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Re: Calculating a simple integral

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  • Subject: [mg131129] Re: Calculating a simple integral
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Thu, 13 Jun 2013 02:38:55 -0400 (EDT)
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On 11 Jun., 08:23, Andrzej Kozlowski <akozlow... at gmail.com> wrote:
> No, it's similar to:
>
> Integrate[(1 -
>     Cos[x])/(x^2*(x^2 - 4*Pi^2)^2), {x, -Infinity, Infinity}]
>
> 3/(32*Pi^3)
>
> On 10 Jun 2013, at 10:11, djmpark <djmp... at comcast.net> wrote:
>
>
>
> > Doesn't this have a singularity at 2 Pi that produces non-convergence? It's
> > similar to:
>
> > Integrate[1/x^2, {x, \[Epsilon], \[Infinity]},
> > Assumptions -> \[Epsilon] > 0]
>
> > 1/\[Epsilon]
>
> > That diverges as epsilon -> 0.
>
> > Are you sure you copied the integral correctly?
>
> > David Park
> > djmp... at comcast.net
> >http://home.comcast.net/~djmpark/index.html
>
> > From: dsmirno... at gmail.com [mailto:dsmirno... at gmail.com]
>
> > If there is a way to calculate with Mathematica the following integral:
>
> > in = -((-1 + Cos[kz])/(kz^2 (kr^2 + kz^2)^2 (kz^2 - 4 \[Pi]^2)^2))
> > Integrate[in, {kz, -Infinity, Infinity}, Assumptions -> kr > 0]
>
> > Another system calculates the same integral instantly. :)
>
> > Thanks for any suggestions.

Sorry, but I made indeed a calculation error!
Correcting it the partial fraction decomposition leads to Dmitry's
result.
Furthermore, calculating first the indefinite integral and then taking
limits leads to a false result.
Direct calculation of the integral leads to MeierG functions which are
useless because we cannot enter any numerical value.
So, rather than provding the correct result Mathematica comes up with
different false result depending on the method used, and we cannot tel
which one is correct without "research" work.
Summarizing, I need to restate my criticism of Mathematica with
respect to integration (I'm using version 8).

Regards,
Wolfgang



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