Methods | Part IB, 2002

Consider the real Sturm-Liouville problem

Ly(x)=(p(x)y)+q(x)y=λr(x)y,\mathcal{L} y(x)=-\left(p(x) y^{\prime}\right)^{\prime}+q(x) y=\lambda r(x) y,

with the boundary conditions y(a)=y(b)=0y(a)=y(b)=0, where p,qp, q and rr are continuous and positive on [a,b][a, b]. Show that, with suitable choices of inner product and normalisation, the eigenfunctions yn(x),n=1,2,3y_{n}(x), \quad n=1,2,3 \ldots, form an orthonormal set.

Hence show that the corresponding Green's function G(x,ξ)G(x, \xi) satisfying

(Lμr(x))G(x,ξ)=δ(xξ),(\mathcal{L}-\mu r(x)) G(x, \xi)=\delta(x-\xi),

where μ\mu is not an eigenvalue, is

G(x,ξ)=n=1yn(x)yn(ξ)λnμG(x, \xi)=\sum_{n=1}^{\infty} \frac{y_{n}(x) y_{n}(\xi)}{\lambda_{n}-\mu}

where λn\lambda_{n} is the eigenvalue corresponding to yny_{n}.

Find the Green's function in the case where

Lyy,\mathcal{L} y \equiv y^{\prime \prime},

with boundary conditions y(0)=y(π)=0y(0)=y(\pi)=0, and deduce, by suitable choice of μ\mu, that

n=01(2n+1)2=π28.\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}}=\frac{\pi^{2}}{8} .

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