Do you mean integration over 2-dimensional space from minus to plus infinity in both coordinates (x,y)? Then it makes sense to transform it to polar coordinates x=r cos phi, y=r sin phi) and to check if there exists integrable function F, so that for the integrated function |f(r, phi)|

Dear Yuri Yegorov,

Regrettably, I mean integration (x, y) from finite to infinity. So your suggestion is not useful for me.

Dear George,

Of course, If f(x, y) \geq c holds, the double integral diverges.But I want to know the condition for convergence.

Sincerely, Susumu

Dear Susumu, if your area is just infinite surarea of the 1st quadrant, my message is still true. Integration over angle will be from 0 to \pi/2. The key is convergence of integral with r. For example, if your function f(r, \phi) behaves like r^{-2}, then the integral of \int_{a}^{+\infty}  f r dr is diverging logarithmically. For faster decay with distance it converges.

Dear Yuri Yegorov,

Your suggestion is right. But my area is direct product of semi-infinite interval.

Dear George,

Thank you for introducing helpful information.

Dear Susumu,

In the specific class of examples you asked, some criteria for convergence and for divergence can be given and it will depend on whether a>0 or a