lec_fourierseries
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Periodic FunctionsEven and Odd Functions
Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Fourier Series
Dr. Yosza Dasril
Universiti Teknikal Malaysia Melaka (UTeM), Hang Tuah Jaya 76100, Melaka
Dr. Yosza Dasril Fourier Series
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Periodic FunctionsEven and Odd Functions
Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Outline
1 Periodic Functions
2 Even and Odd Functions
3 Fourier SeriesTrigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
4 Fourier Series for Even and Odd FunctionsFourier Cosine Series
Fourier Sine Series5 Half-Range Expansions: Fourier Cosine & Fourier Sine Series
Half-Range Expansions in Fourier Cosine SeriesExamples
6
Exercises 2.1Dr. Yosza Dasril Fourier Series
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Periodic FunctionsEven and Odd Functions
Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Periodic Functions
A function f(x) is said to be periodic if f(x + T) = f(x) x and for some positive number T.
Definition
A function f(x) is periodic with period T for all domain values x if
f(x + T) = f(x), forT > 0 (1)
The smallest value of T is called the fundamental period of f. Aconstant function is a periodic function with an arbitrary periodbut no fundamental period.
Dr. Yosza Dasril Fourier Series
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Periodic FunctionsEven and Odd Functions
Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Periodic Functions
Example
1. The graph of sine could be present as
Figure: 1.1 sine graph
Solution The periods of sin(x) are 2, 4, 6,... wheresin(x + 2) = sin(x + 4) = sin(x + 6) = sin(x).
So, the fundamental period of sin(x) is 2.Dr. Yosza Dasril Fourier Series
P i di F i
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Periodic FunctionsEven and Odd Functions
Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Periodic Functions
Notes 1. If T is the period of f(x) then nT is also period of f forany integer n.
Notes 2. Function h(x) = af(x) + bg(x) has period T if f(x) andg(x) have period T. Here a and b are constants.
Example
2. if h(x) = a cos x + bsin x, then
h(x + 2) = a cos(x + 2) + bsin(x + 2)
= a cos x + bsin x = h(x)
Dr. Yosza Dasril Fourier Series
P i di F ti
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Periodic FunctionsEven and Odd Functions
Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Periodic Functions
Notes 3. If f(x) is a periodic function of period T, then f(ax)with a = 0, is a periodic function of period T
a.
Example3. sin(2x) has period 22 = . cos(3x) has period
23 , and so on.
Exercise 1.1
For the given functions, determine whether it is a periodic function
or not.
a) f(x) = cos(x)
b) g(x) = x2
Dr. Yosza Dasril Fourier Series
Periodic Functions
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Periodic FunctionsEven and Odd Functions
Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Periodic Functions
Notes 4. The period of the sum of a number of periodic functionsis the least common multiple of the periods.
Example
4. f(x) = sin x+ 12 sin(2x) +13 sin(3x) +
14 sin(4x). Note that sin x,
sin2x, sin 3x, sin 4x have periods 2, , 22 and
2 respectively.Then the period of f(x) is 2 which is the L.C.M. of these periods.
Exercise 1.1
For the given functions, determine whether it is a periodic functionor not.
a) f(x) = cos(x)
b) g(x) = x2
Dr. Yosza Dasril Fourier Series
Periodic Functions
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Periodic FunctionsEven and Odd Functions
Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Outline
1 Periodic Functions
2 Even and Odd Functions
3 Fourier SeriesTrigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
4 Fourier Series for Even and Odd FunctionsFourier Cosine Series
Fourier Sine Series5 Half-Range Expansions: Fourier Cosine & Fourier Sine Series
Half-Range Expansions in Fourier Cosine SeriesExamples
6 Exercises 2.1
Dr. Yosza Dasril Fourier Series
Periodic Functions
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Periodic FunctionsEven and Odd Functions
Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Even and Odd Functions
Definition
A function f(x) is said to be even if
f(x) = f(x). (2)The graph of an even function is symmetric about the y-axis.
Figure: 1.2 Even function
Dr. Yosza Dasril Fourier Series
Periodic Functions
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Periodic FunctionsEven and Odd Functions
Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Even and Odd Functions
Definition
A function f(x) is said to be odd if
f(x) = f(x). (3)
The graph of an odd function is symmetric about the origin.
