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FOURIER SERIES

FOURIER SERIES

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FOURIER SERIES

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  1. FOURIER SERIES 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). PERIODICFUNCTIONS

  2. 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 nxand cosnxis 2/n.

  3. 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

  4. where the Fourier coeffecients are

  5. If is defined in the interval (c,c+2 ), the coefficients can be determined equivalently from

  6. 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)

  7. 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

  8. METHOD OF OBTAINING FOURIER SERIES OF 1. 2. 3. 4.

  9. SOLVED PROBLEMS 1. Expand f(x)=x2,0<x<2 in Fourier series if the period is 2 . Prove that

  10. SOLUTION Period = 2 = 2  thus =  and choosing c=0

  11. At x=0 the above Fourier series reduces to X=0 is the point of discontinuity

  12. By Dirichlet conditions, the series converges at x=0 to (0+4 2)/2 = 2 2

  13. 2. Find the Fourier series expansion for the following periodic function of period 4. Solution

  14. EVEN AND ODD FUNCTIONS 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

  15. 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

  16. If a periodic function is even in , its Fourier series expansion contains only cosine terms

  17. Case 2. When is an odd function

  18. If a periodic function is odd in ,its Fourier expansion contains only sine terms

  19. SOLVED PROBLEMS 1.For a function defined by obtain a Fourier series. Deduce that Solution is an even function

  20. SOLUTION

  21. 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

  22. PROBLEM 2 Is the function even or odd. Find the Fourier series of f(x)

  23. SOLUTION is odd function

  24. 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)

  25. 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]

  26. 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

  27. if

  28. in

  29. the above series reduces to At is a point of continuity The given series converges to

  30. 2) Expand in half range (a) sine Series (b) Cosine series. SOLUTION (a) Extend the definition of given function to that of an odd function of period 4 i.e

  31. Here

  32. (b) Extend the definition of given function to that of an even function of period 4

  33. Exercise problems 1. Find Fourier series of in 2. Find Fourier series of

  34. 3.Find the Fourier series of in 4.Find the Fourier series of (-2 ,2) in

  35. 5.Represent function In (0,L) by a Fourier cosine series 6.Determine the half range sine series for