# Green's Fnt. For 2-D Helmholtz Eqn.

1. Apr 23, 2012

### dmriser

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

Show that the Green's function for the two-dimensional Helmholtz equation,

2 G + k2 G = δ(x)

with the boundary conditions of an outgoing wave at infinity, is a Hankel function of the first kind.

Here, x is over 2d.

2. Relevant equations

The eigenvalue expansion?

3. The attempt at a solution

Unfortunately I am not sure where to start. I have solved the one dimensional case with the same boundary conditions, but I have no experience with PDE's (aside from Schrodingers). Since I have provided no attempt, anything would be of help, including references where I can find some help. Afrken isn't helping very much, and google hasn't turned out much information either. The 3-dimensional case is easily found, but i'm not sure it translates directly. Thank's in advance

2. Apr 23, 2012

### sunjin09

Working in polar coordinates, from symmetry consideration, G(r,θ)=G(r), so that the PDE reduces to ODE (Bessel equation), which can be solved similarly as the 1D case, except that instead of using Fourier transform, use Bessel transform, and evaluate the integral representation of G using residue techniques (shifting the poles according to radiation condition), as was done in the 1D case.

3. Apr 23, 2012

### dmriser

This approach worked, thank you for your help!