Figure: 1.3 Odd function
Dr. Yosza Dasril Fourier Series
Periodic Functions
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Even and Odd Functions
Properties of Even and Odd Functions
a) The product of two even functions is even
b) The product of two odd functions is even
c) The product of an even function and an odd functionis odd
d) The sum (difference) of two even functions is even
e) The sum (difference) of two odd functions is odd
f) If f(x) is even function, thenL
Lf(x)dx = 2
L
0 f(x)dx
g) If f(x) is odd function, then
L
Lf(x)dx = 0
Dr. Yosza Dasril Fourier Series
Periodic Functions
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Even and Odd Functions
Exercise 1.2
Determine whether the function is even, odd, or neither
a) f(x) = x4 + x2
b) g(x) = x4 + x
c) f(x) = cos(x), < x <
d) f(x) = ex + ex
e) f(x) = sin(x), < x <
f) g(x) = 2x5 + x
g) f(x) = 2x5 + x2
h) g(x) = tan(x)
Dr. Yosza Dasril Fourier Series
Periodic Functions
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Outline
1 Periodic Functions
2 Even and Odd Functions
3 Fourier SeriesTrigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
4 Fourier Series for Even and Odd FunctionsFourier Cosine Series
Fourier Sine Series5 Half-Range Expansions: Fourier Cosine & Fourier Sine Series
Half-Range Expansions in Fourier Cosine SeriesExamples
6 Exercises 2.1
Dr. Yosza Dasril Fourier Series
Periodic FunctionsO
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Fourier Series
A Fourier series decomposes any periodic function or periodicsignal into the sum of a (possibly infinite) set of simple oscillating
functions, namely sines and cosines (or complex exponentials).The study of Fourier series is a branch of Fourier analysis. Fourierseries were introduced by Joseph Fourier (1768 1830) for thepurpose of solving the heat equation in a metal plate.
Fourier Theorem: A periodic function that satisfies certainconditions can be express as the sum of a number of sine functionsof different amplitudes, phases & periods.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsE d Odd F i
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Fourier Series
That is, if f(x) is a periodic function with period T, then
f(x) = A0 + A1 sin(x + 1) + A2 sin(2x + 2)
+ + An sin(nx + n) + (4)where the As & s are constant and = 2
Tis the frequency of
f(x).
A1 sin(x + 1) is called the fundamental mode, their
frequency is = original function f(x).An sin(nx + n) is the nth harmonic, their frequency (n)
An denotes the amplitude of the nth harmonic
n is its phase angle, measuring the lead of the nth harmonicwith reference to a pure sine wave of the same frequency.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsE d Odd F ti s
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Fourier Series
Since
An sin(nx + n) (An cos n)sin(nx) + (An sin(n)) cos(nx)
bn sin(nx) + an cos(nx)
where bn = An cos(n) and an = An sin(n).The expansion of (4) may be written as
f(x) = 12
a0 +
n=1
an cos(nx) +
n=1
bn sin(nx) (5)
where a0 = 2A0. Equation (5) is called Fourier series expansionand an & bn is called Fourier coefficients.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Fourier Series
In electrical engineering an and bn is called respectively as thein-phase and phase quadrature components of the nth harmonic.This terminology arising from the use of the phasor notation
einx = cos(nx) + isin(nx)
This is an alternative form of the Fourier series (4) with amplitude
and phase are An =
a2n + b2n, n = tan
anbn
.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Trigonometric Series
If the period T of the periodic function f(x) is taken to be 2then = 1, and the series (5) becomes.
f(x) =
1
2 a0 +
n=1
an cos(nx) +
n=1
bn sin(nx) (6)
where the constants an and bn are called the coefficients.
Notes 5. Let n and m be integers, n = 0, m = 0, for m = n.
a)+2
cos mx cos nxdx = 0.
b)+2
sin mx sin nxdx = 0.
c) +2
sin mx cos nxdx = 0.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
T i i S i
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Trigonometric Series
d) +2
cos mxdx = 0.
e)
+2
sin mxdx = 0.
