1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

2.8 Expansion in Series of Eigenfunctions 183

(xφ′)′+λ^2

( 1

x

)

φ= 0 , 1 <x<b,

φ( 1 )= 0 ,φ(b)= 0.

Find the expansion of the functionf(x)=xin terms of these eigenfunc-

tions. To what values does the series converge atx=1andx=b?

2. Ifφ 1 ,φ 2 ,...are the eigenfunctions of a regular Sturm–Liouville problem
   and are orthogonal with weight functionp(x)onl<x<rand iff(x)is
   sectionally smooth, then
   ∫r

l

f^2 (x)p(x)dx=

∑∞

n= 1

anc^2 n,

where

an=

∫r

l

φn^2 (x)p(x)dx

andcnis the coefficient offas given in the theorem. Show why this should

be true, and conclude thatcn√an→0asn→∞.

3. Verify that the eigenvalues and eigenfunctions of the problem

(exφ′)′+exγ^2 φ= 0 , 0 <x<a,

φ( 0 )= 0 ,φ(a)= 0

are

γn^2 =

(

nπ

a

) 2

+^1

4

,φn(x)=exp

(

−x

2

)

sin

(

nπx

a

)

.

Find the coefficients for the expansion of the functionf(x)=1, 0<x<a,

in terms of theφn.

4. Ifφ 1 ,φ 2 ,...are eigenfunctions of a regular Sturm–Liouville problem, the
   numbers√anare callednormalizing constants, and the functionsψn=
   φn/√anare callednormalized eigenfunctions. Show that
   ∫r

l

ψn^2 (x)p(x)dx= 1 ,

∫r

l

ψn(x)ψm(x)p(x)dx= 0 , n=m.

5. Find the formula for the coefficients of a sectionally smooth functionf(x)
   in the series

f(x)=

∑∞

n= 1

bnψn(x), l<x<r,

where theψnare normalized eigenfunctions.