11.3) with z replacing t. This yields
w(z) ≈ e −z2
+ 1
πi
∞�
he −t2 n 1 − (−1)n cos π(α − z)/h
tn − z
n=−∞
(13.11.8)
This form is particularly useful in obtaining Re w(z) when |y| ≪1. Note that in evaluating
(13.11.7) the exponential inside the summation is a constant and needs to be evaluated only
once; a similar comment holds for the cosine in (13.11.8).
There are a variety of formulas that can now be derived from either equation (13.11.7)
or (13.11.8) by choosing particular values of α. Eight interesting choices are: α =0, x, iy,
or z, plus the values obtained by adding h/2 to each of these. Since the error bound (13.11.3)
assumed a real value of α, the choices involving a complex α are useful only if the imaginary
part of z is not too large. This is not the place to catalog all sixteen possible formulas, and we
give only two particular cases that show some of the important features.
First of all let α =0in equation (13.11.8), which yields,
w(z) ≈ e −z2
+ 1
πi
∞�
n=−∞
he −(nh)2 1 − (−1)n cos(πz/h)
nh − z
(13.11.9)
This approximation is good over the entire z-plane. As stated previously, one has to treat the
case where one denominator becomes small by expansion in a power series. Formulas for
the case α =0were discussed briefly in [5]. They are similar, but not identical, to formulas
derived by Chiarella and Reichel [6], using the method of Goodwin [7].
Next, let α = z in (13.11.7), which yields
w(z) ≈ e −z2
− 2 � e
πi
n odd
−(z−nh)2
(13.11.10)
n
the sum being over all odd integers (positive and negative). Note that we have made the
substitution n →−nin the summation. This formula is simpler than (13.11.9) and contains
half the number of terms, but its error is worse if y is large. Equation (13.11.10) is the source
of the approximation formula (6.10.3) for Dawson’s integral, used in §6.10.
CITED REFERENCES AND FURTHER READING:
Rybicki, G.B. 1989, Computers in Physics, vol. 3, no. 2, pp. 85–87. [1]
Hardy, G.H. 1933, Journal of the London Mathematical Society, vol. 8, pp. 227–231. [2]
Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathematics
Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by
Dover Publications, New York). [3]
Gautschi, W. 1970, SIAM Journal on Numerical Analysis, vol. 7, pp. 187–198. [4]
Armstrong, B.H., and Nicholls, R.W. 1972, Emission, Absorption and Transfer of Radiation in
Heated Atmospheres (New York: Pergamon). [5]
Chiarella, C., and Reichel, A. 1968, Mathematics of Computation, vol. 22, pp. 137–143. [6]
Goodwin, E.T. 1949, Proceedings of the Cambridge Philosophical Society, vol. 45, pp. 241–245.
[7]
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. Further reproduction, or any copying of machinereadable
files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website
http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).