For m = n
a)+2
cos mx cos nxdx =+2
cos2 mxdx = .
b) +2 sin2 mxdx = .c)+2
cos mx sin mxdx = 0.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
T i t i S i
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Eulers (Fourier-Euler) Formulae
Let f(x), a periodic function with period 2 defined in the interval
(, + 2), be the sum of a trigonometric series i.e.,
f(x) =1
2a0 +
n=1
an cos(nx) + bn sin(nx)
(7)
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
Trigonometric Series
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Eulers (Fourier-Euler) Formulae
To determine the coefficient a0, integrate (7) w.r.t x from to + 2. Then
+2
f(x)dx =
+2
a02
dx +
+2
n=1
an cos nx + bn sin nx
=a0
2
x+2
+
n=1
an +2
cos nx dx
+bn
+2
sin nx dx
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
Trigonometric Series
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Eulers (Fourier-Euler) Formulae
Form the Notes 5, the last two integrals for all n will be zero. Thus
+2
f(x) dx =a02 2 = a0.
Hence
a0 =
1
+2 f(x) dx (8)
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
Trigonometric Series
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Eulers (Fourier-Euler) Formulae
To determine coefficient an, for n = 1, 2, .... Multiplying both sidesof (7) by cos mx and integrating w.r.t x in [, + 2], we get
+2
f(x)cos mxdx =+2
ao2 cos mx dx++2
n=1
an cos nxcos mx
dx +
+2
n=1
bn sin nxcos mx
dx
For (m = n), all integrals vanish except for an.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
F i S iTrigonometric Series
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2
Eulers (Fourier-Euler) Formulae
Thus
an =1
+2
f(x)cos nx dx, for n = 1, 2, ... (9)
Similarly,
bn =1
+2
f(x)sin nx dx, for n = 1, 2,... (10)
These are knows as Euler formulae, and a0, an and bn are knownas Fourier coefficients of f(x).
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
F i S iTrigonometric Series
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
gEulers (Fourier-Euler) FormulaeFunctions of Period 2
Functions of Period 2
Dirichlet Conditions
Let f(x) be a periodic function with period 2. Let f(x) piecewisecontinuous, and bounded in the interval (, + 2) with finite
number of critical points. Then at the points of continuity, theFourier series of f(x) in eq. (7) converges to f(x) (LHS of (7)).At the pints of discontinuity, x0, the Fourier series of f(x)
converges to the arithmetic mean of the left and right hand limits
of f(x) at x0 i.e.
f(x0) =12 (f(x0) 0) + f(x0) + 0).
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
Fourier SeriesTrigonometric Series
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
gEulers (Fourier-Euler) FormulaeFunctions of Period 2
Functions of Period 2
Notes 6: Leibnitzs rule:
u v dx = uv1 uv2 + uv3 uv4 + ..., where denotesdifferentiation and suffix integration w.r.t x.
cos n = (1)n
sin n = 0
cos(2n + 1)2 = 0 and
sin(2n + 1)2 = (1)n.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
Fourier SeriesTrigonometric Series
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Eulers (Fourier-Euler) FormulaeFunctions of Period 2
Functions of Period 2
Example
2. Find the Fourier series of
f(x) = 0, x 0
x2, 0 x
Which is assumed to be periodic with period 2.
Solution The Fourier series is given by (7). Here
a0 = 1
f(x)dx =
0
0dx +
0x2dx
=1
x3
3
0=
3
3=
2
3
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
Fourier SeriesTrigonometric Series
( )
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Eulers (Fourier-Euler) FormulaeFunctions of Period 2
Functions of Period 2
an =1
f(x)cos nx dx
=
0
0cos nx dx+
0x2 cos nx dx
=1
2
n2
cos n
=2
n2(1)n, for n = 1, 2,...
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
Fourier SeriesTrigonometric SeriesE l (F i E l ) F l
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Eulers (Fourier-Euler) FormulaeFunctions of Period 2
Functions of Period 2
bn =1
f(x)sin nx dx =1
0x2 sin nx dx
=
0
0cos nx dx+
0x2 cos nx dx
=1
2
ncos n +
2
n3cos(n 1)
=
n(1)n + 2
n3
(1)n 1
, for n = 1, 2,...
