1 CHAPTER 5 : FOURIER SERIES  Introduction  Periodic Functions  Fourier Series of a function  Dirichlet Conditions  Odd and Even Functions  Relationship Between Even and Odd functions to Fourier series

2 5.1 : Introduction Fourier Series ? A Fourier series is a representation of a function as a series of constants times sine and/or cosine functions of different frequencies.

3 5.2 : Periodic Functions A function f(x) is said to be periodic if its function values repeat at regular intervals of the independent variables. For the following example, a function f(x) has the period p. p x y x 1 x 1 + p x 1 + 2p x 1 + 3p

4 In general, a function f(x) is called periodic if there is some positive number p such that ; f(x) = f(x + np) for any integer n.This number p is called a period of f(x).

5 5.3 : Fourier Series of a Function If f(x) is defined within the interval c < x < c+2L. The Fourier Series corresponding to f(x) is given by where

6 5.4 : Dirichlet Conditions If a function f(x) defined within the interval c < x < c+2L, the following conditions must be satisfied; 1. f(x) is defined and single-valued. 2. f(x) is continuous or finite discontinuity in the corresponding periodic interval. 3. f(x) and f ’ (x) are piecewise continuous.

7 Example 5.1: Determine whether the Dirichlet conditions are satisfied in the following cases : i. Yes. ii. No, because there is infinite discontinuity at

8 iii. No, the function is not defined for x < 0. iv. Yes.

9 Example 5.2: 1. Find the Fourier series for the function defined within the interval Answer :

10 Solution The Fourier series is where

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14 Graph of a Function What will we obtain as more terms are included in the series ? y 4 x -  0    

15 Graph of a Function 4  O x

16 Graph of a Function

17 Graph of a Function

18 Graph of a Function The above figure show that the graph is merely to the shape …..

19 The above function is defined within the interval As the number of terms increases, the graph of Fourier series gradually approaches the shape of the original square waveform

20 Example 5.3: Find the Fourier series for defined within the interval Answer : O f(x) x

21 Solution The Fourier series is where

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25 Find the Fourier series for defined within the interval Answer : x f (x)  5  3  O Example 5.4:

26 Solution The Fourier series is

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30 Exercise Find the Fourier series for defined by Answer :

: Odd and Even Functions A function f(x) is said to be even if : f(-x) = f(x) i.e the function value for a particular negative value of x is the same as that for the corresponding positive value of x. A function f(x) is said to be odd if : f(-x) = - f(x) i.e the function value for a particular negative value of x is numerically equal to that for the corresponding positive value of x but opposite in sign.

32 Example 5.5 : Even function : y = f(x) = x 2 is an even function because f(-2) = 4 = f(2) f(-6) = 36 = f(6)

33 Odd function : y = f(x) = x 3 is an odd function because f(-2) = -8 = -f(2) f(-3) = -27 = -f(3)

34 Properties of An Even function The graph of an even function is symmetrical about the y-axis. Hence, the areas under curves is twice the area from 0 to a : y x  a 0 a y = x 2

35 Properties of An Odd function The graph of an odd function is symmetrical about the origin. Thus, the integral is 0 because the areas cancel. y x 0 a y = x 3  a

36 Moreover : If n is positive integer, thus x 2n is an even function, and x 2n+1 is an odd function. Function is an even function. Function is an odd function.

37 The product of two even functions is even. Example : The product of two odd functions is even Example : The product of an even function and an odd function is odd. Example :

: Relationship between Even and Odd functions to Fourier series We could simply obtain the Fourier series for a function, f(x) defined within the interval –a<x<a. In this case, c = -a and c+2L = a, thus L = a, if we could identify whether f(x) is an even or odd function and use the properties of these functions in order to find the coefficients of Fourier series, a 0, a n and b n

39 i. If f(x) is an even function,  and is an even function, thus  an even function.  and is an odd function, thus  an odd function.

40 If f(x) is an even function

41 ii. If f(x) is an odd function, thus :  and an even function, thus  is an odd function.  and an odd function, thus  is an even function

42 If f(x) is an odd function