Lec_fourierSeries

8/2/2019 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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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 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.

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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 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)


P i di F ti


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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


Periodic Functions


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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


Periodic Functions
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Outline







6 Exercises 2.1


Periodic Functions
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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


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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


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


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


Periodic Functions
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Exercises 2.1


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)


Periodic Functions
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Exercises 2.1

Trigonometric SeriesEulers (Fourier-Euler) FormulaeFunctions of Period 2

Outline







6 Exercises 2.1


Periodic FunctionsO


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Exercises 2.1


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.


Periodic FunctionsE d Odd F i


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Exercises 2.1


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.


Periodic FunctionsE d Odd F ti s


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Exercises 2.1


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.




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Exercises 2.1


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

.




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Exercises 2.1


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.



T i i S i
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Exercises 2.1



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.



T i t i S i
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Exercises 2.1


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)



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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



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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)



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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.



F i S iTrigonometric Series
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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).



F i S iTrigonometric Series
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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).



Fourier SeriesTrigonometric Series
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gEulers (Fourier-Euler) FormulaeFunctions 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.



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Eulers (Fourier-Euler) FormulaeFunctions 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




( )
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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,...



Fourier SeriesTrigonometric SeriesE l (F i E l ) F l
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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,...

=



Fourier SeriesTrigonometric SeriesE l (F i E l ) F l
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Exercises 2.1



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.



Fourier SeriesTrigonometric SeriesEulers (Fourier Euler) Formulae
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Exercises 2.1

Euler s (Fourier-Euler) FormulaeFunctions 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


Fourier SeriesTrigonometric SeriesEulers (Fourier Euler) Formulae
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Exercises 2.1

Euler s (Fourier-Euler) FormulaeFunctions 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.


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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Exercises 2.1

Euler s (Fourier Euler) FormulaeFunctions of Period 2


Example

4. Find the Fourier series for the given function

f(x) =

1, < x < 0x, 0 < x <

Solution



F i S i f E d Odd F ti

Trigonometric SeriesEulers (Fourier-Euler) Formulae


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Exercises 2.1

Euler s (Fourier Euler) FormulaeFunctions 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



Fo rier Series for E en and Odd F nctionsFourier Cosine SeriesFo rier Sine Series
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Exercises 2.1

Fourier Sine Series

Outline



3 Fourier SeriesTrigonometric Series





6 Exercises 2.1



Fourier Series for Even and Odd FunctionsFourier Cosine SeriesFourier Sine Series
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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.



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Exercises 2.1

Fourier Sine Series


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.





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Exercises 2.1

Fourier Sine Series


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


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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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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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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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.



Fourier Series for Even and Odd FunctionsH lf R E i F i C i & F i Si S i

Fourier Cosine SeriesFourier Sine Series
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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



Fourier Series for Even and Odd FunctionsH lf R E i F i C i & F i Si S i

Fourier Cosine SeriesFourier Sine Series


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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).



Fourier Series for Even and Odd FunctionsHalf Range Expansions: Fourier Cosine & Fourier Sine Series



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Outline





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



Fourier Series for Even and Odd FunctionsHalf Range Expansions: Fourier Cosine & Fourier Sine Series

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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: 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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Half Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2.1


Figure: 2.1 Graph of f(x)

Figure: 2.2 g(x) is an Even Periodic Continuation of f(x)




Half-Range Expansions: Fourier Cosine & Fourier Sine Series



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g pExercises 2.1


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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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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Exercises 2.1


Figure: 2.3 Graph of f(x)

Figure: 2.4 h(x) is an Odd Periodic Continuation of f(x)




Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesE i 2 1



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Exercises 2.1


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.




Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesE i 2 1



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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).




Half-Range Expansions: Fourier Cosine & Fourier Sine SeriesExercises 2 1



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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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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.


Periodic Functions



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.


Periodic Functions



Exercises 2.1
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Outline

1 Periodic Functions2 Even and Odd Functions



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


Periodic Functions



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 < .

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Lec_fourierSeries

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