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Humanities & Science Department

A.E.M. (2130002) Darshan Institute of engineering & Technology

Unit-2Fourier series & Integral2130002 – Advanced Engineering Mathematics

A.E.M. (2130002) Darshan Institute of engineering & Technology 2

Why Fourier series ?

We know that Taylor’s series representation of functions are valid

only for those functions which are continuous and differentiable.

As we know that discontinuous periodic functions can not be

expanded in Taylor’s series, but such functions can easily be

expanded in terms of infinite series containing “sine” and “cosine”

terms by Fourier series.

Fourier series applies to all continuous, periodic and also

discontinuous in their values.


Dirichlet Condition for existence of Fourier series of is bounded. is single valued. has finite number of maxima and minima in the interval. has finite number of discontinuity in the interval.

e.g.


Fourier Series in the interval The Fourier series for the function f(x) in the interval (c,c+2L) is

defined by

Where the constants, and are given by


Types of intervals

Fourier Series in the interval

Fourier Series in the interval


Fourier Series in the interval The Fourier series for the function in the interval is defined by



Basic formulae


Types of Functions

Polynomial function

Exponential function

Trigonometric function


Polynomial function Leibnitz’s Formula (When Function is Polynomial Function)

Where, are successive derivatives of u and are successive integrals of v.

Choice of and is as per LIATE order.

Where,

L means Logarithmic Function

I means Invertible Function

A means Algebraic Function

T means Trigonometric Function

E means Exponential Function


Polynomial function Leibnitz’s Formula (When Function is Polynomial Function)

e.g.

=

𝑥2

2 𝑥

2

(− cos𝑛𝑥𝑛 )(− sin𝑛𝑥𝑛2 )( cos𝑛𝑥𝑛3 )

−

+¿

+¿


Trigonometric Values


Example 1

Find the Fourier series for in

Solution : We have,


𝑥2 2 𝑥 2( sin𝑛𝜋 𝑥𝑛𝜋 ) (− cos𝑛𝜋 𝑥(𝑛𝜋 )2 ) (− sin𝑛𝜋 𝑥(𝑛𝜋)3 )− +¿


𝑥2 2 𝑥 2(− cos𝑛𝜋 𝑥𝑛𝜋 ) (− sin𝑛𝜋 𝑥(𝑛𝜋)2 ) ( cos𝑛𝜋 𝑥(𝑛𝜋)3 )− +¿


Hence the Fourier series of is given by

Substituting ,

We get the required Fourier series of in the interval


Example 3Find the Fourier series for in

Solution : We have,


Hence, the Fourier series of is given by

Substituting ,



Example 4


Solution : We have,


Since,

(∵ cos𝑛𝜋=(−1 )𝑛 ,cos (1+2𝑛)𝜋=(−1 )1+2𝑛 )



Substituting ,

We get required Fourier series of in the interval


Example 3 Find the Fourier series for in

Solution : We have,

Differentiation :-




Substituting ,



Example 4


Solution : We have,



Substituting ,



Example 7 Determine the Fourier series to represent

the periodic function as shown in the figure.

4𝜋2𝜋

𝜋

𝑓 (𝑥)

𝑥

(2𝜋 ,𝜋)

(0,0)


Example 7Solution :

Line passing through two points &


Here, with period Now,



Substituting ,


A.E.M. (2130002) Darshan Institute of engineering & Technology46

Example 6 Find the Fourier Series for the function given by

Solution : We have,



Substituting ,



Remark At a point of discontinuity the sum of the series is equal to

average of left and right hand limits of at the point of discontinuity, say

e.g.


Even Function

A function is said to be Even Function if

If is an even function defined in

then

e.g.

Every constant is an Even Function. ,Graph is symmetric about axis.


Even FunctionA function is said to be Even Function if

+ve values of -ve values of

+ve values of function-ve values of function

Graph of


Even FunctionGraph is symmetric about axis.

Graph of


Odd FunctionA function is said to be Odd Function if

If is an odd function defined in

then

e.g.

, Graph is symmetric about Origin.


Odd FunctionA function is said to be Even Function if

+ve values of

-ve values of +ve values of function

-ve values of function

Graph of


odd FunctionGraph is symmetric about .

Graph of


Fourier Series For Even Function

Let, be a periodic function defined in

is even,

Where,

𝒇 (𝒙 )=𝒂𝟎

𝟐 +∑𝒏=𝟏

∞

𝒂𝒏𝐜𝐨𝐬 (𝒏𝝅 𝒙𝑳 )


Fourier Series For Odd Function

Let, be a periodic function defined in

is odd,

Where,


Example 1 Find the Fourier series of the periodic function Where

Solution : We have,

The given function is odd function as it is defined on symmetric

interval and it will satisfy .

Fourier Series for Odd Function is given by


𝑥 (1 )(− cos𝑛𝜋 𝑥(𝑛𝜋) ) (− sin𝑛𝜋 𝑥(𝑛𝜋)2 )−



Substituting ,



Example 3 Find the Fourier series for in .

