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Compute the following integral with NIntegrate?

Posted 2 months ago
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Hi, I am trying to compute this integral for different a with 0.0001<a<0.5,

 int[a_?NumericQ, r_?NumericQ]:=NIntegrate[Exp[-a/Sqrt[1-x^2]]/(x^2-r^2)^2, {x, 0, r, 1},Method ->"PrincipalValue"]

where 0<r<1. For high a, the calculation don't display errors,

int[100, 0.8]
(*  1.1619041*10^-44  *)

But for a={0.1,0.01,0.0001} I get this message error.

int[0.1, 0.8]

NIntegrate::inumri: The integrand E^(-(0.1/Sqrt[1-Power[<<2>>]]))/(-0.64+(0.8 -x)^2)^2+E^(-(0.1/Sqrt[1-Power[<<2>>]]))/(-0.64+(0.8 +x)^2)^2 has evaluated to Overflow, Indeterminate, or Infinity for all sampling points in the region with boundaries {{0.,2.7247563*10^-30}}.

I still can't control the accuracy of the calculation. I proceeded by increasing WorkingPrecision, PrecisionGoal, MaxRecursion but it gives each time different values, I do not even know if it gives the true value.

Please, how to increase the precision of this computation?

Problem not solved yet on stackexchange.

3 Replies

Have you tried other NIntegrate methods?

Posted 2 months ago

Isn't the principal value divergent (to +∞)? There's a pole of order 2 at x == r.

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