Kreyszig - Chapter 11 Fourier Analysis - Engineering Mathematics PDF Download

 Page 1


KREYSZIG E
Advanced Engineering Mathematics
(10th ed., Wiley 2011)
Chapter 11 - Fourier analysis
Page 2


KREYSZIG E
Advanced Engineering Mathematics
(10th ed., Wiley 2011)
Chapter 11 - Fourier analysis
.
Page 3


KREYSZIG E
Advanced Engineering Mathematics
(10th ed., Wiley 2011)
Chapter 11 - Fourier analysis
.
474
CHAPTER 11
Fourier Analysis
This chapter on Fourier analysis covers three broad areas: Fourier series in Secs. 11.1–11.4,
more general orthonormal series called Sturm–Liouville expansions in Secs. 11.5 and 11.6
and Fourier integrals and transforms in Secs. 11.7–11.9.
The central starting point of Fourier analysis is Fourier series. They are infinite series
designed to represent general periodic functions in terms of simple ones, namely, cosines
and sines. This trigonometric system is orthogonal, allowing the computation of the
coefficients of the Fourier series by use of the well-known Euler formulas, as shown in
Sec. 11.1. Fourier series are very important to the engineer and physicist because they
allow the solution of ODEs in connection with forced oscillations (Sec. 11.3) and the
approximation of periodic functions (Sec. 11.4). Moreover, applications of Fourier analysis
to PDEs are given in Chap. 12. Fourier series are, in a certain sense, more universal than
the familiar Taylor series in calculus because many discontinuous periodic functions that
come up in applications can be developed in Fourier series but do not have Taylor series
expansions.
The underlying idea of the Fourier series can be extended in two important ways. We
can replace the trigonometric system by other families of orthogonal functions, e.g., Bessel
functions and obtain the Sturm–Liouville expansions. Note that related Secs. 11.5 and
11.6 used to be part of Chap. 5 but, for greater readability and logical coherence, are now
part of Chap. 11. The second expansion is applying Fourier series to nonperiodic
phenomena and obtaining Fourier integrals and Fourier transforms. Both extensions have
important applications to solving PDEs as will be shown in Chap. 12.
In a digital age, the discrete Fourier transform plays an important role. Signals, such
as voice or music, are sampled and analyzed for frequencies. An important algorithm, in
this context, is the fast Fourier transform. This is discussed in Sec. 11.9.
Note that the two extensions of Fourier series are independent of each other and may
be studied in the order suggested in this chapter or by studying Fourier integrals and
transforms first and then Sturm–Liouville expansions.
Prerequisite: Elementary integral calculus (needed for Fourier coefficients).
Sections that may be omitted in a shorter course: 11.4–11.9.
References and Answers to Problems: App. 1 Part C, App. 2.
11.1 Fourier Series
Fourier series are infinite series that represent periodic functions in terms of cosines and
sines. As such, Fourier series are of greatest importance to the engineer and applied
mathematician. To define Fourier series, we first need some background material.
A function is called a periodic function if is defined for all real x, except f ( x) f (x)
c11-a.qxd  10/30/10  1:24 PM  Page 474
Page 4