=
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
Fourier SeriesTrigonometric SeriesE l (F i E l ) F l
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Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Eulers (Fourier-Euler) FormulaeFunctions of Period 2
Functions of Period 2
Substituting a0, an and bn, we get
f(x) =2
6+ 2
n=1
2
n2(1)n cos nx +
n=1
n(1)n +
2
n3(1)n 1 sin nx.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
Fourier SeriesTrigonometric SeriesEulers (Fourier Euler) Formulae
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Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Euler s (Fourier-Euler) FormulaeFunctions of Period 2
Functions of Period 2
Example
3. Obtain the Fourier series expansion of the periodic functionf(x) of period 2 defined by
f(x) = x, (0 < x < 2), f(x) = f(x + 2)
Solution: The graph of f(x) over interval 4 < x < 4 isshown in Fig.1.3
Figure: 1.4 Graph of f(x) = x over (4, 4)Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd Functions
Fourier SeriesTrigonometric SeriesEulers (Fourier Euler) Formulae
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Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Euler s (Fourier-Euler) FormulaeFunctions of Period 2
Functions of Period 2
a0 =1
0+20
xdx = 2
an =1
20
xcos(nx)dx
=1
xsin(nx)
n
+cos(nx)
n2
2
0
=1
2
nsin(2nx) +
1
n2cos(2nx)
cos(0)
n2
= 0.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd FunctionsFourier Series
F i S i f E d Odd F i
Trigonometric SeriesEulers (Fourier-Euler) Formulae
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Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Euler s (Fourier Euler) FormulaeFunctions of Period 2
Functions of Period 2
Example
4. Find the Fourier series for the given function
f(x) =
1, < x < 0x, 0 < x <
Solution
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd FunctionsFourier Series
F i S i f E d Odd F ti
Trigonometric SeriesEulers (Fourier-Euler) Formulae
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Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Euler s (Fourier Euler) FormulaeFunctions of Period 2
Functions of Period 2
Example
5. Given g(x) =
0, 5 < x < 03, 0 < x < 5
1) Determine the Fourier coefficients
2) Find the Fourier series for the given function.
Solution
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd FunctionsFourier Series
Fo rier Series for E en and Odd F nctionsFourier Cosine SeriesFo rier Sine Series
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Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Fourier Sine Series
Outline
1 Periodic Functions
2 Even and Odd Functions
3 Fourier SeriesTrigonometric Series
Eulers (Fourier-Euler) FormulaeFunctions of Period 2
4 Fourier Series for Even and Odd FunctionsFourier Cosine Series
Fourier Sine Series5 Half-Range Expansions: Fourier Cosine & Fourier Sine Series
Half-Range Expansions in Fourier Cosine SeriesExamples
6 Exercises 2.1
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsFourier Cosine SeriesFourier Sine Series
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Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Fourier Sine Series
Fourier Series for Even and Odd Functions
Now, we consider f(x) be a periodic function with period arbitraryperiod 2L defined in the interval c < x < c + 2L. Introduce a newvariable z as
x
2L=
z
2or x =
Lz
or z =
x
L. (11)
At x = c z = 2cL
= d say.At x = c + 2L z =
L(c + 2L) = c
L+ 2 = d + 2.
Thus as c < x < c + 2L, the new variable z lies in the intervald < z < d + 2.So z varies in the interval (d, d + 2L) of length 2.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsFourier Cosine SeriesFourier Sine Series
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Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Fourier Sine Series
Fourier Series for Even and Odd Functions
Substituting for x from (11), we get
f(x) = f
Lz
= F(z) (12)
Let the Fourier series of F(z) defined in the interval (d, d + 2)and with period 2 be
F(z) =1
2a0 +
n=1
an cos(nz) + bn sin(nz) (13)where
a0 =1
d+2L
d
F(z)dz.
Dr. Yosza Dasril Fourier Series
Periodic FunctionsEven and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsFourier Cosine SeriesFourier Sine Series
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Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
Fourier Sine Series
Fourier Series for Even and Odd Functions
Changing the variable to x
a0 =1
c+2L
c
f(x)
Ldx
.
Since dz =
L dx. Thus
a0 =1
L
c+2L
c
f(x) dx. (14)
Similarly,
an =1
d+2L
d
F(z) cos(nz)dz
=1
c+2L
c
f(x)cos
nx
L
Ldx
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ou e Se es o e a d Odd u ct o sHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
ou e S e Se es
Fourier Series for Even and Odd Functions
So,
an =1
L
c+2L
c
f(x)cosnx
L
dx (15)
In similar way,
bn =1L
c+2L
c
f(x)sin
nxL
dx (16)
Hence, the Fourier series expansion for a function f(x) with period2L is
f(x) = F(z)
=a02
+n=0
an cos
nxL
+ bn sin
nxL
with coefficients given by (14), (15) and (16).
