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Chapter 4 FOURIER SERIES
Contents
4.1 Introduction
4.2 Fourier Series Expansion
4.3 Functions Defined over a Finite Interval
4.4 Differentiation and Integration of Fourier Series
4.5 Engineering Applications
4.6 Complex Fourier Series4.7 Orthogonal Functions
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Chapter 4 FOURIER SERIES4.1 INTRODUCTION
The representation of a function in the form of a series is fairlycommon practice in mathematics. Probably the most familiarexpansions are power series of the form
f(x) =a0+a1x+ +anxn + =n=0
anxn.
There are frequently advantages in expanding a function in such aseries, since the first few terms of a good approximation are easily
to deal with.The functions represented by power series play important role inanalysis. But the class of these functions is too restricted in manyinstances.
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4.1 INTRODUCTION
It was therefore an even of major importance when Fourier
illustrated by many examples the fact that convergenttrigonometric seriesof the form
f(x) =a0
2 +
n=1(ancos nx+bnsin nx)
with constant coefficients an, bn are capable of representing a wideclass of functions f(x) which includes essentially every function ofspecific interest.
Soon after Fouriers dramatic discovery the Fourier series wererecognized not only as a most powerful tool for physics andmathematics, but just as much as a fruitful source of manybeautiful purely mathematical results.
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4.1 INTRODUCTION
There are also many practical reasons for expanding a function asa trigonometric sum. Iff(t) is a signal, then a decomposition offinto a trigonometric sum gives a description of its componentfrequencies.
A common task in signal analysis is the elimination of highfrequency noise. One approach is to express f as a trigonometricsum
f(t) =a0
2 +
n=1(ancos nt+bnsin nt)
and then set the high-frequency coefficients (the an and bn forlarge n) equal to zero.
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4.1 INTRODUCTION
Another common task in signal analysis is data compression.
The goal here is to send a signal in a way that requires minimadata transmission. One approach is to express the signal f in terms
of a trigonometric expansion and then send only those coefficients,an and bn, that are larger (in absolute value) than some specifictolerance. The coefficients that are small and do not contributesubstantially to f can be thrown away.
There is no danger that an infinite number of coefficients staylarge, because we will show that an 0 and bn 0 as n .
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4.1 INTRODUCTION
In general, as mentioned in previous chapter, periodic functionsfrequently occur as input signals in engineering applications.
Fourier series provide the ideal framework for analysing the
steady-state response to such periodic input signals, since theyenable us to represent the signals as infinite sums of sinusoids.Because of the linear character of the system, the desiredsteady-state response can be determined as the sum of the
individual responses.
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4.1 INTRODUCTION
Functions that are expressible by trigonometric series
f(t) = a0
2 +
n=1
(ancos nt+bnsin nt)
are periodic with the period 2/. But, as we shall see, thisrestriction is inessential as soon as we consider a function merely ina finite interval from which we can easily extend it as a periodicfunction.
The theory of Fourier series is complicated, but we shall see thatthe application of these series is rather simple. The fundamentalquestion in this chapter is devoted to concern the problem ofrepresenting an arbitrary periodic function in the form of a sum ofsine and cosine functions.
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4.2 FOURIER SERIES EXPANSION4.2.1 PERIODIC FUNCTIONS
Recall that a function f(x) is called a periodic function if there issome positive number T, called a period off(x), such thatf(x+T) =f(x) for all x in the domain off.
If f is a continuous periodic function which is not identically equalto a constant, it possesses the least positive period called itsfundamental period.
When speaking about the period of a function we usually mean itsfundamental period.
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4.2 FOURIER SERIES EXPANSION4.2.1 PERIODIC FUNCTIONS
To provide a measure of the number of repetitions per unit ofx,
we define the frequency of a periodic function to be the reciprocalof its period, so that
frequency = 1
period=
1
T
The term circular frequency is also used in engineering, and isdefined by
circular frequency = 2
frequency =2
T
and is measured in radians per second.
It is common to drop the term circular and refer to this simplyas the frequency when the context is clear.
