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Re: Calculating a simple integral

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  • Subject: [mg131124] Re: Calculating a simple integral
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Thu, 13 Jun 2013 02:37:15 -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.

I have found a rather simple exact expression for the the integral in
question

in = -((-1 + Cos[z])/(z^2 (r^2 + z^2)^2 (z^2 - 4 \[Pi]^2)^2))
ix = Integrate[in, {z, -Infinity, Infinity}, Assumptions -> r > 0]

= (32*Pi^4*(r^2 + 4*(r^4*(4 + r) + Pi^2*(1 + 4*r^3))) -
   E^r*(-3*r^7 + 12*r^9 + 32*Pi^2*r^5*(-1 + 5*r^2) +
      128*Pi^6*(1 - r + 4*r^3) +
      16*Pi^4*r^2*(2 + r*(-7 + 4*r*(8 + 9*r)))))/(E^
    r*(64*r^5*(4*Pi^3 + Pi*r^2)^3))

which only contains elementary functions (no cumbersome and error-
prone MeierG).

Derivation
~~~~~~~~
My starting point was Andrzej's observation that a simplified version
of the integrand is easily integrable:
Indeed, let

in2 = -((-1 + Cos[z])/(z^2 (z^2 - 4 \[Pi]^2)^2));

then

Integrate[in2, {z, -Infinity, Infinity}, Assumptions -> r > 0]
3/(32*Pi^3)

Now we decompose the term which completes the integrand,

ir = 1/(r^2 + z^2)^2;

into partial fractions:

ir1 = 1/((I*r + z)^2*((-I)*r + z)^2);
Simplify[ir == ir1]
True

Apart[ir1];
ip = List @@ %
 {-(1/(4*r^2*((-I)*r + z)^2)), -(I/(4* r^3*((-I)*r + z))), -(1/
(4*r^2*(I*r + z)^2)),  I/(4*r^3*(I*r + z))}

Integrating directly a term involving a quadratic denominator leads to
the "unpleasant" MeierG-functions we would like to avoid.

But we can get rid of the second power by diferentiating (like 1/x' =
-1/x^2) as follows

g = 1/((-I)*r + z);
ip[[1]] == (I/(4*r^2))*D[g, r]
True

Hence we need to calculate just one integral (which takes 77 seconds
on my PC)

ig = Integrate[in2*g, {z, -Infinity, Infinity}, Assumptions -> r > 0]
(I*(20*Pi^2*r^3 + 3*r^5 + 32*Pi^4*(-1 + E^(-r) + r)))/(32*
Pi^3*(4*Pi^2*r + r^3)^2)

Now the sums of the first and the third term of ip leads to

ix0 = Simplify[(I/(4*r^2))*D[ig*4*r^2, r] - (I/(4*r^3))*ig]

 (-   16*Pi^4*r^2*(-2*(1 + 16*r^2 + 4*r^3) +
      E^r*(2 - 7*r + 32*r^2 + 36*r^3)))/(E^ r*(128*Pi^3*r^5*(4*Pi^2 +
r^2)^3))

Because this quantity is real the final result is twice this
expression (ix = 2 ix0).
End of derivation.

The idea is clear, but I'm not sure that no handling error has crept
in. So please check my result.

BTW the function ix(r) reminds me of some potential between two
particles ...

Regards,
Wolfgang



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