I am searching for some theorems and books about convergence of multiple integrals of the form: $$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\;f(x,y)\;\mathrm{d}x\,\mathrm{d}y. $$ In particular, maybe it is trivial, I want to know if this integral converges if $$ f(x,y)=\frac{1}{(x²y(x-y))^{1/3}(c+(1-x)^{1/3})(x^{1/3}+(x-y)^{1/3})} $$ where $c$ is a real number. If it converges, does anybody have an idea how to calculate it?
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$\begingroup$ It seems it does not converge. The denominator is $O(x^2+y^2)$ at infinity, so outside a ball f is greater than $C/(x^2+y^2)$, not integrable at infinity. $\endgroup$– Pietro MajerJul 18, 2012 at 5:21
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$\begingroup$ The complex value when $x>1$ or $y>x$ is a bit disturbing. Is this really what your application gives? $\endgroup$– Brendan McKayJul 18, 2012 at 5:42
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$\begingroup$ Do you know any text book with theorems related to this kind of integrals? I know some theorems about multiple integrals on finite domain but not on this kind of domain. $\endgroup$– Pan AkryJul 18, 2012 at 9:05
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