Solving Cylindrical Coordinate Equation with $\delta$ Functions

[mtszyk]

Homework Statement

Solve in cylindrical coordinates
$-m\delta(z)\delta(\rho-a)=\partial^2_z A(z,\rho)+\partial^2_\rho A(z,\rho)+\frac{1}{\rho}\partial_\rho A(z,\rho)-\frac{1}{\rho^2}A(z,\rho)$

Where $\partial_i$ is the partial derivative with respect to $i$.

Homework Equations

It is suggested to use an integral representation for a delta function (only for the $\rho$ delta), which I think is:
$\delta(\rho-a)=\frac{1}{2 \pi}\int_{-\infty}^{\infty}e^{i k (\rho-a)}\,\mathrm{d}k$

But it could refer to a representation using bessel functions, because I expect there to be bessel functions in the answer.

The Attempt at a Solution

My first attempt went with assuming that because there is delta functions, I can initially set those equal to zero and use the separation of variables, and expect the final answer to be in the form of whatever I get. Turns out that doesn't quite work.

Now I'm a bit lost, anyone have a direction to point me in?

 

[mighty2000]

Thank you for your post. Solving this equation in cylindrical coordinates can be tricky, but here are some steps that may help you find a solution:

1. Start by substituting the integral representation for the \rho delta function into the equation. This will give you an integral equation with an unknown function A(z,\rho).

2. Use the properties of delta functions to simplify the equation as much as possible. For example, you can use the fact that \delta(z) = 0 for z \neq 0 to simplify the equation.

3. Next, try using the method of separation of variables to separate the equation into two simpler equations, one for z and one for \rho. This will involve assuming that the solution can be written as A(z,\rho) = Z(z)R(\rho), where Z(z) is a function of z alone and R(\rho) is a function of \rho alone.

4. Once you have separated the equation, you can solve each equation separately using standard techniques for solving ordinary differential equations.

5. Finally, combine the solutions for Z(z) and R(\rho) to get the overall solution for A(z,\rho).

I hope this helps guide you in the right direction. Best of luck with your solution!