Subject: Re: CCL:complex exp(x2) integrals

From: Tom Sundius <[email-masked]>
Date: Wed, 7 Nov 2001 17:09:49 +0200 (EET)

On Tue, 6 Nov 2001, Armando Navarro wrote:

> Dear Members:
> Can anybody point me to routines (or simply numerical algorithms) for
solution of complex integrals of the form:
>
> int (exp(x2)dx) , with limits between 0 and z, being z a complex number
>of the form a+bi.
> The text in the paper says that this integral can be related to
>complex error function, however I was not capable
>to establish the relationship.

The complex error function of Faddeyeva and Terent'ev is defined as
w(z)=exp(-z^2)(1+(2i/sqrt(pi))*int_0^z (exp(t^2)dt)). It is also expressed
as exp(-z^2)*erfc(-iz), where erfc is the complementary error function
(erfc z = 1 - erf z). There is a chapter about the error function in
the Handbook of Mathematical Tables by Abramowitz and Stegun. This chapter
was written by Walter Gautschi, who also wrote an efficient algorithm to
compute it to ten decimals (see SIAM J. Numer. Analysis 7, 187-198 (1970)).
An even more efficient algorithm has been published by Poppe and Wijers
(Fortran version in http://www.netlib.org/toms/680)

The real part of the complex error function is related to the Voigt
function (see, for instance, T. Sundius: J.Raman Spectrosc. 1, 471-488 (1973)).

    Greetings,

      TS

Tom Sundius
University of Helsinki, Dept of Physical Sciences   phone +358-9-191 50672
P.O.Box 64, FIN-00014 Helsinki, Finland             fax   +358-9-191 50610