This preview shows page 1. Sign up to view the full content.
Unformatted text preview: on the fact that
for f, g ∈ V0 , L(f ), g = f , L(g ) , so that if we thought of L as represented by a matrix, the matrix would be
symmetric. This identity can be veriﬁed by integration by parts; indeed,
b L(f ), g = b ρ(x)σ (x)L(f )(x)g (x)dx =
a a
b =− K (x)
a d
dx K (x) df
(x) g (x)
dx df
dg
(x) (x)dx = · · · = f , L(g ) ,
dx
dx where the steps represented by dots are just like the ﬁrst steps, but in reverse
order.
It follows that if fi (x) and fj (x) are eigenfunctions corresponding to distinct
eigenvalues λi and λj , then
λi fi , fj = L(f ), g = f , L(g ) = λj fi , fj ,
and hence
(λi − λj ) fi , fj = 0.
Since λi − λj = 0, we conclude that fi and fj are perpendicular with respect to
the inner product ·, · , as claimed.
Thus to determine the cn ’s, we can use exactly the same orthogonality techniques that we have used before. Namely, if we normalize the eigenfunctions
so that they have unit length and are orthogonal to each other with respect to
, ·, · , then
cn = h, fn ,
or equivalently, c...
View
Full
Document
This document was uploaded on 01/12/2014.
 Winter '14
 Equations

Click to edit the document details