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4.2 FOURIER SERIES EXPANSION4.2.1 PERIODIC FUNCTIONS
Periodic Extension of a Function
Let f(x) be any function defined on [a, b]. Set T =b a. Thefunction defined by
(x) =
f(x) if a
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4.2 FOURIER SERIES EXPANSION4.2.1 PERIODIC FUNCTIONS
Integral of a Periodic Function over a Period
Iff(x) is a periodic integrable function with periodic Twe have
+T
f(x)dx=
T0
f(x)dx for any < <
Thus, the integral of a periodic function with period T taken overan arbitrary interval of length T has one and the same value.
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4.2 FOURIER SERIES EXPANSION4.2.1 PERIODIC FUNCTIONS
In physics the simplest periodic processes are described by means
of the function
(t) =A sin(t+ ),
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4.2 FOURIER SERIES EXPANSION4.2.1 PERIODIC FUNCTIONS
The fundamental question in this section is devoted to concern theproblem of representing an arbitrary periodic function in the formof a sum of harmonics:
f(x) =A0+A1sin(x+1)+A2sin(2x+2)+ +Ansin(nx+n) .(1)
The term A1sin(x+ 1) is called the first harmonic or thefundamental mode, and it has the same frequency as thefunction f(x).
The term Ansin(nx+ n) is called the nth harmonic, and it hasfrequency n, which is n times that of the fundamental.
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4.2 FOURIER SERIES EXPANSION4.2.1 PERIODIC FUNCTIONS
The expression (1) may be written as
f(x) =a0
2 +
n=1
ancos nx+bnsin nx
. (2)
The functional series on the right hand side of (2) is called atrigonometric series.
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4.2 FOURIER SERIES EXPANSION4.2.1 PERIODIC FUNCTIONS
Statement of the Key Problem The main aim of the present
chapter is to elucidate the following questions:
(1) What are the periodic functions with period 2lwhich can beexpressed in trigonometric series
f(x) =a0
2 +
n=1
ancos
nx
l +bnsin
nx
l
, (3)
i.e., can be represented as a sum of this kind?
(2) How can we determine the coefficients a0, an and bn,n= 1, 2,... of expression (3) if it is valid?
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4.2 FOURIER SERIES EXPANSION4.2.2 FOURIER SERIES
Recall that
A functionf(x) is said to have a jump discontinuity atx0 (a, b) iff(x) is discontinuous at x0 and the one-sided limits
limxx0
f(x) and limxx+0
f(x)
exist as finite numbers. If the discontinuity occurs at an endpoint,x0 =a (orb), a jump discontinuity occurs if the one-sided limit off(x) as x a+ (x b) exists as a finite number.
A functionf is said to be piecewise continuous on [a, b] if it
is continuous at every point in [a, b] except possibly for a finitenumber of points at which fhas a jump discontinuity.
A functionf is said to be odd iff(x) =f(x) for all x in thedomain off. f is even iff(x) =f(x) for all x in the domain off.
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4.2 FOURIER SERIES EXPANSION4.2.2 FOURIER SERIES
Knowing that a function is even or odd can be useful in evaluating
definite integrals.
Theorem
If f is an even piecewise continuous function on [
a, a], then a
a
f(x)dx= 2
a0
f(x)dx.
If f is an odd piecewise continuous function on [a, a], then aa
f(x)dx= 0.
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4.2 FOURIER SERIES EXPANSION4.2.2 FOURIER SERIES
Example 2.1 Show that
(a)
l
lsin mx
l cos nx
l dx= 0.
(b)ll
sin mxl
sin nxl
dx=
0, m =n,l, m=n.
(c)ll
cos mxl
cos nxl
dx=
0, m =n,l, m=n.
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4.2 FOURIER SERIES EXPANSION4.2.2 FOURIER SERIES
It is easy to verify that if each of the functions f1, f2,..., fn isperiodic with period T, then so is any linear combinationc1f1(x) +c2f2(x) + +cnfn(x).
For example, the sum 5 + 3 sin x 8sin3x+ 7 cos 3x hasperiod 2 since each term has period 2.
Furthermore, if the infinite series
a0
2 +
n=1
ancosnxl
+bnsinnx
l
converges to f(x) for all x, then f(x) will be periodic with period2l.
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4.2 FOURIER SERIES EXPANSION4.2.2 FOURIER SERIES
Let f(x) be an arbitrary integrable function defined on [l, l].
