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CHAPTER 5 : FOURIER SERIES
Introduction Periodic Functions Fourier Series of a function Dirichlet Conditions Odd and Even Functions Relationship Between Even and Odd
functions to Fourier series
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5.1 : Introduction
Fourier Series ?
A Fourier series is a representation of a function as a series of constants times sine and/or cosine functions of different frequencies.
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5.2 : Periodic Functions
A function f(x) is said to be periodic if its function values repeat at regular intervals of the independent variables.
For the following example, a function f(x) has the period p.
p
x
y
x1 x1 + p x1 + 2p x1 + 3p
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In general, a function f(x) is called periodic if there is some positive number p such that ;
f(x) = f(x + np)
for any integer n.This number p is called a period of f(x).
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5.3 : Fourier Series of a Function
If f(x) is defined within the interval c < x < c+2L. The Fourier Series corresponding to f(x) is given by
where
1
0 sincos2 n
nn xL
nbx
L
na
axf
Lc
c
n
Lc
c
n
Lc
c
xdxL
nxf
Lb
xdxL
nxf
La
dxxfL
a
2
2
2
0
sin1
cos1
1
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5.4 : Dirichlet Conditions
If a function f(x) defined within the interval c < x < c+2L, the following conditions must be satisfied;
1. f(x) is defined and single-valued.
2. f(x) is continuous or finite discontinuity in the corresponding periodic interval.
3. f(x) and f ’(x) are piecewise continuous .
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Example 5.1:
Determine whether the Dirichlet conditions are satisfied in the following cases :
i.Yes.
ii.No, because there is infinite discontinuity
at
xx
xf ; 1
xxxf ; 2
.0x
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Example 5.2:
1. Find the Fourier series for the function defined within the interval
Answer :
2
22
2
,0
,4
,0
x
x
x
xf
xxxxxxf 9cos
9
17cos
7
15cos
5
13cos
3
1cos
82
. x
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Solution
The Fourier series is
where
1
0 sincos2 n
nn nxbnxaa
xf
.sin)(1
cos)(1
)(1
0
dxnxxfb
dxnxxfa
dxxfa
n
n
. Therefore, .2 and get We LLcc
12
.2
sin8
sin14
cos41
,cos)(1
2
2
2
2
n
n
nxn
dxnx
dxnxxfan
.0
cos14
sin41
sin)(1
2
2
2
2
nxn
dxnx
dxnxxfbn
13
.9cos9
17cos
7
15cos
5
13cos
3
1cos
82
9cos9
87cos
7
85cos
5
83cos
3
8cos
82
cos2
sin8
2
4
)sincos(2
is seriesFourier theTherefore,
1
1
0
xxxxx
xxxxx
nxn
n
nxbnxaa
xf
n
nnn
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Graph of a Function
2
22
2
,0
,4
,0
x
x
x
xf
What will we obtain as more terms are included in the series ?
2
y
4
x- 0
2
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Graph of a Function
xxxxxf 7cos
7
15cos
5
13cos
3
1cos
82
4
x
O
The above figure show that the graph is merely to the shape …..
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The above function is defined within the interval
As the number of terms increases, the graph of Fourier series gradually approaches the shape of the original square waveform.
. x
x-5 -3 - 0 3 5
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Example 5.3:
Find the Fourier series for defined within the interval
Answer :
2xxf . x
12
2
cos1
43 n
n
nxn
xf
O
f(x)
x
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Solution
The Fourier series is
where
1
0 sincos2 n
nn nxbnxaa
xf
.sin)(1
cos)(1
)(1
0
dxnxxfb
dxnxxfa
dxxfa
n
n
. Therefore, .2 and get We LLcc
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.cos)1(
43
cos4
)1(3
2
2
1
)sincos(2
)(
is seriesFourier The
12
2
12
2
1
0
n
n
n
n
n
nn
nxn
nxn
nxbnxaa
xf
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Find the Fourier series for defined within the interval
Answer :
xxf .11 x
1
1
sin12
n
n
xnn
xf
x
f (x)
5 3 1 1 3 5
O
Example 5.4:
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Solution
The Fourier series is
.sincos2
sincos2
1
0
1
0
nnn
nnn
xnbxnaa
xL
nbx
L
na
axf
.1 Therefore, .12 and 1get We LLcc
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.0
.0
)cos(11
)cos(1
1
,cos)(1
1
1
1
1
2
n
Lc
c
n
a
xnnn
dxxnx
dxxL
nxf
La
.2
)1(
.cos2
)sin(1
1
sin)(1
1
1
1
2
nb
nn
dxxnx
dxxL
nxf
Lb
nn
Lc
c
n
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.sin)1(2
sin2
)1(
)sincos(2
is seriesFourier theTherefore,
1
1
1
1
1
0
n
n
n
n
n
nn
xnn
xnn
xnbxnaa
xf
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5.5 : Odd and Even Functions
A function f(x) is said to be even if :
f(-x) = f(x)
i.e the function value for a particular negative value of x is the same as that for the corresponding positive value of x.
A function f(x) is said to be odd if :
f(-x) = - f(x)
i.e the function value for a particular negative value of x is numerically equal to that for the corresponding positive value of x but opposite in sign.
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Example 5.5 :
Even function :
y = f(x) = x2 is an even function
because
f(-2) = 4 = f(2)
f(-6) = 36 = f(6)
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Properties of An Even function
The graph of an even function is symmetrical about the y-axis.
Hence, the areas under curves is twice the area from 0 to a :
aa
a
dxxfdxxf0
2
y
x a 0 a
y = x2
.0
0
a
a
dxxfdxxf
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Properties of An Odd function
The graph of an odd function is symmetrical about the origin.
Thus, the integral is 0 because the areas cancel.
0
a
a
dxxf
y
x 0 a
y = x3
a
.0
0
a
a
dxxfdxxf
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Moreover :If n is positive integer, thus
x2n is an even function,and x2n+1 is an odd function.
Function is an even function.
Function is an odd function.
x
L
cos
x
L
sin
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The product of two even functions is even.Example :
The product of two odd functions is even Example :
The product of an even function and an odd function is odd. Example :
.642 xxx
.43 xxx
.532 xxx
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5.6 : Relationship between Even and Odd functions to Fourier series
We could simply obtain the Fourier series for a function, f(x) defined within the interval –a<x<a. In this case, c = -a and c+2L = a, thus L = a, if we could identify whether f(x) is an even or odd function and use the properties of these functions in order to find the coefficients of Fourier series, a0, an and bn
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i. If f(x) is an even function,
and is an even function, thus
an even function.
and is an odd function, thus
an odd function.
xa
ncos
xa
nxf
cos)(
xa
nxf
sin)(
xa
nsin
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If f(x) is an even function
aa
a
n dxxa
nxf
adxx
a
nxf
aa
0
cos2
cos1
aa
a
dxxfa
dxxfa
a0
0
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.0sin1
a
a
n dxxa
nxf
ab
series. CosinesFourier called is
cos2
series,Fourier 1
0
nn x
a
na
axf
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ii. If f(x) is an odd function, thus :
and an even function, thus
is an odd function.
and an odd function, thus
is an even function
xa
ncos
xa
nxf
cos)(
xa
nxf
sin)(
xa
nsin