1. 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).

2. 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.

4. 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

5. where the Fourier coeffecients are

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

7. 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)

8. 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

9. METHOD OF OBTAINING FOURIER SERIES OF1. 2. 3. 4.

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

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

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

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

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

19. 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

20. 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

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

22. Case 2. When is an odd function

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

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

25. SOLUTION

26. 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

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

28. SOLUTION
is odd function

32. 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)

33. 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]

34. 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

37. if

38. in

39. the above series reduces toAt
is a point of continuity
The given series converges to

41. 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

43. Here

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

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

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

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

51. 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

53. CASE-I
If f(x) is defined in (0,2l) then Parseval’s Identity is given by

54. CASE-II If half range cosine series in (o,l) for f(x) is.
Then Parseval’s Identity is given by .

55. CASE-III If the half range Sine sereies in (0,l) for f(x) is
Then Parseval,s Identity is given by

56. 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 .

57. Equation(2) becomes

58. Equation(3) becomes

59. Equation(4) becomes
Equation(5)becomes

60. SOLVED PROBLEMS 1) Find the Fourier series of periodic function in
Hence deduce the sum of series
Assuming that

61. SOLUTION in

62. if
is odd function
is odd function

63. Using the Parseval’s Identity

65. 2)By using sine series for in
Show that
SOLUTION
for

66. By Parseval’s Identity

67. 3)Prove that in
and deduce that
SOLUTION
In Half range cosine series

69. By Parseval’s Identity

70. COMPLEX FORM OF FOURIER SERIES
The Fourier series of a periodic function of period 2l is

71. The Fourier series can be represented in the following way

72. SOLVED PROBLEM
1.Find the complex form of the Fourier series of the periodic function

73. SOLUTION

74. 2.Find the complex form of Fourier seriesof f(x)=sinx in (0,)

75. SOLUTION

77. HARMONIC ANALYSIS

79. 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