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**1. Homework Statement**

The homogeneous Helmholtz equation

[tex]\bigtriangledown^2\psi+\lambda^2\psi=0[/tex]

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

[tex]\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)[/tex]

may be written as

[tex]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}[/tex]

**2. Homework Equations**

[tex]\int(\psi \bigtriangledown^2 G-G\bigtriangledown^2 \psi) d\tau_2=\int(\psi \bigtriangledown G-G\bigtriangledown\psi) d\sigma[/tex]

**3. 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?