Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL TRANSFORMS

13.4 Exercises

13.1 Find the Fourier transform of the functionf(t)=exp(−|t|).

(a) By applying Fourier’s inversion theorem prove that

π

2

exp(−|t|)=

∫∞

0

cosωt

1+ω^2

dω.

(b) By making the substitutionω=tanθ, demonstrate the validity of Parseval’s

theorem for this function.

13.2 Use the general definition and properties of Fourier transforms to show the
following.

(a) Iff(x) is periodic with periodathen ̃f(k) = 0, unlesska=2πnfor integern.

(b) The Fourier transform oftf(t)isid ̃f(ω)/dω.

(c) The Fourier transform off(mt+c)is

eiωc/m

m

̃f

(ω

m

)

.

13.3 Find the Fourier transform ofH(x−a)e−bx,whereH(x) is the Heaviside function.
13.4 Prove that the Fourier transform of the functionf(t) defined in thetf-plane by
straight-line segments joining (−T,0) to (0,1) to (T,0), withf(t) = 0 outside
|t|<T,is

̃f(ω)=√T

2 π

sinc^2

(

ωT

2

)

,

where sincxis defined as (sinx)/x.

Use the general properties of Fourier transforms to determine the transforms

of the following functions, graphically defined by straight-line segments and equal

to zero outside the ranges specified:

(a) (0,0) to (0. 5 ,1) to (1,0) to (2,2) to (3,0) to (4. 5 ,3) to (6,0);

(b) (− 2 ,0) to (− 1 ,2) to (1,2) to (2,0);

(c) (0,0) to (0,1) to (1,2) to (1,0) to (2,−1) to (2,0).

13.5 By taking the Fourier transform of the equation

d^2 φ

dx^2

−K^2 φ=f(x),

show that its solution,φ(x), can be written as

φ(x)=

− 1

√

2 π

∫∞

−∞

eikx ̃f(k)

k^2 +K^2

dk,

where ̃f(k) is the Fourier transform off(x).
13.6 By differentiating the definition of the Fourier sine transform ̃fs(ω) of the function
f(t)=t−^1 /^2 with respect toω, and then integrating the resulting expression by
parts, find an elementary differential equation satisfied by ̃fs(ω). Hence show that
this function is its own Fourier sine transform, i.e.f ̃s(ω)=Af(ω), whereAis a
constant. Show that it is also its own Fourier cosine transform. Assume that the
limit asx→∞ofx^1 /^2 sinαxcan be taken as zero.
13.7 Find the Fourier transform of the unit rectangular distribution

f(t)=

{

1 |t|< 1 ,

0otherwise.