- #1

- 699

- 5

## Homework Statement

The generalized Green function is

$$

G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}.

$$

Show [itex]G_g[/itex] satisfies the equation

$$

(\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x')

$$

where [itex]\delta(x - x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')[/itex]

and the condition that

$$

\int_0^{\ell}u_m(x)G_g(x, x')dx = 0.

$$

## Homework Equations

## The Attempt at a Solution

From a previous problem, I found

$$

u_n(x) = \sqrt{\frac{2}{\ell}}\sin(k_nx).

$$

I then end up with

\begin{gather}

(\mathcal{L} + k_m^2)G_g(x, x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx') - u_m(x)u_m(x') \\

\sum_{n\neq m}^{\infty}\sin(k_nx)\sin(k_nx') = \left(\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')\right) - \sin(k_mx)\sin(k_mx')

\end{gather}

What is going wrong?