DefinitionThe numbers a0, an and bn, n= 1, 2, ..., determined by
an =1
l l
l
f(x)cosnx
l dx, n= 0, 1, 2, ..., (4)
bn =1
l
ll
f(x)sinnx
l dx, n= 1, 2, ..., (5)
are called the Fourier coefficients of the function f(x). The series
a02
+n=1
ancos
nx
l +bnsin
nx
l
is called the Fourier series of the function f(x).
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4.2 FOURIER SERIES EXPANSION4.2.2 FOURIER SERIES
Formulae (4)(5) are called the Euler formulae for the Fourier
coefficients an and bn.
Note Iff(x) is a periodic function of period T = 2/ then itsFourier coefficients can be given by
an = 2
T
+T
f(x)cos nxdx (n= 0, 1, 2, . . . )
bn = 2
T
+T
f(x)sin nxdx (n= 1, 2, . . . )
The limits of integration in these formulas may be specified overany period. In practice, it is common to specify f(x) over eitherthe period
12 T
x
12 Tor the period 0
x
T.
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4.2 FOURIER SERIES EXPANSION
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4.2 FOURIER SERIES EXPANSION4.2.2 FOURIER SERIES
To every function f(x) integrable on the interval [l, l] therecorresponds its Fourier series. This correspondence can be writtenin the form
f(x) a02
+
n=1
ancosnx
l +bnsin
nx
l
where the series on the right-hand side is the trigonometric serieswhose coefficients are determined by Euler formulas.
Example 2.2 Obtain the Fourier series of the periodic functionf(t) of period 2 defined by
f(t) =t (0< t
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4.2 FOURIER SERIES EXPANSION4.2.3 FOURIER SERIES OF EVEN AND ODD FUNCTIONS
Theorem
(a) The Fourier series for an even function is a pure cosine series;that is, it contains no sine terms.
(b)The Fourier series for an odd function is a pure sine series; thatis, it contains no cosine terms.
Example 2.3 Letf(x) =x,l
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S S S4.2.4 CONVERGENCE OF FOURIER SERIES
Let
f(x+ 0) = limh0+ f(x+h) and f(x 0) = limh0f(x h).The following theorem covers most of the situations that arise inapplications.
Theorem(Pointwise Convergence of Fourier Series)If f and f are piecewise continuous on [l, l], then on (l, l), theFourier series of f
a0
2 +
n=1
ancosnxl
+bnsinnx
l
converges to f(x) if f is continuous at x, and to12 [f(x+ 0) +f(x 0)] if f is discontinuous at x . For x= l , theseries converges to 12 [f(
l+ 0) +f(l
0)].
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4.2 FOURIER SERIES EXPANSION
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4.2.4 CONVERGENCE OF FOURIER SERIES
In other words, when f and f are piecewise continuous on [
l, l],
the Fourier series converges to f(x) whenever f is continuous at xand converges to the average of the left- and right-hand limits atpoints where f is discontinuous.
Example 2.5 Compute the Fourier series for
f(x) =
0,
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4.2.4 CONVERGENCE OF FOURIER SERIES
Example 2.7 Find the expansion of the function
f(x) = |x|, x into Fouriers series
Remark: It can be proved that iff is piecewise continuous and
f(x) =c0
2 +
n=1
cncos
nx
l +dnsin
nx
l
on [
l, l], then this series must be the Fourier series off.
Therefore, a function fcan be expanded in one, and only one,Fourier series on the interval [l, l].
Example 2.8 Find the Fourier series of the functionf(x) = cos2 xon the interval [
, ].
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4.3 FUNCTIONS DEFINED OVER A FINITE INTERVAL
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4.3.1 FULL-RANGE SERIES
Consider a function fdefined in an interval [0, l] and suppose thatit is required to expand f into a trigonometric series.
3.3.1 FULL-RANGE SERIES
To obtain a full-range Fourier series representation off(x) (that is,a series consisting of both cosine and sine terms) we define theperiodic extension (x) off(x) by
(x) =
f(x), if 0
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4.3.1 FULL-RANGE SERIES
The Fourier coefficients are
an =2l
l
0f(x)cos2n
l xdx (n= 0, 1, 2, . . . )
bn =2
l
l0
f(x)sin2n
l xdx (n= 1, 2, . . . )
Example 2.9 Find a full-range Fourier series expansion off(x) =xvalid in the finite interval 0< x< . Draw graphs ofboth f(x) and the periodic function represented by the Fourier
series obtained.
In applications the most important role is played by the expansionsinto Fouriers sine and cosine series.
