(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

[tex] 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 [/tex]

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

2. Relevant equations

A hint is given; it says to note that [tex] \exp i(\xi x+\eta y+\zeta z)=\exp i\mathbf{R}\cdot\mathbf{r} [/tex] whereRis the vector to the point [tex] (\xi ,\eta ,\zeta ) [/tex],ris the vector to the point (x,y,z), and [tex] \theta [/tex] is the angle betweenRandr. Then change over to spherical polars [tex] R, \theta, \phi [/tex] withras the polar axis.

3. The attempt at a solution

Following the hint, I get

[tex] 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 [/tex]

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Integral involved in solving Poisson's equation

**Physics Forums | Science Articles, Homework Help, Discussion**