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Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Fourier Cosine Series
For an even function f(x), in (L, L), we only need to determinethe values of a0 and an, (bn = 0). Then the Fouriers seriesbecomes
f(x) =
a0
2 +
n=1
an cosnx
L
(17)
where
a0 =2
L L
0f(x)dx (18)
an =2
L
L
0f(x)cos
nxL
dx (19)
Equation (17) is called the Fourier Cosine Series.
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Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Fourier Cosine Series
Example
6. Find the Fourier Cosine Series for the given function
f(x) =
1, 0 < x < 1x, 1 < x < 2
Solution
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Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Fourier Sine Series
For an odd function f(x), the values a0 = an = 0. We need only todetermine the value of bn.
f(x) =n=1
bn sinnx
L
(20)
where
bn =
2
LL
0 f(x)sinnx
L
dx (21)
Equation (20) is called Fourier Sine Series.
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Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Fourier Sine Series
Example
5. Find the Fourier Sine Series for the given function
g(x) =
1 x, 0 < x < 10, 1 < x < 2
Solution
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Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Fourier Series for Even and Odd Functions
Procedure
1) Identify whether the given function f is even or odd
function in the given interval.2) If f is even, calculate only a0 and ans from (12) and
(13). It doesnt need to calculate bns. The Fourierseries is given by (11).
3) If f is odd, calculate only bns from (15). TheFourier sine series is given by (14).
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Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Outline
1 Periodic Functions
2 Even and Odd Functions
3 Fourier SeriesTrigonometric Series
Eulers (Fourier-Euler) FormulaeFunctions of Period 2
4 Fourier Series for Even and Odd FunctionsFourier Cosine SeriesFourier Sine Series
5 Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesHalf-Range Expansions in Fourier Cosine SeriesExamples
6 Exercises 2.1
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Half-Range Expansions in Fourier Cosine SeriesExamples
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Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Half-Range Expansions: Fourier Cosine & Fourier Sine
Series
So far we have considered the Fourier expansion of a functionwhich is periodic with periods 2 and 2L, defined in an interval c
to c + 2L of length 2L. This time we consider the procedure toexpand a non-periodic function f(x) defined in half of the aboveinterval, say (0, L) of length L. Such expansion are known as halfrange expansion or half range Fourier series.In particular, a half range expansion constraining only cosine termsis known as half range Fourier cosine series of f(x) in interval(0, L). In similar way half range Fourier sine expansion containsonly sine terms.
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Half-Range Expansions in Fourier Cosine SeriesExamples
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Half Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Half-Range Expansions in Fourier Cosine Series
Note that the given function f(x) is neither periodic nor even norodd. In order to obtain a Fourier cosine series for f(x) in theinterval (0, L), we construct a new function g(x) such that:
i) g(x) f(x) in the interval (0, L), see Fig. 2.1 below.ii) g(x) is even function in (L, L) and is periodic with
period 2L. Fig. 2.1.
Here
g(x) = f(x), in (0, L)
= f(x), in (L, 0)
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Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Half-Range Expansions in Fourier Cosine SeriesExamples
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Half Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
Half-Range Expansions in Fourier Cosine Series
Figure: 2.1 Graph of f(x)
Figure: 2.2 g(x) is an Even Periodic Continuation of f(x)
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine Series
Half-Range Expansions in Fourier Cosine SeriesExamples
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g pExercises 2.1
Half-Range Expansions in Fourier Cosine Series
Function g(x) is called as Even Periodic Continuation (Expansion)of f(x). The Fourier cosine series for g(x) valid in (L, L) or infact x is readily obtained as
g(x) =a
02 +
n=1
an cosnx
L
(22)
where
a0 =2
L
L
0
g(x)dx =2
L
L
0
f(x)dx (23)
an =2
L
L
0g(x)cos
nxL
dx =
2
L
L
0f(x)cos
nxL
dx (24)
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine Series
Half-Range Expansions in Fourier Cosine SeriesExamples
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g pExercises 2.1
Half-Range Expansions in Fourier Sine Series
Important Note: The series expansion of f(x) given by (22) isvalid for f(x) only in the interval (0, L) but not outside this
interval.A similar way, to obtain the half range Fourier sine series for f(x)in (0, L), define h(x) such that:
i) h(x) f(x) in (0, L), see Fig. 2.3 and
ii) h(x) is an odd function in (
L,
L), periodic withperiod 2L. See Fig. 2.4.
