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How do I solve this integral?

Posted 10 years ago
POSTED BY: M. R.
2 Replies

The series expansion at y=1 indicates all is fine there provided a!=-b. I suspect it will take careful analysis of behavior at infinity in order to figure out a full set of conditions for which you get convergence. From some manipulations I did by hand, I think what's required is that re(a+2b)>0.

POSTED BY: Daniel Lichtblau

It seems that Integrate is not getting the full region of integration as the solution as you notice. The ConditionalExpression does not mean that the integral is necessarily divergent or undefined outside of the region that it specifies--just that the solution that is given is valid inside of that region.

But perhaps someone with integration expertise at WRI can comment on why the result doesn't cover all permissible values of a and b.

Of course you can turn off the generation of the ConditionalExpression by using,

Integrate[
 y^(2*a + b - 1)/Sqrt[y^(2*(a + b)) - 1] - y^(a - 1), {y, 1, Infinity}, GenerateConditions -> False]

but this doesn't solve your question.

POSTED BY: David Reiss
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