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4.3 FUNCTIONS DEFINED OVER A FINITE INTERVAL
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4.3.1 FULL-RANGE SERIES
A functionf(x) can also be extended to the interval [
l, 0) in such
a way that the extended function F(x) is even. This isaccomplished by defining
F(x) = f(x), 0 x l,f(
x),
l
x
0.
Since F is even, it is called the even extension off. Thus, theeven periodic extension off(t) is defined by
F(x) = f(x), 0 x l,f(x), l x 0
F(x+ 2l) =F(x).
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4.3.2 HALF-RANGE COSINE AND SINE SERIES
Similarly, the function f(x) can be extended to the interval [l, 0)as an odd function defined in the interval [
l, l]. The function
G(x) =
f(x), 0
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4.3.2 HALF-RANGE COSINE AND SINE SERIES
Definition
Let fbe piecewise continuous on the interval [0, l]. The Fouriercosine series off on [0, l] is
a02 +
n=1
ancosnx
l , (6)
where
an =2
l l
0f(x)cos
nx
l dx, n= 0, 1, 2,....
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4.3 FUNCTIONS DEFINED OVER A FINITE INTERVAL4 3 2 HALF RANGE COSINE AND SINE SERIES
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4.3.2 HALF-RANGE COSINE AND SINE SERIES
Definition
The Fourier sine series off on [0, l] is
n=1bnsin
nx
l , (7)
where
bn =2
l
l0
f(x)sinnx
l dx, n= 1, 2,....
(6) and (7) are called half-range Fourier series expansions forf(x). (6) is half-range cosine series expansion and (7)half-range sine series expansion off(x).
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4.3 FUNCTIONS DEFINED OVER A FINITE INTERVAL4 3 2 HALF RANGE COSINE AND SINE SERIES
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4.3.2 HALF-RANGE COSINE AND SINE SERIES
TheoremIf f and f are piecewise continuous on [0, l], then on this interval,f(x) can be expanded in the Fourier cosine series
a02 +
n=1
ancos
nx
l
or in the Fourier sine series
n=1
bnsin
nx
l .
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4.3 FUNCTIONS DEFINED OVER A FINITE INTERVAL4 3 2 HALF RANGE COSINE AND SINE SERIES
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4.3.2 HALF-RANGE COSINE AND SINE SERIES
To illustrate the various extensions, lets consider the functionf(x) =x, 0
x
.
The period extension (x) is given by
(x) =f(x) =x, 0< x<
(x+ ) = (x).
Then the full-range Fourier series representation off(x) is
2
n=1
1
nsin 2nx,
which consists of both odd functions (the sine terms) and evenfunction (the constant term) since the extension (x) is neither aneven nor an odd function.
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4.3 FUNCTIONS DEFINED OVER A FINITE INTERVAL4 3 2 HALF RANGE COSINE AND SINE SERIES
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4.3.2 HALF-RANGE COSINE AND SINE SERIES
The even extension off is the functionF(x) =
|x|,
x
.Thus, the half-range cosine series expansion for f(x) is
2 4
n=11
(2n 1)2 cos(2n 1)x.
The odd extension off is G(x) =x,
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4.3.2 HALF-RANGE COSINE AND SINE SERIES
The preceding three extensions are natural extensions off. Thereare many other ways of extending f(x). For example, the function
H(x) =
0, x
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4.4.1 DIFFERENTIATION OF A FOURIER SERIES
The term-by-term differentiation of a Fourier series is not alwayspermissible. The following theorem gives sufficient conditions for
using termwise differentiation.
Theorem(Differentiation of Fourier Series)
Let f be a continuous function on(,) and periodic of period2l. Let f(x) and f(x) be piecewise continuous on [
l, l]. Then,
the Fourier series of f can be obtained from the Fourier series forf by termwise differentiation. In particular, if
f(x) =a0
2 +
n=1
ancosnx
l +bnsin
nx
l ,then
f(x) n=1
n
l
ansinnx
l +bncos
nx
l
.
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4.4 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES4 4 1 DIFFERENTIATION OF A FOURIER SERIES
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4.4.1 DIFFERENTIATION OF A FOURIER SERIES
Example 4.1 Consider the process of differentiating term byterm the Fourier series expansion of the function
f(t) =t2, t f(t+ 2) =f(t).
