Convergence of several improper integrals

[jiazhen]

There are several improper integrals which keeps puzzling me. Let's talk about them in xoy plane. For simplicity purpose, I need to define r=sqrt(x^2+y^2). The integrals are ∫∫(1/r)dxdy, ∫∫(x/r^2)dxdy, ∫∫(x^2/r^4)dxdy, and ∫∫(x^3/r^6)dxdy. Here ‘^’ is power symbol. The integration area D contains the point (0,0) as its interior point, which makes the integrals improper. My questions are:
1. Are there detailed and strict proves that these integrals are convergent when the integration area D contains the point (0,0) as its interior point?
2. If these integrals are convergent, then how do they behave when the origin point (0,0) moves across the boundary of D? Are there 'jumps' of the integration results? People familiar with potential theory could understand that what I exactly want are the boundary conditions.
3. As we know, according to Green's theorem, when the integration area D does not contain the point (0,0), the integrals above could be expressed as line integrals against the boundary of D. My question is: when the point (0,0) is inside D, is it possible to express the areal integrals with some kinds of line integrals?

 

[aesir]

It seems that you already understood that (0,0) is the problematic point.

If you choose a symmetric domain about (0,0) you can calculate those integral analytically.

Any other domain is just a small symmetric one plus something else.