Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 13 Integral transforms - all with Video Answers

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Chapter Questions

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Problem 1

Find the Fourier transform of the function $f(t)=\exp (-|t|)$.
(a) By applying Fourier's inversion theorem prove that
$$
\frac{\pi}{2} \exp (-|t|)=\int_{0}^{\infty} \frac{\cos \omega t}{1+\omega^{2}} d \omega
$$
(b) By making the substitution $\omega=\tan \theta$, demonstrate the validity of Parseval's theorem for this function.

Eduard Sanchez

Numerade Educator

08:33

Problem 2

Use the general definition and properties of Fourier transforms to show the following.
(a) If $f(x)$ is periodic with period $a$ then $f(k)=0$ unless $k a=2 \pi n$ for integer $n$.
(b) The Fourier transform of $t f(t)$ is $i d f(\omega) / d \omega$.
(c) The Fourier transform of $f(m t+c)$ is
$$
\frac{e^{\mathrm{i\omegac} / \mathrm{m}}}{m} \hat{f}\left(\frac{\omega}{m}\right)
$$

Eduard Sanchez

Numerade Educator

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Problem 3

Find the Fourier transform of } H(x-a) e^{-b x} \text {, where } H(x) \text { is the Heaviside function. }

Zia Ullah

Numerade Educator

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Problem 4

Prove that the Fourier transform of the function $f(t)$ defined in the $t$-plane by straight-line segments joining $(-T, 0)$ to $(0,1)$ to $(T, 0)$, with $f(t)=0$ outside $|t|<T$, is
$$
\tilde{f}(\omega)=\frac{T}{\sqrt{2 \pi}} \operatorname{sinc}^{2}\left(\frac{\omega T}{2}\right)
$$
where sinc $x$ is defined as $(\sin x) / x$.
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 $(2,0)$.

Eduard Sanchez

Numerade Educator

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Problem 5

By taking the Fourier transform of the equation
$$
\frac{d^{2} \phi}{d x^{2}}-K^{2} \phi=f(x)
$$
show that its solution $\phi(x)$ can be written as
$$
\phi(x)=\frac{-1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \frac{e^{\mathbb{d} \times \tilde{f}}(k)}{k^{2}+K^{2}} d k
$$
where $\tilde{f}(k)$ is the Fourier transform of $f(x)$.

Eduard Sanchez

Numerade Educator

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Problem 6

By differentiating the definition of the Fourier sine transform $f_{s}(\omega)$ of the function $f(t)=t^{-1 / 2}$ with respect to $\omega$, and then integrating the resulting expression by parts, find an elementary differential equation satisfied by $\hat{f}_{s}(\omega)$. Hence show that this function is its own Fourier sine transform, i.e. $f_{s}(\omega)=A f(\omega)$, where $A$ is a constant. Show that it is also its own Fourier cosine transform. (Assume that the, limit as $x \rightarrow \infty$ of $x^{1 / 2} \sin \alpha x$ can be taken as zero.

Eduard Sanchez

Numerade Educator

04:10

Problem 7

(a) Find the Fourier transform of the unit rectangular distribution
$$
f(t)= \begin{cases}1 & |t|<1 \\ 0 & \text { otherwise }\end{cases}
$$
(b) Determine the convolution of $f$ with itself and, without further integration, deduce its transform.
(c) Deduce that
$$
\begin{aligned}
&\int_{-\infty}^{\infty} \frac{\sin ^{2} \omega}{\omega^{2}} d \omega=\pi \\
&\int_{-\infty}^{\infty} \frac{\sin ^{4} \omega}{\omega^{4}} d \omega=\frac{2 \pi}{3}
\end{aligned}
$$

