(a) Show that the Green function G(x, y; x', y’)

(a) Show that the Green function G(x, y; x', y’) appropriate for Dirichlet boundary conditions for a square two-dimensional region, 0 ≤ x ≤ l, 0 ≤ y ≤ l, has an expansion

G(x, y; x', y') = 2 2 8,(y, y') sin(n Tx) sin(nTx') n=1


Where gn(y,  y’) satisfies

(a) Show that the Green function G(x, y; x', y')


(b) Taking for gn(y, y') appropriate linear combinations of sinh(nπy') and cosh(nπy') in the two regions, у' < у and у' > у, in accord with the boundary conditions and the discontinuity in slope required by the source delta function, show that the explicit form of G is

(a) Show that the Green function G(x, y; x', y')


Where y<(y>) is the smaller (larger) of у and y'.