Hints and answers

INTEGRAL TRANSFORMS
13.27
The function fa (x) is defined as unity for 0 < x < a and zero otherwise. Find its
Laplace transform f¯a (s) and deduce that the transform of xfa (x) is
1 1 − (1 + as)e−sa .
s2
Write fa (x) in terms of Heaviside functions and hence obtain an explicit expression for
x
ga (x) =
fa (y)fa (x − y) dy.
0
13.28
Use the expression to write ḡa (s) in terms of the functions f¯a (s) and f¯2a (s), and
their derivatives, and hence show that ḡa (s) is equal to the square of f¯a (s), in
accordance with the convolution theorem.
Show that the Laplace transform of f(t − a)H(t − a), where a ≥ 0, is e−as f̄(s) and
that, if g(t) is a periodic function of period T , ḡ(s) can be written as
T
1
e−st g(t) dt.
−sT
1−e
0
(a) Sketch the periodic function defined in 0 ≤ t ≤ T by
2t/T
0 ≤ t < T /2,
g(t) =
2(1 − t/T ) T /2 ≤ t ≤ T ,
and, using the previous result, find its Laplace transform.
(b) Show, by sketching it, that
2
(−1)n (t − 12 nT )H(t − 12 nT )]
[tH(t) + 2
T
n=1
∞
is another representation of g(t) and hence derive the relationship
tanh x = 1 + 2
∞
(−1)n e−2nx .
n=1
13.5 Hints and answers
13.1
13.3
13.5
13.7
13.9
13.11
13.13
13.15
Note that the integrand has different analytic forms for t < 0 and t ≥ 0.
1/2
2 −1
(2/π)
√ (1 + ω ) . 2
(1/ 2π)[(b − ik)/(b + k 2 )]e−a(b+ik) .
4 (k) = −k 2 φ̃(k) to obtain an algebraic equation for φ̃(k) and then
Use or derive φ
use√the Fourier inversion formula.
(2/ 2π)(sin ω/ω).
The√convolution is 2 − |t| for |t| < 2, zero otherwise. Use the convolution theorem.
(4/ 2π)(sin2 ω/ω 2 ).
Apply Parseval’s theorem to f and to f ∗ f.
The Fourier coefficient is T −1 , independent of n. Make the changes of variables
t → ω, n → −n and T → 2π/X and apply the translation theorem.
(b) Recall that the infinite integral involved in defining f̃(ω) has a non-zero
integrand
√ only in |t| < T2 /2. 2
(a) (1/ 2π){p/[(γ + iω)
√ + p ]}.
(b) Show that Q = 2π Ĩ(0) and use the convolution theorem. The required
relationship√is a1 p1 /(γ12 + p21 ) + a2 p2 /(γ22 + p22 ) = 0.
g̃(ω) = 1/[ 2π(α + iω)2 ], leading to g(t) = te−αt .
466
13.5 HINTS AND ANSWERS
13.17
13.19
13.21
13.23
13.25
13.27
3 (k) ∝ [−2π/(ik)] {exp[−(µ − ik)r] − exp[−(µ + ik)r]} dr.
V
Note that the lower limit in the calculation of a(z) is 0, for z > 0, and |z|, for
z < 0. Auto-correlation a(z) = [(1/(2λ3 )] exp(−λ|z|).
−1/2
by parts.
Prove the result for t1/2 by integrating
√ that for t
(a) Use (13.62) with n = 2 on L
t ; (b) use (13.63);
(c) consider L [exp(±at)
cos bt] and use the translation property, subsection 13.2.2.
(a) Note that | lim g(t)e−st dt| ≤ | lim g(t) dt|.
(b) (s2 + as + b)ȳ(s) = {c(s2 + 2ω 2 )/[s(s2 + 4ω 2 )]} + (a + s)y(0) + y (0).
For this damped system, at large t (corresponding to s → 0) rates of change
are negligible and the equation reduces to by = c cos2 ωt. The average value of
cos2 ωt is 12 .
s−1 [1 − exp(−sa)]; ga (x) = x for 0 < x < a, ga (x) = 2a − x for a ≤ x ≤ 2a,
ga (x) = 0 otherwise.
467