n=1

with bn the coe¬ecients that appear in the expansion of f , solves the problem.

That this is so may be seen on substituting this expansion for • into the

268 CHAPTER 9. INTEGRAL EQUATIONS

integral equation and using second of the principal-part identities. Note that

that this identity provides no way to generate a term with T0 ; hence the

constraint. Next we observe that the expansion for • is generated term-by-

term from the expansion for f by substituting this into the integral form of

the solution and using the ¬rst principal-part identity.

Similarly, we can solve the for •(y) in

P 1

1

y ∈ [’1, 1],

•(x) dx = f (y), (9.78)

x’y

π ’1

where now • is permitted to be singular at x = ±1. The solution is now

√

1 1 C

1

•(y) = √ dx + √

1 ’ x2 f (x)

P , (9.79)

x’y

π 1 ’ y2 1 ’ y2

’1

where C is an arbitrary constant. To see this, expand

∞

f (x) = an Un’1 (x), (9.80)

n=1

and then

∞

1

•(x) = √ an Tn (x) + CT0 , (9.81)

1 ’ x2 n=1

satis¬es the equation for any value of the constant C. Again the expansion

for • is generated from that of f by use of the second principal-part identity.

Explanation of the Principal-Part Identities

Suppose we want to solve

un+1 + un’1 ’ (2 cos φ)un = δn0 , () (9.82)

for un . The eigenfunctions for the homogeneous problem

un+1 + un’1 = »un (9.83)

are

un = e±inθ , (9.84)

with eigenvalues » = 2 cos θ. The solution to (9.82) is therefore given by

einθ dθ 1

π

ei|n|φ ,

un = = Im φ > 0. (9.85)

2 cos θ ’ 2 cos φ 2π 2i sin φ

’π

9.6. SOME FUNCTIONAL ANALYSIS 269

The expression for the integral can be con¬rmed by noting that it is the

evaluation of the Fourier coe¬cient of the elementary double geometric series

∞

2i sin φ

e’inθ ei|n|φ = , Im φ > 0. (9.86)

2 cos θ ’ 2 cos φ

n=’∞

By using einθ = cos nθ + i sin nθ and observing that the sine term integrates

to zero, we have

cos nθ π

π

dθ = (cos nφ + i sin nφ), (9.87)

cos θ ’ cos φ i sin φ

0

where n > 0, and again we have taken Im φ > 0. Now take φ on to the real

axis and apply the Plemelj formula. We ¬nd

cos nθ sin nφ

π

P dθ = π . (9.88)

cos θ ’ cos φ sin φ

0

This is the ¬rst principal-part integral indentity. The second integral,

sin θ sin nθ

π

dθ = ’πcos nφ,

P (9.89)

cos θ ’ cos φ

0

can be obtained by using the ¬rst, coupled with the addition theorems for

the sine and cosine.

9.6 Some Functional Analysis

Here is a quick overview of some functional analysis for those readers who

know what it means for a set to be compact.

9.6.1 Bounded and Compact Operators

i) A linear operator K : L2 ’ L2 is bounded i¬ there is a positive number

M such that

Kx ¤ M x , ∀x ∈ L2 . (9.90)

If K is bounded then smallest such M is the norm of K, whch we

denote by K . Thus

Kx ¤ K x. (9.91)

270 CHAPTER 9. INTEGRAL EQUATIONS

For a ¬nite-dimensional matrix, K is the largest eigenvalue of K. A

linear operator is a continuous function of its argument i¬ it is bounded.

“Bounded” and “continuous” are therefore synonyms. Linear di¬eren-

tial operators are never bounded, and this is the source of most of the

complications in their theory.

ii) If the operators A and B are bounded, then so is AB and

AB ¤ A B . (9.92)

iii) A linear operator K : L2 ’ L2 is compact (or completely continuous)

i¬ it maps bounded sets to relatively compact sets (sets whose closure is

compact). Equivalently, K is compact i¬ the image sequence, Kxn , of

every bounded sequence of functions, xn , contains a convergent subse-

quence. Compact ’ continuous, but not vice versa. Given any positive

number M , a compact self-adjoint operator has only a ¬nite number of

eigenvalues with » outside the interval [’M, M ]. The eigenvectors un

with non-zero eigenvalues span the range of the operator. Any vector

can therefore be written

u = u0 + ai ui , (9.93)

i

where u0 lies in the null space of K. The Green function of a linear

di¬erential operator de¬ned on a ¬nite interval is usually compact.

iv) If K is compact then

H =I +K (9.94)

is Fredholm. This means that H has a ¬nite dimensional kernel and

co-kernel, and that the Fredholm alternative applies.

v) An integral kernel is Hilbert-Schmidt i¬

|K(ξ, ·)|2 dξd· < ∞. (9.95)

This means that K can be expanded in terms of a complete orthonormal

set {φm } as

∞

Anm φn (x)φ— (y)

K(x, y) = (9.96)

m

n,m=1