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FOURIER SERIES PERIODIC FUNCTIONS
A function f(x) is said to be periodic with period T if f(x+T)=f(x) x , where T is a positive constant . The least value of T>0 is called the period of f(x).
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Ex.2 The period of sin nx and cos nx is 2/n.
f(x+2T) =f ((x+T)+T) =f (x+T)=f(x) f(x+nT)=f(x) for all x Ex.1 f(x)=sin x has periods 2, 4, 6, …. and 2 is the period of f(x). Ex.2 The period of sin nx and cos nx is 2/n.
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FOURIER SERIES Let be defined in the interval and outside the interval by i.e assume that has the period .The Fourierseries corresponding to is given by
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where the Fourier coeffecients are
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If is defined in the interval
(c,c+2 ), the coefficients can be determined equivalently from
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DIRICHLET CONDITIONS Suppose that
f(x) is defined and single valued except possibly at finite number of points in (-l,+l) f(x) is periodic outside (-l,+l) with period 2l f(x) and f’(x) are piecewise continuous in(-l,+l)
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Then the Fourier series of f(x) converges to
f(x) if x is a point of continuity b)[f(x+0)+f(x-0)]/2 if x is a point of discontinuity
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METHOD OF OBTAINING FOURIER SERIES OF
1. 2. 3. 4.
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SOLVED PROBLEMS 1. Expand f(x)=x2,0<x<2 in Fourier series if the period is 2 . Prove that
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Period = 2 = 2 thus = and choosing c=0
SOLUTION Period = 2 = 2 thus = and choosing c=0
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At x=0 the above Fourier series reduces to
X=0 is the point of discontinuity
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By Dirichlet conditions, the series converges at x=0 to (0+4 2)/2 = 2 2
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2. Find the Fourier series expansion for the following periodic function of period 4.
Solution
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EVEN AND ODD FUNCTIONS f(-x)=-f(x) Ex: x3, sin x, tan x,x5+2x+3
A function f(x) is called odd if f(-x)=-f(x) Ex: x3, sin x, tan x,x5+2x+3 A function f(x) is called even if f(-x)=f(x) Ex: x4, cos x,ex+e-x,2x6+x2+2
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EXPANSIONS OF EVEN AND ODD PERIODIC FUNCTIONS
If is a periodic function defined in the interval , it can be represented by the Fourier series Case1. If is an even function
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If a periodic function is even in
, its Fourier series expansion contains only cosine terms
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Case 2. When is an odd function
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If a periodic function is odd in
,its Fourier expansion contains only sine terms
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SOLVED PROBLEMS 1.For a function defined by
obtain a Fourier series. Deduce that Solution is an even function
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SOLUTION
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At x=0 the above series reduces to
x=0 is a point of continuity, by Dirichlet condition the Fourier series converges to f(0) and f(0)=0
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PROBLEM 2 Is the function even or odd. Find the Fourier series of f(x)
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SOLUTION is odd function
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HALF RANGE SERIES COSINE SERIES
A function defined in can be expanded as a Fourier series of period containing only cosine terms by extending suitably in (As an even function)
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SINE SERIES A function defined in can be expanded
as a Fourier series of period containing only sine terms by extending suitably in [As an odd function]
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SOLVED PROBLEMS Obtain the Fourier expansion of (x sinx )as a
cosine series in .Hence find the value of SOLUTION Given function represents an even function in
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if
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in
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the above series reduces to
At is a point of continuity The given series converges to
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SOLUTION 2) Expand in half range (a) sine Series (b) Cosine series.
Extend the definition of given function to that of an odd function of period 4 i.e
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Here
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(b) Extend the definition of given function to that of an even function of period 4
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Exercise problems 1. Find Fourier series of in 2.
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3.Find the Fourier series of
(-2 ,2) in
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5.Represent function In (0,L) by a Fourier cosine series 6.Determine the half range sine series for
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PARSEVAL’S IDENTITY To prove that
Provided the Fourier series for f(x) converges uniformly in (-l, I). The Fourier Series for f(x) in (-l,l) is Multiplying both sides of (1) by f(x)and integrating term from – l to l ( which is justified because f(x) is uniformly convergent) 51
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CASE-I If f(x) is defined in (0,2l) then Parseval’s Identity is given by
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CASE-II If half range cosine series in (o,l) for f(x) is
. Then Parseval’s Identity is given by .
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CASE-III If the half range Sine sereies in (0,l) for f(x) is
Then Parseval,s Identity is given by
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RMS VALUE OF FUNCTION If a function y=f(x) is defined in ( c , c+2l ),then is called the root mean square value (RMS value) of y in ( c , c+2l ).It is denoted by .
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Equation(2) becomes
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Equation(3) becomes
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Equation(4) becomes Equation(5)becomes
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SOLVED PROBLEMS 1) Find the Fourier series of periodic function in
Hence deduce the sum of series Assuming that
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SOLUTION in
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if is odd function is odd function
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Using the Parseval’s Identity
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2)By using sine series for in
Show that SOLUTION for
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By Parseval’s Identity
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3)Prove that in and deduce that SOLUTION In Half range cosine series
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By Parseval’s Identity
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COMPLEX FORM OF FOURIER SERIES
The Fourier series of a periodic function of period 2l is
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The Fourier series can be represented in the following way
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SOLVED PROBLEM 1.Find the complex form of the Fourier series of the periodic function
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SOLUTION
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2.Find the complex form of Fourier seriesof f(x)=sinx in (0,)
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SOLUTION
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HARMONIC ANALYSIS
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1.Find first two harmonics of Fourier Series from the following table
The term a1cosx+b1sinx is called the fundamental or first harmonic, the term a2cosx+b2sinx is called the second harmonic and so on. Solved Problem 1.Find first two harmonics of Fourier Series from the following table
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