KREYSZIG E
Advanced Engineering Mathematics
(10th ed., Wiley 2011)
Chapter 11 - Fourier analysis
.
474
CHAPTER 11
Fourier Analysis
This chapter on Fourier analysis covers three broad areas: Fourier series in Secs. 11.1–11.4,
more general orthonormal series called Sturm–Liouville expansions in Secs. 11.5 and 11.6
and Fourier integrals and transforms in Secs. 11.7–11.9.
The central starting point of Fourier analysis is Fourier series. They are infinite series
designed to represent general periodic functions in terms of simple ones, namely, cosines
and sines. This trigonometric system is orthogonal, allowing the computation of the
coefficients of the Fourier series by use of the well-known Euler formulas, as shown in
Sec. 11.1. Fourier series are very important to the engineer and physicist because they
allow the solution of ODEs in connection with forced oscillations (Sec. 11.3) and the
approximation of periodic functions (Sec. 11.4). Moreover, applications of Fourier analysis
to PDEs are given in Chap. 12. Fourier series are, in a certain sense, more universal than
the familiar Taylor series in calculus because many discontinuous periodic functions that
come up in applications can be developed in Fourier series but do not have Taylor series
expansions.
The underlying idea of the Fourier series can be extended in two important ways. We
can replace the trigonometric system by other families of orthogonal functions, e.g., Bessel
functions and obtain the Sturm–Liouville expansions. Note that related Secs. 11.5 and
11.6 used to be part of Chap. 5 but, for greater readability and logical coherence, are now
part of Chap. 11. The second expansion is applying Fourier series to nonperiodic
phenomena and obtaining Fourier integrals and Fourier transforms. Both extensions have
important applications to solving PDEs as will be shown in Chap. 12.
In a digital age, the discrete Fourier transform plays an important role. Signals, such
as voice or music, are sampled and analyzed for frequencies. An important algorithm, in
this context, is the fast Fourier transform. This is discussed in Sec. 11.9.
Note that the two extensions of Fourier series are independent of each other and may
be studied in the order suggested in this chapter or by studying Fourier integrals and
transforms first and then Sturm–Liouville expansions.
Prerequisite: Elementary integral calculus (needed for Fourier coefficients).
Sections that may be omitted in a shorter course: 11.4–11.9.
References and Answers to Problems: App. 1 Part C, App. 2.
11.1 Fourier Series
Fourier series are infinite series that represent periodic functions in terms of cosines and
sines. As such, Fourier series are of greatest importance to the engineer and applied
mathematician. To define Fourier series, we first need some background material.
A function is called a periodic function if is defined for all real x, except f ( x) f (x)
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SEC. 11.1 Fourier Series 475
x
f(x)
p
Fig. 258. Periodic function of period p
possibly at some points, and if there is some positive number p, called a periodof ,
such that
(1) for all x.
(The function is a periodic function that is not defined for all real x but
undefined for some points (more precisely, countably many points), that is 
.)
The graph of a periodic function has the characteristic that it can be obtained by periodic
repetition of its graph in any interval of length p (Fig. 258).
The smallest positive period is often called the fundamental period. (See Probs. 2–4.)
Familiar periodic functions are the cosine, sine, tangent, and cotangent. Examples of
functions that are not periodic are , to mention just a few.
If has period p, it also has the period 2p because (1) implies 
, etc.; thus for any integer 
(2) for all x.
Furthermore if and have period p, then with any constants a and
b also has the period p.
Our problem in the first few sections of this chapter will be the representation of various
functions of period in terms of the simple functions
(3)
All these functions have the period . They form the so-called trigonometric system.
Figure 259 shows the first few of them (except for the constant 1, which is periodic with
any period).
2p
1,  cos x, sin x,  cos 2x, sin 2x,
Á
,  cos nx, sin nx,
Á
.
2p f (x)
af (x) bg (x) g (x) f (x)
f (x np) f (x)
n 1, 2, 3,
Á
, f ([x p] p) f (x p) f (x)
f (x 2p) f (x)
x, x
2
, x
3
, e
x
, cosh x, and ln x
Á
3p>2,
xp>2,
f (x) tan x
f (x p) f (x)
f (x)
0 2p pp
cos x
0 2p pp
sin x
0 2p pp
sin 2x
0 2p pp
sin 3x
0 2p pp
cos 2x
0 2p pp
cos 3x
Fig. 259. Cosine and sine functions having the period 2 (the first few members of the
trigonometric system (3), except for the constant 1)
p
c11-a.qxd  10/30/10  1:24 PM  Page 475
Page 5


