1. DEPARTMENT OF MATHEMATI CS [ YEAR OF ESTABLISHMENT – 1997 ] DEPARTMENT OF MATHEMATICS, CVRCE

2. MATHEMATICS-II LECTURE-14 Fourier series of Even and Odd functions. [Chapter – 10.4] TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8 th EDITION]

3. DEPARTMENT OF MATHEMATICS, CVRCE OUTLINES  Concept of an even function and their geometrical representation.  Concept of an odd function and their geometrical representation.  Fourier series of an even function and odd function.  Some problem based on even and odd functions.

4. DEPARTMENT OF MATHEMATICS, CVRCE Even Function Definition A function f(x) is said to be even, if Thus,

5. DEPARTMENT OF MATHEMATICS, CVRCE  The graph of an even function is always symmetric about y-axis.  contains only even powers of x, that is, any equation remains unchanged when x is replaced by –x in the equation.   The sum and the product of two even functions is even. Some Important Notes

6. DEPARTMENT OF MATHEMATICS, CVRCE Geometrical representation of an even function For an even function, the curve is symmetric about y-axis. So any equation remains unchanged when x is replaced by –.

7. DEPARTMENT OF MATHEMATICS, CVRCE Odd Function A function f(x) is said to be odd, if Definition

8. DEPARTMENT OF MATHEMATICS, CVRCE Some Important Notes  The graph of an odd function is always symmetric about the origin (i.e., lies in the opposite quadrants Ist and IIIrd ).  contains only odd powers of x.   The sum of two odd functions is odd.  The product of two odd functions is even.  The product of odd function and even function is odd.

9. DEPARTMENT OF MATHEMATICS, CVRCE Geometrical representation of an odd function For an odd function, the curve is symmetric with respect to the origin. Now, let us consider the function which is an odd. When x is positive, then y is positive; also when x is negative, then y is negative. This lies in the opposite quadrants Ist and IIIrd.

10. DEPARTMENT OF MATHEMATICS, CVRCE Examples of Graphical Representation of Odd Function

11. DEPARTMENT OF MATHEMATICS, CVRCE FOURIER SERIES OF EVEN AND ODD FUNCTIONS We know that the Fourier series of a periodic function f(x) with period T given in the range –T/2 ≤ x ≤ T/2

12. DEPARTMENT OF MATHEMATICS, CVRCE FOURIER SERIES OF EVEN AND ODD FUNCTIONS Case – I : If the function f(x) is even [Since f(x)cosx is even] [Since f(x)sinx is odd]

13. DEPARTMENT OF MATHEMATICS, CVRCE Therefore, the Fourier series of a even periodic function f(x) with period T given in the range –T/2 ≤ x ≤ T/2 is given by FOURIER SERIES OF EVEN AND ODD FUNCTIONS

14. DEPARTMENT OF MATHEMATICS, CVRCE FOURIER SERIES OF EVEN AND ODD FUNCTIONS Case – II : If the function f(x) is odd [Since f(x)cosx is odd] [Since f(x)sinx is even]

15. DEPARTMENT OF MATHEMATICS, CVRCE Therefore, the Fourier series of a odd periodic function f(x) with period T given in the range –T/2 ≤ x ≤ T/2 is given by FOURIER SERIES OF EVEN AND ODD FUNCTIONS

16. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS Example(1): Expand the function as a Fourier series in the interval Solution The given function is Therefore, the given function is an even function and its Fourier expansion contains only cosine series in the interval which is given by

17. DEPARTMENT OF MATHEMATICS, CVRCE We know that the Fourier series of a even periodic function f(x) with period T given in the range –T/2 ≤ x ≤ T/2 is given by Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS Here T = 2 

18. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS Therefore, the Fourier series expansion of the given function f(x) is

19. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

20. DEPARTMENT OF MATHEMATICS, CVRCE Substituting the values of a 0 and a n ’s in eqn.(2), we obtain the required Fourier series as Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

21. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS Example(2): Express the function as a Fourier series in the interval. Solution: Here, given that the function. Therefore, the given function is an odd function. We know that the Fourier series of odd periodic function f(x) with period T given in the range –T/2 ≤ x ≤ T/2 is given by

22. DEPARTMENT OF MATHEMATICS, CVRCE Here T = 2  Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

23. DEPARTMENT OF MATHEMATICS, CVRCE Substituting the values of b n ’s in eqn.(1), we obtain the desired Fourier series as Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

24. DEPARTMENT OF MATHEMATICS, CVRCE (3) Obtain the Fourier series for the function given by Deduce that (3) Obtain the Fourier series for the function given by Deduce that Solution Since, and Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

25. DEPARTMENT OF MATHEMATICS, CVRCE Therefore, is an even function in the interval It is clear from the graph ABC, which is symmetrical about the y-axis. Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

26. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS We know that the Fourier series of a even periodic function f(x) with period T given in the range –T/2 ≤ x ≤ T/2 is given by Here T = 2 

27. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS Therefore, the Fourier series expansion of the given function f(x) is

28. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS and

29. DEPARTMENT OF MATHEMATICS, CVRCE Substituting the values of a n ’s in eqn.(1), we obtain the desired Fourier series as Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

30. DEPARTMENT OF MATHEMATICS, CVRCE Putting in eqn.(3), we have which is the required result. Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS DEPARTMENT OF MATHEMATICS, CVRCE

31. Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS (4) Examine whether the following function is odd or even or neither. Solution

32. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS (5) Examine whether the function is odd or even or neither. Solution

33. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS (6) Examine whether the function is odd or even or neither. Solution

34. Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS (7) Examine whether the following periodic function of period 2  is odd or even or neither. Solution:

35. Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS (8) Examine whether the following periodic function of period 2  is odd or even or neither. Solution: The given function is

36. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS 8. State whether the given function is even or odd. Find it’s Fourier series. Solution : Since f(-x)  f(x) and f(-x)  -f(x)

37. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS Thus, we can say that the given function is neither odd nor even. So, the required Fourier series is given by the formula as where

38. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS So, the required Fourier series is

39. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

40. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

41. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

42. DEPARTMENT OF MATHEMATICS, CVRCE. Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

43. DEPARTMENT OF MATHEMATICS, CVRCE Substituting the values of a n ’s and b n ’s in eqn.(1), we obtain the required Fourier series as: Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

44. 9. State whether the given function is even or odd. Find it’s Fourier series. Solution : Thus, we can say that the given function is an even function.

45. DEPARTMENT OF MATHEMATICS, CVRCE Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS We know that the Fourier series of a even periodic function f(x) with period T given in the range –T/2 ≤ x ≤ T/2 is given by Here T = 2 

46. Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS Therefore, the Fourier series expansion of the given function f(x) is and

47. DEPARTMENT OF MATHEMATICS, CVRCE Substituting the values of a n ’s in the eqn.(1), we get the desired Fourier series of the given function as: Problems involving FOURIER SERIES OF EVEN AND ODD FUNCTIONS

48. DEPARTMENT OF MATHEMATICS, CVRCE Assignments (1) Express the function as the Fourier series in the interval (2) Find the Fourier series expansion of the function (3) Find the Fourier series expansion of the function Hence deduce that

49. DEPARTMENT OF MATHEMATICS, CVRCE (4) Find the Fourier series expansion of the following function is given by Assignment Contd…