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4.4 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES4.4.2 INTEGRATION OF FOURIER SERIES
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4.4.2 INTEGRATION OF FOURIER SERIES
Theorem(Integration of Fourier Series)
Let f be piecewise continuous on [l, l] with Fourier series
f(x)
a0
2
+
n=1
ancosnxl
+bnsinnx
l,
then, for any x in [l, l], we have
x
l
f(t)dt= x
l
a0
2dt+
n=1
x
lancosnt
l +bnsin
nt
ldt.
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.4 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES4.4.2 INTEGRATION OF FOURIER SERIES
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Example 4.2 The Fourier series expansion of the function
f(t) =t2
( t ), f(t+ 2) =f(t)is
t2 =2
3 + 4
n=1(1)n cos nt
n2 , t .
Integrating this result between the limits and x gives x
t2dt=
x
2
3dt+ 4
n=1
x
(1)n cos ntn2
dt
that is,
1
3x3 =
2
3x+ 4
n=1
(1)n sin nxn3
, x .
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.4 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES4.4.2 INTEGRATION OF FOURIER SERIES
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Rearranging, we have
x3 2x= 12n=1
(1)n sin nxn3
, x .
and the right-hand side may be taken to be the Fourier seriesexpansion of the function
g(x) =x3 2x, x
g(x+ 2) =g(x).
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.4 DIFFERENTIATION AND INTEGRATION OF FOURIER SERIES4.4.2 INTEGRATION OF FOURIER SERIES
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Example 4.3 Use the Fourier series
x2 =2
3 + 4
n=1(1)n
n2 cos nx, x
to deduce the value of the series
k=1(1)k+1
(2k 1)3 .
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.5 ENGINEERING APPLICATIONS
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We know that forced oscillations of a body of mass m on a springof modulus kare governed by the differential equation
my
+cy
+ky=r(t) (8)
where y(t) is the displacement from rest, cthe damping constant,kthe spring constant (spring modulus), and r(t) the external forcedepending on time t.
In (8), letm= 1 (gm), c= 0.05 (gm/sec), and k= 25 (gm/sec2),so that (8) becomes
y + 0.05y + 25y=r(t)
where r(t) is measured in gmcm/sec2. Let
r(t) =
t+ 2 , if
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Consider a Fourier series
f(x) = a02 +
n=1
ancos nx+bnsin nx
.
Let us replace cos nxand sin nxby their expressions
cos nx= e
jnx
+e
jnx
2 , sin nx= e
jnx
ejnx
2j .
Then
f(x) =
a0
2 +
n=1
an
ejnx +ejnx
2 +bnejnx
ejnx
2j
= a0
2 +
1
2
n=1
ejnx(an jbn) +ejnx(an+jbn)
. (9)
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.6 COMPLEX FOURIER SERIES4.6.1 COMPLEX REPRESENTATION
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Denote cn = (an jbn)/2 and use Eulers formulas for an and bnto obtain:
cn = an jbn
2 = 1
2
f(x)cos nxdxj 1
2
f(x)sin nxdx
= 1
2
f(x)
cos nxjsin nx
dx.
Thus,cn =
1
2
f(x)ejnxdx.
Let
cn = cn =an+jbn
2=
1
2
f(x)
cos nx+jsin nx
dx
= 1
2
f(x)ejnxdx.
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.6 COMPLEX FOURIER SERIES4.6.1 COMPLEX REPRESENTATION
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We now write down series (9) in the form
f(x) =c0+n=1
cnejnx +
n=1
cnejnx
or, briefly,
f(x) =
n=
cnejnx where cn =
1
2
f(x)ejnxdx (10)
(10) is the so-called complex form of the Fourier series, morebriefly, the complex Fourier series, off(x).
Thecn are called the complex coefficients off(x).
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.6 COMPLEX FOURIER SERIES4.6.1 COMPLEX REPRESENTATION
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The sum
cnejnx +cne
jnx =cnejnx +cnejnx = 2Recnejnx
=Re
(an jbn)(cos nx+jsin nx)
=ancos nx+bnsin nx.
is a real function representing the nth harmonic. It can be
rewritten as
ancos nx+bnsin nx=Ancos(nxn)where An =
a2n+b
2n = 2|cn| is the amplitude; since tan n = bnan ,
the initial phase is expressed as n =Argcn =
Arg cn.
