Please help to improve/correct the following arguments.
I want to show that the integral $$I(x,y,z)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{z\over 2\pi [(\alpha-x)^2+(\beta-y)^2+z^2]^{3\over2}}f(\alpha,\beta)\;d\alpha\; d\beta$$
where $f$ is an arbitrary continuous function and $I$ is the solution of Laplace's equation $\nabla^2 I=0$, has the following properties:
1) Tends to $0$ as $x^2+y^2\to\infty$. Argument: As $x^2+y^2\to\infty$, the integrand tends to $0$, hence the integral tends to $0$. Concern: Does the arbitrariness of $f$ screw things up?
2) That $I(x,y,0)=f(x,y)$. Argument: For $\alpha,\beta\neq x,y$ respectively, the integrand is $0$ when $z=0$, but when $\alpha,\beta= x,y$ respectively, we have a singularity. This resembles the sampling property of the delta function, but how would I know that I can use that argument here, since we don't really have a delta function. Please help!
Thank you.