Solving a 2nd order ODE using Green's Function

[ultimateguy]

Homework Statement

The homogeneous Helmholtz equation

$$\bigtriangledown^2\psi+\lambda^2\psi=0$$

has eigenvalues $$\lambda^2_i$$ and eigenfunctions $$\psi_i$$. Show that the corresponding Green's function that satisfies

$$\bigtriangledown^2 G(\vec{r}_1, \vec{r}_2)+\lambda^2 G(\vec{r}_1, \vec{r}_2)=-\delta(\vec{r}_1-\vec{r}_2)$$

may be written as

$$G(\vec{r}_1, \vec{r}_2)=\sum_{i=1}^{\infty}\frac{\psi_i(\vec{r}_1)\psi_i(\vec{r}_2)}{\lambda^2_i-\lambda^2}$$

Homework Equations

$$\int(\psi \bigtriangledown^2 G-G\bigtriangledown^2 \psi) d\tau_2=\int(\psi \bigtriangledown G-G\bigtriangledown\psi) d\sigma$$

The Attempt at a Solution

I'm using Arfken's Mathematical methods for physicists, and it isn't very good at explaining the examples it uses. I just need some kind of jump start to get me going. Do I need to use the equation in the relevant equations section?

 

[HallsofIvy]

You might want to start by writing out the definition of "Green's function"- that's far more important than examples.

Green's function for a linear, nonhomogeneous, differential equation, L(Y)=f(x), where L( ) is a linear differential operator, with given boundary conditions, is a function G(r, r') such that
1) L(G(r,r'))= 0 for all r not equal to r'
2) limit as r->r' from the right of L(G(r, r')) minus the limit as r->r' from the left of L(G(r,r'))= 1.
3) G(r, r') satisfies the boundary conditions.
2)

 

[ultimateguy]

How do I know the boundary conditions? The problem doesn't state any. Do I just assume there aren't any?