Amit Srivastava

Numerade Educator

04:02

Problem 8

Calculate the Fraunhofer spectrum produced by a diffraction grating, uniformly illuminated by light of wavelength $2 \pi / k$, as follows. Consider a grating with $4 N$ equal strips each of width $a$ and alternately opaque and transparent. The aperture function is then
$$
f(y)= \begin{cases}A & \text { for }(2 n+1) a \leq y \leq(2 n+2) a, \quad-N \leq n<N \\ 0 & \text { otherwise }\end{cases}
$$
(a) Show, for diffraction at angle $\theta$ to the normal to the grating, that the required Fourier transform can be written
$$
\tilde{f}(q)=(2 \pi)^{-1 / 2} \sum_{r=-N}^{N-1} \exp (-2 i a r q) \int_{a}^{2 a} A \exp (-i q u) d u
$$
where $q=k \sin \theta$
(b) Evaluate the integral and sum to show that
$$
\tilde{f}(q)=(2 \pi)^{-1 / 2} \exp (-i q a / 2) \frac{A \sin (2 q a N)}{q \cos (q a / 2)}
$$
and hence that the intensity distribution $I(\theta)$ in the spectrum is proportional to
$$
\frac{\sin ^{2}(2 q a N)}{q^{2} \cos ^{2}(q a / 2)}
$$
(c) For large values of $N$, the numerator in the above expression has very closely spaced maxima and minima as a function of $\theta$ and effectively takes its mean value, $1 / 2$, giving a low-intensity background. Much more significant peaks in $I(\theta)$ occur when $\theta=0$ or the cosine term in the denominator vanishes. Show that the corresponding values of $|\tilde{f}(q)|$ are
$$
\frac{2 a N A}{(2 \pi)^{1 / 2}} \text { and } \frac{4 a N A}{(2 \pi)^{1 / 2}(2 m+1) \pi} \quad \text { with } m \text { integral. }
$$
Note that the constructive interference makes the maxima in $I(\theta) \propto N^{2}$, not $N$. Of course, observable maxima only occur for $0 \leq \theta \leq \pi / 2$

Khoobchandra Agrawal

Numerade Educator

13:37

Problem 9

By finding the complex Fourier series for its LHS show that either side of the equation
$$
\sum_{n=-\infty}^{\infty} \delta(t+n T)=\frac{1}{T} \sum_{n=-\infty}^{\infty} e^{-2 \sin / T}
$$
can represent a periodic train of impulses. By expressing the function $f(t+n X)$, in which $X$ is a constant, in terms of the Fourier transform $\hat{f}(\omega)$ of $f(t)$, show that
$$
\sum_{n=-\infty}^{\infty} f(t+n X)=\frac{\sqrt{2 \pi}}{X} \sum_{n=-\infty}^{\infty} \hat{f}\left(\frac{2 n \pi}{X}\right) e^{2 \pi \operatorname{mit} / X}
$$
This result is known as the Poisson summation formula.

Matthew Allcock

Numerade Educator

02:08

Problem 10

In many applications in which the frequency spectrum of an analogue signal is required, the best that can be done is to sample the signal $f(t)$ a finite number of times at fixed intervals and then use a discrete Fourier transform $F_{k}$ to estimate discrete points on the (true) frequency spectrum $\tilde{f}(\omega)$.
(a) By an argument that is essentially the converse of that given in section $13.1$, show that, if $N$ samples $f_{n}$, beginning at $t=0$ and spaced $t$ apart, are taken, then $f(2 \pi k /(N \tau)) \approx F_{k} \tau$ where
$$
F_{k}=\frac{1}{\sqrt{2 \pi}} \sum_{n=0}^{N-1} f_{n} e^{-2 \pi m k / N}
$$
(b) For the function $f(t)$ defined by
$$
f(t)= \begin{cases}1 & \text { for } 0 \leq t<1 \\ 0 & \text { otherwise }\end{cases}
$$
from which eight samples are drawn at intervals of $\tau=0.25$, find a formula for $\left|F_{k}\right|$ and evaluate it for $k=0,1, \ldots, 7 .$
(c) Find the exact frequency spectrum of $f(t)$ and compare the actual and estimated values of $\sqrt{2 \pi}|\hat{f}(\omega)|$ at $\omega=k \pi$ for $k=0,1, \ldots, 7$. Note the relatively good agreement for $k<4$ and the lack of agreement for larger values of $k$

James Kiss

Numerade Educator

03:33

Problem 11

For a function $f(t)$ that is non-zero only in the range $|t|<T / 2$, the full frequency spectrum $\hat{f}(\omega)$ can be constructed, in principle exactly, from values at discrete sample points $\omega=n(2 \pi / T)$. Prove this as follows.
(a) Show that the coefficients of a complex Fourier series representation of $f(t)$ with period $T$ can be written as
$$
c_{n}=\frac{\sqrt{2 \pi}}{T} \hat{f}\left(\frac{2 \pi n}{T}\right)
$$
(b) Use this result to represent $f(t)$ as an infinite sum in the defining integral for $\tilde{f}(\omega)$, and hence show that
$$
\hat{f}(\omega)=\sum_{n=-\infty}^{\infty} \hat{f}\left(\frac{2 \pi n}{T}\right) \operatorname{sinc}\left(n \pi-\frac{\omega T}{2}\right)
$$
where sinc $x$ is defined as $(\sin x) / x$.