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine Series
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Exercises 2.1
Half-Range Expansions in Fourier Sine Series
Figure: 2.3 Graph of f(x)
Figure: 2.4 h(x) is an Odd Periodic Continuation of f(x)
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Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesE i 2 1
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Exercises 2.1
Half-Range Expansions in Fourier Sine Series
The new function, h(x) as called as an odd periodic expansion off(x). The Fourier sine series and their coefficients are given by
h(x) =
n=1
bn sinnxL (25)
where
bn =2
L
L
0
h(x)sinnxLdx = 2
L
L
0
f(x)sinnxLdx (26)
As usual, this expansion is valid for f(x) only in the interval (0, L).As conclusion, a given non periodic function f(x) can be expandedin cosine or sine series.
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Exercises 2.1
Examples
Example
2.1. If f(x) = 1 xL
in 0 < x < L, find:
a) Fourier cosine series, andb) Fourier sine series of f(x). Sketch the corresponding
continuation of f(x).
Solution. a) Construct a new function g(x) such that
i) g(x) = f(x) in (0, L)
ii) g(x) is even periodic function in (L, L).
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Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2 1
Half-Range Expansions in Fourier Cosine SeriesExamples
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Exercises 2.1
Example
Define
g(x) = f(x) = 1 xL
in 0 < x < L
= 1 +x
Lin L < x < 0.
and g(x + 2L) = g(x)
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Exercises 2.1
Example
We get,
a0 =2
L
L
0
1
x
L
dx = 1.
an =2
L
L
0
1
x
L
cos
nxL
dx
=2
L1 x
L L
n
sinnx
L
1
L
L2n22
cos
nx
L
L0
=2
n
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Exercises 2.1
Example
Since
g(x) =a02
+n=1
an cosnx
L
We get
g(x) =1
2+
2
2]n=1
(1 (1)n)
n2cos
nxL
Thus the required Fourier cosine series of f(x) in (0,
L) is
f(x) = g(x) =1
2+
2
2]n=1
(1 (1)n)
n2cos
nxL
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Fourier SeriesFourier Series for Even and Odd Functions
Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1
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Exercises 2.1
Examples
b) Fourier sine series of f(x) in (0, L). Define new function h(x)such that
i) h(x) = f(x) in (0, L) and
ii) h(x) is odd periodic function. Define
h(x) = f(x) = 1x
Lin (0, L)
= 1 xL in (L, 0)
and h(x + 2L) = h(x). Thus h(x) is odd periodic function in(L, L) with period 2L.
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Examples
The Fourier sine series expansion of f(x) in the interval (L, L) isdetermined as follows.
bn =2
L
L
0
1
x
L
sinnx
L
dx
=2
L1
x
L(1)
cos
nx
L L
n
1
L
L2
n22sin nx
L
L
0= 2
n.
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Exercises 2.1
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Examples
So,
h(x) =2
n=1
1
n
sinnxL.
Thus the required Fourier sine series of f(x) in the interval (L, L)is
h(x) = f(x) =2
n=1
1
n
sinnxL.
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Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
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Outline
1 Periodic Functions2 Even and Odd Functions
3 Fourier SeriesTrigonometric Series
Eulers (Fourier-Euler) FormulaeFunctions of Period 2
4 Fourier Series for Even and Odd FunctionsFourier Cosine SeriesFourier Sine Series
5 Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesHalf-Range Expansions in Fourier Cosine SeriesExamples
6 Exercises 2.1
Dr. Yosza Dasril Fourier Series
Periodic Functions
Even and Odd FunctionsFourier Series
Fourier Series for Even and Odd FunctionsHalf-Range Expansions: Fourier Cosine & Fourier Sine Series
Exercises 2.1
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Exercises
1) Expand f(x) = x in (0, ) by (a) Fourier sine series.(b) Fourier Cosine series.
2) Find the two half range expansion of
f(x) =
2kxL
, if0 < x < L22k(Lx)
L, ifL2 < x < L
3) Obtain the Fourier cosine series of f(x) = sin x in the
interval 0 < x < .
Dr. Yosza Dasril Fourier Series
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