Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL TRANSFORMS

13.27 The functionfa(x) is defined as unity for 0<x<aand zero otherwise. Find its
Laplace transformf ̄a(s) and deduce that the transform ofxfa(x)is
1
s^2

[

1 −(1 +as)e−sa

]

.

Writefa(x) in terms of Heaviside functions and hence obtain an explicit expres-

sion for

ga(x)=

∫x

0

fa(y)fa(x−y)dy.

Use the expression to write ̄ga(s) in terms of the functionsf ̄a(s)andf ̄ 2 a(s), and
their derivatives, and hence show that ̄ga(s) is equal to the square off ̄a(s), in
accordance with the convolution theorem.
13.28 Show that the Laplace transform off(t−a)H(t−a), wherea≥0, ise−as ̄f(s)and
that, ifg(t) is a periodic function of periodT, ̄g(s) can be written as

1

1 −e−sT

∫T

0

e−stg(t)dt.

(a) Sketch the periodic function defined in 0≤t≤Tby

g(t)=

{

2 t/T 0 ≤t<T/ 2 ,

2(1−t/T) T/ 2 ≤t≤T,

and, using the previous result, find its Laplace transform.

(b) Show, by sketching it, that

2

T

[tH(t)+2

∑∞

n=1

(−1)n(t−^12 nT)H(t−^12 nT)]

is another representation ofg(t) and hence derive the relationship

tanhx=1+2

∑∞

n=1

(−1)ne−^2 nx.

13.5 Hints and answers

13.1 Note that the integrand has different analytic forms fort<0andt≥0.
(2/π)^1 /^2 (1 +ω^2 )−^1.
13.3 (1/

√

2 π)[(b−ik)/(b^2 +k^2 )]e−a(b+ik).

13.5 Use or deriveφ ̃′′(k)=−k^2 φ ̃(k)toobtainanalgebraicequationforφ ̃(k)andthen
use the Fourier inversion formula.
13.7 (2/

√

2 π)(sinω/ω).

The convolution is 2−|t|for|t|<2, zero otherwise. Use the convolution theorem.

(4/

√

2 π)(sin^2 ω/ω^2 ).
Apply Parseval’s theorem tofand tof∗f.
13.9 The Fourier coefficient isT−^1 , independent ofn. Make the changes of variables
t→ω,n→−nandT→ 2 π/Xand apply the translation theorem.
13.11 (b) Recall that the infinite integral involved in defining ̃f(ω) has a non-zero
integrand only in|t|<T/2.

13.13 (a) (1/

√

2 π){p/[(γ+iω)^2 +p^2 ]}.

(b) Show thatQ=

√

2 π ̃I(0) and use the convolution theorem. The required

relationship isa 1 p 1 /(γ^21 +p^21 )+a 2 p 2 /(γ^22 +p^22 )=0.

13.15 g ̃(ω)=1/[

√

2 π(α+iω)^2 ], leading tog(t)=te−αt.