The sequence of complex numbers cn is called the spectralsequenceof the function f(x); the real sequence 2|cn| is termedthe amplitude spectrumoff(x) and the sequence n = Arg cnthe phase spectrum.
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.6 COMPLEX FOURIER SERIES4.6.1 COMPLEX REPRESENTATION
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In the general case we deal with a function f(x) defined in aninterval [
l, l] for which the Fourier trigonometric expansion is
constructed. In this case Fourier series is written in complex formas
f(x) =
n=
cnejnx where cn =
1
2l l
l
f(x)ejnxdx and n =n
l
Iff(x) is a periodic function of period T its complex form of theFourier series expansion is
f(x) =
n=
cnejnx where cn =
1
T
+T
f(x)ejnxdx and n =2n
T
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.6 COMPLEX FOURIER SERIES4.6.1 COMPLEX REPRESENTATION
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Example 6.1 Find the Fourier series of the functionf(x) =ex
on the interval [, ].
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.6 COMPLEX FOURIER SERIES4.6.2 THE MULTIPLICATION THEOREM AND PARSEVALS THEOREM
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Theorem
Let f and g be piecewise continuous functions on[l, l]. If an, bnare the Fourier series coefficients of f(x) andn, n those of g(x),
then
1
2l
ll
f(x)g(x)dx=1
40a0+
1
2
n=1
(nan+ nbn)
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.6 COMPLEX FOURIER SERIES4.6.2 THE MULTIPLICATION THEOREM AND PARSEVALS THEOREM
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In terms of complex coefficients the multiplication theorem can bestated as follows.
If f and g are piecewise continuous functions on[l, l], then
12l
l
l
f(x)g(x)dx=
n=
cndn
where the cn and dn are the coefficients in the complex
Fourier series expansion of f and g respectively.
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.6 COMPLEX FOURIER SERIES4.6.2 THE MULTIPLICATION THEOREM AND PARSEVALS THEOREM
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If we put g(x) =f(x) in the above result, the following importanttheorem immediately follows.
Theorem(Parsevals theorem)
If f is piecewise continuous on[l, l] and has Fourier coefficientsan, bn, then
1
2l
ll
[f(x)]2dx=1
4a20+
1
2
n=1
(a2n+b2n) (11)
If cn are the coefficients in the complex Fourier series expansion off then
1
2l
ll
[f(x)]2dx=
n=
|cn|2
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.6 COMPLEX FOURIER SERIES4.6.2 THE MULTIPLICATION THEOREM AND PARSEVALS THEOREM
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Parseval equality (11) implies that the series
a202
+n=1
a2n+b
2n
is convergent and, consequently, we have
limn
an =1
l limn
ll
f(x)cosnx
l dx= 0
and
limn
bn =1l
limn
l
l
f(x)sinnxl
dx= 0.
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.6 COMPLEX FOURIER SERIES4.6.2 THE MULTIPLICATION THEOREM AND PARSEVALS THEOREM
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Example 6.2 Given the Fourier series
t2 =2
3 + 4
n=1
(1)nn2
cos nt, t
deduce the value ofn=1
1
n4.
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.1 ORTHOGONAL SYSTEM OF FUNCTIONS
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Definition
Two real functions f(x) and g(x) that are piecewise continuous in
the interval [a, b] are said to be orthogonal in this interval if
ba
f(x)g(x)dx= 0.
A set of functions
1(x), 2(x), . . . , n(x), . . .
each of which is piecewise continuous on [a, b], is said to be an
orthogonal set on this interval if ba
m(x)n(x)dx=
0 if m =n>0 if m=n.
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.1 ORTHOGONAL SYSTEM OF FUNCTIONS
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Definition
An orthogonal set{
n(x)}
is said to be orthonormal if
ba
n(x)
2dx= 1 for all n= 1, 2, . . .
Example 7.1 (a) The set
1
2, cos
x
l , sin
x
l , cos
2x
l , sin
2x
l , . . . , cos
nx
l , sin
nx
l , . . .
is orthogonal on [l, l]. The set12l
, 1
lcos
x
l ,
1l
sinx
l ,
1l
cos2x
l ,
1l
sin2x
l , . . . ,
1l
cosnx
l ,
forms an orthonormal set on the same interval.Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.1 ORTHOGONAL SYSTEM OF FUNCTIONS
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(b) The system ofLegendre polynomials
P0(x) = 1, Pn(x) = 1
2nn!
dn
dxn
(x2 1)n, n= 1, 2, . . .is orthogonal on [1, 1], while the set
12
P0(x),
2n+ 1
2 Pn(x), n= 1, 2, . . .
is an orthonormal set on [1, 1].