Amit Srivastava

Numerade Educator

03:29

Problem 12

A signal obtained by sampling a function $x(t)$ at regular intervals $T$ is passed through an electronic filter, whose response $g(t)$ to a unit $\delta$-function input is represented in a tg-plot by straight lines joining $(0,0)$ to $(T, 1 / T)$ to $(2 T, 0)$ and is zero for all other values of $t .$ The output of the filter is the convolution of the input, $\sum_{-\infty}^{\infty} x(t) \delta(t-n T)$, with $g(t)$.
Using the convolution theorem, and the result given in exercise $13.4$, show that the output of the filter can be written
$$
y(t)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} x(n T) \int_{-\infty}^{\infty} \sin c^{2}\left(\frac{\omega T}{2}\right) e^{-i \omega[(n+1) T-t]} d \omega
$$

Arpit Gupta

Numerade Educator

03:33

Problem 13

(a) Find the Fourier transform of
$$
f(\gamma, p, t)= \begin{cases}e^{-\gamma \mathrm{r}} \sin p t & t>0 \\ 0 & t<0\end{cases}
$$
where $\gamma(>0)$ and $p$ are constant parameters. (b) The current $I(t)$ flowing through a certain system is related to the applied voltage $V(t)$ by the equation
$$
I(t)=\int_{-\infty}^{\infty} K(t-u) V(u) d u
$$
where
$$
K(\tau)=a_{1} f\left(\gamma_{1}, p_{1}, \tau\right)+a_{2} f\left(\gamma_{2}, p_{2}, \tau\right)
$$
The function $f(\gamma, p, t)$ is as given in (a) and all the $a_{i}, \gamma_{i}(>0)$ and $p_{i}$ are fixed parameters. By considering the Fourier transform of $I(t)$, find the relationship that must hold between $a_{1}$ and $a_{2}$ if the total net charge $Q$ passed through the system (over a very long time) is to be zero for an arbitrary applied voltage.

Amit Srivastava

Numerade Educator

04:33

Problem 14

Prove the equality
$$
\int_{0}^{\infty} e^{-2 a t} \sin ^{2} a t d t=\frac{1}{\pi} \int_{0}^{\infty} \frac{a^{2}}{4 a^{4}+\omega^{4}} d \omega
$$

Eduard Sanchez

Numerade Educator

03:06

Problem 15

A linear amplifier produces an output that is the convolution of its input and its response function. The Fourier transform of the response function for a particular amplifier is
$$
\tilde{K}(\omega)=\frac{i \omega}{\sqrt{2 \pi}(\alpha+i \omega)^{2}}
$$
Determine the time variation of its output $g(t)$ when its input is the Heaviside step function. (Consider the Fourier transform of a decaying exponential function and the result of exercise $13.2(\mathrm{~b})$.)

Narayan Hari

Numerade Educator

11:54

Problem 16

In quantum mechanics, two equal-mass particles having momenta $\mathbf{p}_{j}=\hbar \mathbf{k}_{j}$ and energies $E_{j}=\hbar \omega_{j}$ and represented by plane wavefunctions $\phi_{j}=\exp \left[i\left(\mathbf{k}_{j} \cdot \mathbf{r}_{j}-\omega_{j} t\right)\right]$ $j=1,2$, interact through a potential $V=V\left(\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|\right) .$ In first-order perturbation theory the probability of scattering to a state with momenta and energies $\mathbf{p}_{j}^{\prime}, E_{j}^{\prime}$ is determined by the modulus squared of the quantity
$$
M=\iiint \psi_{f}^{*} V \psi_{l} d \mathbf{r}_{1} d \mathbf{r}_{2} d t
$$
The initial state $\psi_{i}$ is $\phi_{1} \phi_{2}$ and the final state $\psi_{f}$ is $\phi_{1}^{\prime} \phi_{2}^{\prime}$
(a) By writing $\mathbf{r}_{1}+\mathbf{r}_{2}=2 \mathbf{R}$ and $\mathbf{r}_{1}-\mathbf{r}_{2}=\mathbf{r}$ and assuming that $d \mathbf{r}_{1} d \mathbf{r}_{2}=d \mathbf{R} d \mathbf{r}$, show that $M$ can be written as the product of three one-dimensional integrals.
(b) From two of the integrals deduce energy and momentum conservation in the form of $\delta$-functions.
(c) Show that $M$ is proportional to the Fourier transform of $V$, i.e. $\tilde{V}(k)$ where $2 \hbar \mathbf{k}=\left(\mathbf{p}_{2}-\mathbf{p}_{1}\right)-\left(\mathbf{p}_{2}^{\prime}-\mathbf{p}_{1}^{\prime}\right)$