KREYSZIG E
Advanced Engineering Mathematics
(10th ed., Wiley 2011)
Chapter 11 - Fourier analysis
.
474
CHAPTER 11
Fourier Analysis
This chapter on Fourier analysis covers three broad areas: Fourier series in Secs. 11.1–11.4,
more general orthonormal series called Sturm–Liouville expansions in Secs. 11.5 and 11.6
and Fourier integrals and transforms in Secs. 11.7–11.9.
The central starting point of Fourier analysis is Fourier series. They are infinite series
designed to represent general periodic functions in terms of simple ones, namely, cosines
and sines. This trigonometric system is orthogonal, allowing the computation of the
coefficients of the Fourier series by use of the well-known Euler formulas, as shown in
Sec. 11.1. Fourier series are very important to the engineer and physicist because they
allow the solution of ODEs in connection with forced oscillations (Sec. 11.3) and the
approximation of periodic functions (Sec. 11.4). Moreover, applications of Fourier analysis
to PDEs are given in Chap. 12. Fourier series are, in a certain sense, more universal than
the familiar Taylor series in calculus because many discontinuous periodic functions that
come up in applications can be developed in Fourier series but do not have Taylor series
expansions.
The underlying idea of the Fourier series can be extended in two important ways. We
can replace the trigonometric system by other families of orthogonal functions, e.g., Bessel
functions and obtain the Sturm–Liouville expansions. Note that related Secs. 11.5 and
11.6 used to be part of Chap. 5 but, for greater readability and logical coherence, are now
part of Chap. 11. The second expansion is applying Fourier series to nonperiodic
phenomena and obtaining Fourier integrals and Fourier transforms. Both extensions have
important applications to solving PDEs as will be shown in Chap. 12.
In a digital age, the discrete Fourier transform plays an important role. Signals, such
as voice or music, are sampled and analyzed for frequencies. An important algorithm, in
this context, is the fast Fourier transform. This is discussed in Sec. 11.9.
Note that the two extensions of Fourier series are independent of each other and may
be studied in the order suggested in this chapter or by studying Fourier integrals and
transforms first and then Sturm–Liouville expansions.
Prerequisite: Elementary integral calculus (needed for Fourier coefficients).
Sections that may be omitted in a shorter course: 11.4–11.9.
References and Answers to Problems: App. 1 Part C, App. 2.
11.1 Fourier Series
Fourier series are infinite series that represent periodic functions in terms of cosines and
sines. As such, Fourier series are of greatest importance to the engineer and applied
mathematician. To define Fourier series, we first need some background material.
A function is called a periodic function if is defined for all real x, except f ( x) f (x)
c11-a.qxd  10/30/10  1:24 PM  Page 474
SEC. 11.1 Fourier Series 475
x
f(x)
p
Fig. 258. Periodic function of period p
possibly at some points, and if there is some positive number p, called a periodof ,
such that
(1) for all x.
(The function is a periodic function that is not defined for all real x but
undefined for some points (more precisely, countably many points), that is 
.)
The graph of a periodic function has the characteristic that it can be obtained by periodic
repetition of its graph in any interval of length p (Fig. 258).
The smallest positive period is often called the fundamental period. (See Probs. 2–4.)
Familiar periodic functions are the cosine, sine, tangent, and cotangent. Examples of
functions that are not periodic are , to mention just a few.
If has period p, it also has the period 2p because (1) implies 
, etc.; thus for any integer 
(2) for all x.
Furthermore if and have period p, then with any constants a and
b also has the period p.
Our problem in the first few sections of this chapter will be the representation of various
functions of period in terms of the simple functions
(3)
All these functions have the period . They form the so-called trigonometric system.
Figure 259 shows the first few of them (except for the constant 1, which is periodic with
any period).
2p
1,  cos x, sin x,  cos 2x, sin 2x,
Á
,  cos nx, sin nx,
Á
.
2p f (x)
af (x) bg (x) g (x) f (x)
f (x np) f (x)
n 1, 2, 3,
Á
, f ([x p] p) f (x p) f (x)
f (x 2p) f (x)
x, x
2
, x
3
, e
x
, cosh x, and ln x
Á
3p>2,
xp>2,
f (x) tan x
f (x p) f (x)
f (x)
0 2p pp
cos x
0 2p pp
sin x
0 2p pp
sin 2x
0 2p pp
sin 3x
0 2p pp
cos 2x
0 2p pp
cos 3x
Fig. 259. Cosine and sine functions having the period 2 (the first few members of the
trigonometric system (3), except for the constant 1)
p
c11-a.qxd  10/30/10  1:24 PM  Page 475
The series to be obtained will be a trigonometric series, that is, a series of the form
(4)
are constants, called the coefficients of the series. We see that each
term has the period Hence if the coefficients are such that the series converges, its
sum will be a function of period
Expressions such as (4) will occur frequently in Fourier analysis. To compare the
expression on the right with that on the left, simply write the terms in the summation.
Convergence of one side implies convergence of the other and the sums will be the
same.
Now suppose that is a given function of period and is such that it can be
represented by a series (4), that is, (4) converges and, moreover, has the sum . Then,
using the equality sign, we write
(5)
and call (5) the Fourier series of . We shall prove that in this case the coefficients
of (5) are the so-called Fourier coefficients of , given by the Euler formulas
(0)
(6) (a)
(b) .
The name “Fourier series” is sometimes also used in the exceptional case that (5) with
coefficients (6) does not converge or does not have the sum —this may happen but
is merely of theoretical interest. (For Euler see footnote 4 in Sec. 2.5.)
A Basic Example
Before we derive the Euler formulas (6), let us consider how (5) and (6) are applied in
this important basic example. Be fully alert, as the way we approach and solve this
example will be the technique you will use for other functions. Note that the integration
is a little bit different from what you are familiar with in calculus because of the n. Do
not just routinely use your software but try to get a good understanding and make
observations: How are continuous functions (cosines and sines) able to represent a given
discontinuous function? How does the quality of the approximation increase if you take
more and more terms of the series? Why are the approximating functions, called the
f (x)
n 1, 2,
Á
b
n

1
p
 

p
p
f (x) sin nx dx
n 1, 2,
Á
a
n

1
p
 

p
p
 
f (x) cos nx dx
a
0

1
2p
 

p
p
f (x) dx
f (x)
f (x)
f (x) a
0

a

n1
(a
n 
cos nx b
n 
sin nx)
f (x)
2p f (x)
2p.
2p.
a
0
, a
1
, b
1
, a
2
, b
2
,
Á
 a
0

a

n1
 (a
n 
cos nx b
n 
sin nx).
a
0
 a
1 
cos x b
1 
sin x a
2 
cos 2x b
2 
sin 2x
Á
476 CHAP. 11 Fourier Analysis
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