A method for the computation of the error function of a complex variable

Strand, Otto Neall

doi:10.1090/S0025-5718-1965-0170456-8

A method for the computation of the error function of a complex variable
HTML articles powered by AMS MathViewer

by Otto Neall Strand PDF

Math. Comp. 19 (1965), 127-129 Request permission

Abstract:

This paper presents a method of computing $z \equiv \left ( {2/\sqrt \pi } \right )\int _0^z {{e^{ - {u^2}}}du}$, where $z$ is complex. It is shown that $z \equiv 1 - {\text {erf }}z$ has no zeros in the right-hand half plane. An estimate of $|{\text {erfc }}z|$ is derived.

References

J. Barkley Rosser, Theory and Application of $\int _{0^z}e^{-x^{2}}dx$ and $\int _{0^z}e^{-p^{2}y^{2}}dy\int ^y_0 e^{-x^{2}}dx$. Part I. Methods of Computation, Mapleton House, Brooklyn, N.Y., 1948. MR 0027176
H. E. Salzer, Formulas for calculating the error function of a complex variable, Math. Tables Aids Comput. 5 (1951), 67–70. MR 48150, DOI 10.1090/S0025-5718-1951-0048150-3
Joseph Kestin and Leif N. Persen, On the error function of a complex argument, Z. Angew. Math. Phys. 7 (1956), 33–40. MR 76941, DOI 10.1007/BF01600725
Burton D. Fried and Samuel D. Conte, The plasma dispersion function. The Hilbert transform of the Gaussian. , Academic Press, New York-London, 1961. MR 0135916

Tables of the Function $w(z) = {e^{ - {z^2}}}\int _0^z {{e^{{x^2}}}dx}$ in a Complex Region

16

V. N. Faddeyeva and N. M. Terent′ev, Tables of values of the function $w(z)=e^{-z^{2}}(1+2i\pi ^{-1/2}\int _{0}^{z}e^{t^{2}}dt)$ for complex argument, Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford-London-New York-Paris, 1961. Edited by V. A. Fok; translated from the Russian by D. G. Fry. MR 0122013
M. E. Munroe, Introduction to measure and integration, Addison-Wesley Publishing Co., Inc., Cambridge, Mass., 1953. MR 0053186
Theodor Laible, Höhenkarte des Fehler-integrals, Z. Angew. Math. Phys. 2 (1951), 484–486 (German). MR 44894, DOI 10.1007/bf02579766
William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1957. 2nd ed. MR 0088081