Welcome to our community

Be a part of something great, join today!

[SOLVED] Generalized Green function

dwsmith

Well-known member
Feb 1, 2012
1,673
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 \(G_g\) satisfies the equation
\[
(\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x')
\]
where \(\delta(x - x') = \frac{2}{\ell}\sum_{n = 1}^{\infty}\sin(k_nx)\sin(k_nx')\)
and the condition that
\[
\int_0^{\ell}u_m(x)G_g(x, x')dx = 0.
\]
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?