Jackson 2.17 on the Laplace equation

[shaun_chou]

Homework Statement

I have problems solving the related Laplace equations in the problem

Homework Equations

$$\frac{1}{\rho}\frac{\partial}{\partial\rho}\rho\frac{\partial g_m(\rho,\rho^')}{\partial\rho}-m^2g_m(\rho,\rho^')}=-4\pi\frac{\delta(\rho-\rho^')}{\rho}$$

The Attempt at a Solution

My questions are as follows:
1. What's the difference between this equation and $$\frac{1}{\rho}\frac{\partial}{\partial\rho}\rho\frac{\partial g_m(\rho)}{\partial\rho}-m^2g_m(\rho)=-4\pi\frac{\delta(\rho)}{\rho}$$?
2. The solution I found on internet suggests that the solution is different when $$\rho > \rho^'$$ and $$\rho < \rho^'$$. Why?
Thanks a lot for your time.

 

[Ben Niehoff]

1. $\rho'$ is just a constant, for the purposes of this equation. You want it in there, because it's needed in the Green function.

2. To find a solution to the equation with the delta source on the rhs, first find the solution with zero on the rhs. You will have some arbitrary constants in the solution. Then, the idea is to take two different solutions and stitch them together such that they produce the delta source term.