Subject: Summary for complex exp(x2)dx integrals

From: "Armando Navarro" <[email-masked]>
Date: Wed, 14 Nov 2001 10:27:49 +0100

Dear members:
Here the summary for the question on complex exp(x2)dx. Thanks to all people

The original question:
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.

Thanks in advance
Armando Navarro
Facultade de Quimica
Departamento de quimica Organica
Universidade de Santiago de Compostela. Spain
e-mail: [email-masked]

----------------------------------------------
> From Robert Waterland:
Armando:

Equation 7.1.3 in  Handbook of Mathematical Functions by Abramowitz and
Stegun is

w(z) = exp(-z^2) (1+ 2i/sqrt(pi)*INT) = exp(-z^2)erfc(-iz)

where INT is the integral you are looking for,  i.e. int(exp(x^2)dx)
between 0 and z.

So the integral you want is related to the Complementary Error function
with complex argument.

Abramowitz and Stegun have a number of series forms for the Error function
which (I believe) can be used for complex arguments.  See section 7.5
examples 12 to 19 for use of tables and series for complex arguments.
There is also a table (Table 7.9) of w(z) for complex arguments (z=x+iy)
where 0<=x<=1 and 0<=y<=3) so you can test the series formulae to see if
they work.

Rob
--------------------------------------------------------------
> From Gerd Raether:
Please look in the "bible" - numerical recipes - you can download pages
in ps/pdf format from

http://www.nr.com/

For complex integration you might need also the
residue theorem - see a textbook for complex analysis.

Hope this helps somewhat,

Gerd
-----------------------------------------------------------
> From Bruno Sopko:

Try GPL
Bruno Sopko
------------------------------------------------------------
> From Tom Sundius:

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