I have to calculate the following Inverse Fourier-Transform, which describe the potential function for a point force on a half-space:
$$f(x,y,z)=\frac{P}{(2\pi)^2}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\frac{1}{\rho^2}e^{-\rho z}e^{i(u\cdot x+w\cdot y)}\ \mathrm du\ \mathrm dw,$$
where $\rho=\sqrt{u^2+w^2}$.
In a text book I found the following solution:
$$f(x,y,z)=-\frac{P}{2\pi}\ln(R+z),$$
where $R=\sqrt{x^2+y^2+z^2}$.
The problem is that I'm not able to derive the solution myself. I suppose that the $\ln(R+z)$ derives from the $\int \frac{1}{R}dz$, but I don't understand why. Is there someone who can help me?
The result is wrong. If you consider the integral and rescale $x \to \lambda x$, $y\to \lambda y$, $z \to \lambda z$ for $\lambda >0$, you easily see that the result of the integral (if computed literally, however see below) does not change because you can pass the factor $\lambda$ to the variables $u,w$ and $du dw/\rho^2$ is rescaling invariant. Conversely, the proposed solution is not rescaling invariant.
However, the integral is not absolutely convergent so, if the result is correct in some (not quite evident) sense, some regularization has been used which could explain the breaking of rescaling invariance.