# Bessel Function / Helmholtz equation

1. Apr 30, 2012

### rustygecko

1. The problem statement, all variables and given/known data

I'm interested in the solution of an equation given below. (It's not a homework/coursework question, but can be stated in a similar style, so I thought it best to post here.)

2. Relevant equations

$A \nabla^2 f(x)-Bf(x)+C \exp(-2x^2/D^2)=0$
where A,B,C,D are constants.

I know the solution (or the solution that's relevant for me) is:
$f(x)=\frac{D^2C}{4} \int_0^{\infty} \frac{kJ_0(kx)\exp(-D^2k^2/8)}{Ak^2+B}dk$
where $J_0$ is a zeroth-order Bessel function, but I'm not entirely sure how to get there.

3. The attempt at a solution

It seems like a starting point might be solving:
$A \nabla^2 f(x)-Bf(x)=0$
which looks like a Helmholtz equation and then modifying that solution, but I haven't been able to solve that so far.

2. May 2, 2012

### susskind_leon

Are you familiar with Green's functions?
You are dealing with a screened Poisson equation here, which can be solved by means of Green's function (which is where the Bessel function comes in). The solution also depends on the boundary conditions, which you can implicitly defined here.