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Re: RE: RE: ExpIntegralEi

• To: mathgroup at smc.vnet.net
• Subject: [mg18684] Re: [mg18632] RE: [mg18491] RE: [mg18463] ExpIntegralEi
• From: "Kevin J. McCann" <kevinmccann at Home.com>
• Date: Thu, 15 Jul 1999 01:46:02 -0400
• References: <199907140745.QAA00863@i.bekkoame.ne.jp>
• Sender: owner-wri-mathgroup at wolfram.com

Andrzej,

You have hit it exactly. Since the real and imaginary parts of the original
integrand ARE well-defined, well-behaved, etc. There is NO mathematical
ambiguity and certainly no computational ambiguity. I believe that Hendrik's
assertion that both answers are equally valid is incorrect.  This is easily
seen from the definition of a Riemann integral of well-defined, etc.
function.  Conclusion - the numerical integration is correct (to the
accuracy of the numerics), the symbolic answer to the DEFINITE integral is
wrong.

My two cents worth,

Kevin

----- Original Message -----
From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
To: mathgroup at smc.vnet.net
<arnaud at lmt.ens-cachan.fr>; <m.van.almsick at cityweb.de>;
<kevinmccann at home.com>; <h.vanhees at gsi.de>
Subject: [mg18684] Re: [mg18632] RE: [mg18491] RE: [mg18463] ExpIntegralEi
<snip>
> An integral like the one considered here is simply an integral of a
complex
> valued function over a measurable subset of R^2.  This is just a special
> case of a more general concept of an integral of a function with values in
a
> Banach space (called a Bochner integral) and evaluating it simply amounts
to
> evaluating its real and imaginary parts separately and taking the
resulting
> complex number. All that is required for such an integral to be well
defined
> is that the real and imaginary parts of the function  be integrable real
> functions. Such an integral can always quite unambiguously  be evaluated
> simply by evaluating its real and imaginary parts separately. Each of
these
> can be reduced two repeated computation of ordinary integrals by Fubini's
> theorem:

<snip>