Ren Jie Tuieng

Numerade Educator

01:29

Problem 17

For some ion-atom scattering processes, the potential $V$ of the previous example may be approximated by $V=\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|^{-1} \exp \left(-\mu\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|\right)$. Show, using the result of the worked example in subsection $13.1 .10$, that the probability that the ion will scatter from, say, $\mathbf{p}_{1}$ to $\mathbf{p}_{1}^{\prime}$ is proportional to $\left(\mu^{2}+k^{2}\right)^{-2}$ where $k=|\mathbf{k}|$ and $\mathbf{k}$ is as given in part (c) of exercise $13.16$

Khoobchandra Agrawal

Numerade Educator

06:00

Problem 18

The equivalent duration and bandwidth, $T_{e}$ and $B_{c}$, of a signal $x(t)$ are defined in terms of the latter and its Fourier transform $\tilde{x}(\omega):$
$$
\begin{gathered}
T_{\mathrm{e}}=\frac{1}{x(0)} \int_{-\infty}^{\infty} x(t) d t \\
B_{e}=\frac{1}{\tilde{x}(0)} \int_{-\infty}^{\infty} \tilde{x}(\omega) d \omega \\
where neither $x(0)$ nor $x(0)$ is zero. Show that the product $T_{c} B_{e}=2 \pi$ (this is a form of uncertainty principle), and find the equivalent bandwidth of the signal
$$
x(t)=\exp (-|t| / T)
$$
For this signal, determine the fraction of the total energy that lies in the frequency range $|\omega|<B_{e} / 4$. You will need the indefinite integral with respect to $x$ of $\left(a^{2}+x^{2}\right)^{-2}$, which is
$$
\frac{x}{2 a^{2}\left(a^{2}+x^{2}\right)}+\frac{1}{2 a^{3}} \tan ^{-1} \frac{x}{a}
$$

Saman Zulfiqar

Numerade Educator

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Problem 19

Calculate directly the auto-correlation function $a(z)$ for the product of the exponential decay distribution and the Heaviside step function
$$
f(t)=\frac{1}{\lambda} e^{-\hat{L}} H(t)
$$
Use the Fourier transform and energy spectrum of $f(t)$ to deduce that
$$
\int_{-\infty}^{\infty} \frac{e^{j \omega z}}{\lambda^{2}+\omega^{2}} d \omega=\frac{\pi}{\lambda} e^{-\lambda|z|}
$$

Eduard Sanchez

Numerade Educator

01:24

Problem 20

Prove that the cross-correlation $C(z)$ of the Gaussian and Lorentzian distributions
$$
f(t)=\frac{1}{\tau \sqrt{2 \pi}} \exp \left(-\frac{t^{2}}{2 \tau^{2}}\right), \quad g(t)=\left(\frac{a}{\pi}\right) \frac{1}{t^{2}+a^{2}}
$$
has as its Fourier transform the function
$$
\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{\tau^{2} \omega^{2}}{2}\right) \exp (-a|\omega|)
$$
Hence show that
$$
C(z)=\frac{1}{\tau \sqrt{2 \pi}} \exp \left(\frac{a^{2}-z^{2}}{2 t^{2}}\right) \cos \left(\frac{a z}{t^{2}}\right)
$$

Manik Pulyani

Numerade Educator

00:59

Problem 21

Prove the expressions given in table $13.1$ for the Laplace transforms of $t^{-1 / 2}$ and $t^{1 / 2}$, by setting $x^{2}=t s$ in the result
$$
\int_{0}^{\infty} \exp \left(-x^{2}\right) d x=\frac{1}{2} \sqrt{\pi}
$$

James Kiss

Numerade Educator

06:46

Problem 22

Find the functions $y(t)$ whose Laplace transforms are the following,
(a) $1 /\left(s^{2}-s-2\right)$
(b) $2 s /\left[(s+1)\left(s^{2}+4\right)\right]$,
(c) $e^{-(i+t) t_{0}} /\left[(s+\gamma)^{2}+b^{2}\right]$.

Arpit Gupta

Numerade Educator

07:47

Problem 23

Use the properties of Laplace transforms to prove the following without evaluating any Laplace integrals explicitly:
(a) $\mathscr{L}\left[t^{5 / 2}\right]=\frac{15}{8} \sqrt{\pi s^{-7 / 2}}$
(b) $\mathscr{L}[(\sinh a t) / t]=\frac{1}{2} \ln [(s+a) /(s-a)], \quad s>|a|$
(c) $\mathscr{L}[\sinh a t \cos b t]=a\left(s^{2}-a^{2}+b^{2}\right)\left[(s-a)^{2}+b^{2}\right]^{-1}\left[(s+a)^{2}+b^{2}\right]^{-1}$

