Mathematical Methods for Physics and Engineering : A Comprehensive Guide

23.8 EXERCISES

(b) Obtain the eigenvalues and eigenfunctions over the interval [0, 2 π]if

K(x, y)=

∑∞

n=1

1

n

cosnxcosny.

23.8 By taking its Laplace transform, and that ofxne−ax, obtain the explicit solution
of

f(x)=e−x

[

x+

∫x

0

(x−u)euf(u)du

]

.

Verify your answer by substitution.
23.9 Forf(t)=exp(−t^2 /2), use the relationships of the Fourier transforms off′(t)and
tf(t)tothatoff(t) itself to find a simple differential equation satisfied by ̃f(ω),
the Fourier transform off(t), and hence determine ̃f(ω) to within a constant.
Use this result to solve the integral equation
∫∞

−∞

e−t(t−^2 x)/^2 h(t)dt=e^3 x

(^2) / 8
forh(t).
23.10 Show that the equation
f(x)=x−^1 /^3 +λ

∫∞

0

f(y)exp(−xy)dy

has a solution of the formAxα+Bxβ. Determine the values ofαandβ,andshow

that those ofAandBare

1

1 −λ^2 Γ(^13 )Γ(^23 )

and

λΓ(^23 )

1 −λ^2 Γ(^13 )Γ(^23 )

,

where Γ(z) is the gamma function.
23.11 At an international ‘peace’ conference a large number of delegates are seated
around a circular table with each delegation sitting near its allies and diametrically
opposite the delegation most bitterly opposed to it. The position of a delegate is
denoted byθ,with0≤θ≤ 2 π.Thefuryf(θ) felt by the delegate atθis the sum
of his own natural hostilityh(θ) and the influences on him of each of the other
delegates; a delegate at positionφcontributes an amountK(θ−φ)f(φ). Thus

f(θ)=h(θ)+

∫ 2 π

0

K(θ−φ)f(φ)dφ.

Show that ifK(ψ)takestheformK(ψ)=k 0 +k 1 cosψthen

f(θ)=h(θ)+p+qcosθ+rsinθ

and evaluatep,qandr. A positive value fork 1 implies that delegates tend to
placate their opponents but upset their allies, whilst negative values imply that
they calm their allies but infuriate their opponents. A walkout will occur iff(θ)
exceeds a certain threshold value for someθ.Isthismorelikelytohappenfor
positive or for negative values ofk 1?
23.12 By considering functions of the formh(x)=

∫x

0 (x−y)f(y)dy, show that the

solutionf(x)oftheintegralequation

f(x)=x+^12

∫ 1

0

|x−y|f(y)dy

satisfies the equationf′′(x)=f(x).