Integral involved in solving Poisson's equation

[iomtt6076]

Homework Statement

Problem 8.30 from Greenberg's Foundations of Applied Mathematics: We meet the integral

$$I = -\frac{1}{8\pi^3}\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\frac{e^{i(\xi x+\eta y+\zeta z)}}{\xi^2+\eta^2+\zeta^2}\,d\xi\,d\eta\,d\zeta$$

when we solve the Poisson equation by the Fourier transform. Show that $$I=-1/4\pi r$$, where $$r=\sqrt{x^2+y^2+z^2}$$

Homework Equations

A hint is given; it says to note that $$\exp i(\xi x+\eta y+\zeta z)=\exp i\mathbf{R}\cdot\mathbf{r}$$ where R is the vector to the point $$(\xi ,\eta ,\zeta )$$, r is the vector to the point (x,y,z), and $$\theta$$ is the angle between R and r. Then change over to spherical polars $$R, \theta, \phi$$ with r as the polar axis.

The Attempt at a Solution

Following the hint, I get

$$I = \int_0^{2\pi}\int_0^\pi\int_0^\infty\frac{e^{irR\cos\theta}}{R^2}R^2\sin\theta\,dR\,d\theta\,d\phi$$

since the axes are only rotated so the Jacobian is 1 (volumes are not contracted or expanded). However, then I get that the integral doesn't exist. Now I found

https://www.physicsforums.com/showthread.php?t=293550"

where apparently a constant squared is thrown into the denominator, integration methods from complex variables are used, and then the limit is taken as the constant approaches zero. However, our class hasn't done complex variables and in any case I am not yet familiar with such methods.

So have I done something wrong or is there another way to proceed?

Thanks in advance.

 

[iomtt6076]

However, then I get that the integral doesn't exist.

Actually... it does:

$$I = 2\pi\int_0^\infty\,dR\int_0^\pi\sin\theta e^{irR\cos\theta}\,d\theta =2\pi\int_0^\infty\,dR\int_0^\pi (-\frac{1}{irR})\,de^{irR\cos\theta}$$

$$=\frac{4\pi}{r}\int_0^\infty\,dR\frac{1}{R}\frac{e^{irR}-e^{-irR}}{2i}=\frac{4\pi}{r}\int_0^\infty\frac{\sin rR}{R}\,dR=\frac{2\pi^2}{r}$$