Sam Low

Numerade Educator

01:03

Problem 24

Find the solution (the so-called impulse response or Green's function) of the equation
$$
T \frac{d x}{d t}+x=\delta(t)
$$
by proceeding as follows. (a) Show by substitution that
$$
x(t)=A\left(1-e^{-t / T}\right) H(t)
$$
is a solution, for which $x(0)=0$, of
$$
T \frac{d x}{d t}+x=A H(t)
$$
where $H(t)$ is the Heaviside step function.
(b) Construct the solution when the RHS of $\left(^{*}\right)$ is replaced by $A H(t-\tau)$ with $d x / d t=x=0$ for $t<t$, and hence find the solution when the RHS is a rectangular pulse of duration $\tau .$
(c) By setting $A=1 / \tau$ and taking the limit when $\tau \rightarrow 0$, show that the impulse response is $x(t)=T^{-1} e^{-\mathrm{t} / T}$
(d) Obtain the same result much more directly by taking the Laplace transform of each term in the original equation, solving the resulting algebraic equation and then using the entries in table $13.1$.

Raj Bala

Numerade Educator

04:30

Problem 25

(a) If $f(t)=A+g(t)$, where $A$ is a constant and the indefinite integral of $g(t)$ is bounded as its upper limit tends to $\infty$, show that
$$
\lim _{s \rightarrow 0} \bar{f}(s)=A
$$
(b) For $t>0$ the function $y(t)$ obeys the differential equation
$$
\frac{d^{2} y}{d t^{2}}+a \frac{d y}{d t}+b y=c \cos ^{2} \omega t
$$
where $a, b$ and $c$ are positive constants. Find $y(s)$ and show that $s \bar{y}(s) \rightarrow c / 2 b$ as $s \rightarrow 0 .$ Interpret the result in the $t$-domain.

Albert Zhang

Numerade Educator

02:37

Problem 26

By writing $f(x)$ as an integral involving the $\delta$-function $\delta(\xi-x)$ and taking the Laplace transforms of both sides, show that the transform of the solution of the equation
$$
\frac{d^{4} y}{d x^{4}}-y=f(x)
$$
for which $y$ and its first three derivatives vanish at $x=0$ can be written as
$$
\bar{y}(s)=\int_{0}^{\infty} f(\xi) \frac{e^{-s \xi}}{s^{4}-1} d \xi
$$
Use the properties of Laplace transforms and the entries in table $13.1$ to show that
$$
y(x)=\frac{1}{2} \int_{0}^{x} f(\xi)[\sinh (x-\xi)-\sin (x-\xi)] d \xi
$$

Andrija Isakov

Numerade Educator

01:26

Problem 27

The function $f_{a}(x)$ is defined as unity for $0<x<a$ and zero otherwise. Find its Laplace transform $\bar{f}_{a}(s)$ and deduce that the transform of $x f_{a}(x)$ is
$$
\frac{1}{s^{2}}\left[1-(1+a s) e^{-s a}\right]
$$
Write $f_{a}(x)$ in terms of Heaviside functions and hence obtain an explicit expression for
$$
g_{a}(x)=\int_{0}^{x} f_{a}(y) f_{a}(x-y) d y
$$
Use the expression to write $\bar{g}_{a}(s)$ in terms of the functions $\vec{f}_{a}(s)$, and $\bar{f}_{2 a}(s)$ and their derivatives, and hence show that $\bar{g}_{a}(s)$ is equal to the square of $\bar{f}_{a}(s)$, in accordance with the convolution theorem.

Nick Johnson

Numerade Educator

03:01

Problem 28

(a) Show that the Laplace transform of $f(t-a) H(t-a)$, where $a \geq 0$, is $e^{-a s} \vec{f}(s)$.
(b) If $g(t)$ is a periodic function of period $T$, show that $\bar{g}(s)$ can be written as
$$
\frac{1}{1-e^{-s T}} \int_{0}^{T} e^{-s t} g(t) d t
$$
(c) Sketch the periodic function defined in $0 \leq t \leq T$ by
$$
g(t)= \begin{cases}2 t / T & 0 \leq t<T / 2 \\ 2(1-t / T) & T / 2 \leq t \leq T\end{cases}
$$
and, using the result in (b), find its Laplace transform.
(d) Show, by sketching it, that
$$
\frac{2}{T}\left[t H(t)+2 \sum_{n=1}^{\infty}(-1)^{n}\left(t-\frac{1}{2} n T\right) H\left(t-\frac{1}{2} n T\right)\right]
$$
is another representation of $g(t)$ and hence derive the relationship
$$
\tanh x=1+2 \sum^{\infty}(-1)^{n} e^{-2 n x}
$$

James Kiss

Numerade Educator