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>