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.1 ORTHOGONAL SYSTEM OF FUNCTIONS
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If a function (x) is piecewise continuous on [a, b], thenonnegative number
=
b
a
2(x)dx
1/2
is called the normof(x) on [a, b].
For example, on [l, l],
1
2
=
l
2,cosnx
l
= l, cosnxl
= l, n= 1, 2, . . .The norms of Legendres polynomials on the interval [
1, 1] are
Pn(x) = 1
1P2n(x)dx
1/2=
2
2n+ 1, n= 0, 1, 2, . . . .
We note that any orthogonal set{n(x)} can be converted intoan orthonormal set by diving each member n(x)ofitsnorm.
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.1 ORTHOGONAL SYSTEM OF FUNCTIONS
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Examples of other sets of orthogonal functions that are widelyused in practice are Bessel functions, Hermite polynomials,Laguerre polynomials, Jacobi polynomials, Chebyshev polynomialsand Walsh functions.
Over recent years wavelets are another set of orthogonalfunctions that have been widely used, particularly in applicationssuch as processing and data compression.
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.2 GENERALIZED FOURIER SERIES
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Definition
Iff(x) is a piecewise continuous function on the interval [a, b] and{n(x)} is an orthogonal set on this interval then the numbers
cn = 1
n2 ba
f(x)n(x)dx, n= 1, 2, . . .
are called the generalized Fourier coefficients of the functionf(x) with respect to the basic set{n(x)}. The series
n=1 cnn(x)
is called the generalized Fourier series off(x) with respect tothe basic set{n(x)}.
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.2 GENERALIZED FOURIER SERIES
L f ( ) b i i i [ b] A f i f h f
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Let f(x) be piecewise continuous on [a, b]. A function of the form
F(x) =N
k=1
nk(x),
where 1, 2,...,n are constants, can be considered anapproximation of the given function f(x). F(x) is called a
polynomial of order Nwith respect to orthogonal system{n(x)}.
Question: Is the partial sum
SN(x) =N
n=1
cnk(x)
of the generalized Fourier series the bestapproximationoff?.Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.2 GENERALIZED FOURIER SERIES
W d fi th E b t th t l l f
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We define the mean square error ENbetween the actual value off(x) and the approximation F(x) as
EN= 1b a
b
a
f(x) F(x)2dx.
Clearly, EN 0.
Theorem(Minimum Square Error)The square error of F(x) =
Nn=1 nn(x) (with fixed N) relative
to f on[a, b] is minimum if and only if the coefficients of F are thegeneralized Fourier coefficients of f . This minimum value is givenby
EN=
ba
f2dxN
n=1
c2nn2 (12)
where cn are the generalized Fourier coefficients of f .
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.2 GENERALIZED FOURIER SERIES
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The polynomialSN(x) =
Nn=1
cnk(x)
whose coefficients cn are the generalized Fourier coefficients off
with respect to the given orthogonal system{n(x)} is called theFourier polynomial of the function fwith respect to theorthogonal system{n(x)}.
Thus, the mean square error of polynomial F(x) relative to f isminimum if and only if F is the Fourier polynomial of f .
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.2 GENERALIZED FOURIER SERIES
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Since EN 0 and (12) holds for every N, we obtain Besselsinequality
n=1
c2nn2 ba
f2(x)dx (13)
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
4.7 ORTHOGONAL FUNCTIONS4.7.2 GENERALIZED FOURIER SERIES
O i h i i i i h h E 0
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One question that arises in practice is whether or not EN 0 asN
. If this were the case then
n=1
c2nn2 = ba
f2(x)dx (14)
which is the generalized form of Paservals equality, and the set{n(x)} is said to be complete.
In the case of the trigonometric system equality (14) turns into
a202
+n=1
a2n+b
2n
=
1
l
ll
f2(x)dx
Assoc. Prof. N. Dinh & Assoc. Prof. G. Vallet & Dr. N. N. Hai CALCULUS 3 Chapter 4 FOURIER SERIES
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