Solution : We have,

The given function is even function as it is defined on symmetric


Fourier Series for even Function is given by

𝒇 (𝒙 )=𝒂𝟎

𝟐 +∑𝒏=𝟏

∞




Substituting,


𝒇 (𝒙 )=𝒂𝟎

𝟐 +∑𝒏=𝟏

∞



Example 5

Find the Fourier series of ,Where

Solution :

Here, .

The given function is Neither even nor odd function .


Here, is an odd function and is an even function.


Here, is an even function and is an odd function.



Substituting,


𝑓 ( x )=a02

+∑n=1

∞

¿¿


Example 1 Obtain the Fourier series for in the interval

and hence deduce that

Solution :

We have,

Here, .

The given function is even function as it is defined on symmetric




Substituting ,


𝒇 (𝒙 )=𝒂𝟎

𝟐 +∑𝒏=𝟏

∞



Take

𝐂𝐥𝐚𝐢𝐦 (𝐈 ):∑𝒏=𝟏

∞ 𝟏𝒏𝟐=

𝝅𝟐

𝟔


𝐂𝐥𝐚𝐢𝐦 (𝐈𝐈 ):∑𝒏=𝟏

∞ (−𝟏 )𝒏+𝟏

𝒏𝟐 = 𝝅𝟐

𝟏𝟐


Add Eq. and ,

𝐂𝐥𝐚𝐢𝐦 (𝐈𝐈𝐈 ):∑𝐧=𝟏

∞ 𝟏(𝟐𝐧−𝟏 )𝟐

=𝛑𝟐

𝟖


Example 7

Find the Fourier series expansion of the function

. Deduce that.

Solution :Here, .




Substituting ,


𝑓 ( x )=a02

+∑n=1

∞

¿¿


Take, in Since, is discontinuous at ,


Example 6Find the Fourier series to representation in the interval .

Solution :

Here, .



Note that, and



Substituting ,


𝑓 ( x )=a02

+∑n=1

∞

¿¿


Example 5Find the Fourier series of .Solution : Here, .



Here, is an even function and is an odd function.



Substituting ,


𝑓 ( x )=a02

+∑n=1

∞

¿¿


Example 11If ; then expand in a Fourier series.

Solution :

We check function is odd OR even.

Thus, is an even function. So,


Here, & both are an even functions. So, is an even function.



Substituting ,


𝑓 ( x )=a02

+∑n=1

∞

¿¿


HALF RANGE COSINE SERIES IN

Where the constants and are given by


HALF RANGE SINE SERIES IN

Where the constants is given by


Example 1

Find the Fourier cosine series for .

Solution : Here, .

Half range cosine series is given by



Substituting ,



Example 3

Find Half range cosine series for in and

show that Solution : Here, .Half range cosine series is given by


If n is odd, than .

If n is even, than .

Here, is not defined at .

Let us find it separately.


Hence Fourier Half range cosine series of is given by

Substituting ,

We get required Half range cosine series in the interval

𝒇 (𝒙 )=𝒂𝟎

𝟐 +∑𝒏=𝟏

∞



Example 4

Find Half-range cosine series for in .

Solution :

Here, .


Hence the Fourier Half range cosine series of is given by

Substituting ,

We get the required Fourier Half range cosine series of in the

interval

𝒇 (𝒙 )=𝒂𝟎

𝟐 +∑𝒏=𝟏

∞



Example 1

Find the Half range sine series for Solution :

Here,

Fourier Half range sine series of is given by


Hence the Fourier Half range sine series of is given by

Substituting ,

We get the required Fourier Half range sine series of in the

interval


Example 5Find the Half range sine series .

Solution : Here,


Hence the Fourier Half range sine series of is given by

Substituting,

We get the required Fourier Half range sine series of in the

interval


FOURIER INTEGRALS

Fourier Integral of is given by


FOURIER COSINE INTEGRAL

Fourier cosine Integral of is given by


FOURIER SINE INTEGRAL

Fourier sine Integral of is given by


Example 1Using Fourier integral, Prove that

Solution: Here, .Fourier Integral of is given by


Example 3Find the Fourier integral representation of

Solution : Here, Note that, means & means


Take, ,


Example 4Using Fourier Integral, Prove that

Solution :

Here,

Fourier sine Integral of is given by


FOURIER TRANSFORM

Fourier transform of is defined as F ( λ )= 1

√2 π ∫−∞∞

f ( x )e− iλxdx


FOURIER COSINE TRANSFORM

Fourier cosine transform of is defined as

FC ( λ )=√ 2π∫0∞

f ( x ) cos λx dx


FOURIER SINE TRANSFORM

Fourier cosine transform of is defined as

FS ( λ )=√ 2π∫0∞

f ( x )sin λxdx


Example 1Find Fourier transformation of .Solution: Here, .Fourier transform of is defined as

F ( λ )= 1√2 π ∫−∞

∞

f ( x )e− iλxdx

[ppt]powerpoint presentation - darshan institute of … · web viewwhy fourier series ? we know...

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