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

2. MATHEMATICS-II LECTURE-13 Fourier series of a periodic function of any arbitrary period [Chapter – 10.3] TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8 th EDITION]

3. Let f(x) be a periodic function of period T given in the interval [ ,  +T]. Derivation of Fourier series of a periodic function of any arbitrary period Therefore, g(v) is a periodic function of period 2  given in the interval

4. DEPARTMENT OF MATHEMATICS, CVRCE Therefore, the Fourier series of g(v) is given by

5. DEPARTMENT OF MATHEMATICS, CVRCE Setting g(v)= f(x) and v=2  x/T we get.

6. DEPARTMENT OF MATHEMATICS, CVRCE FOURIER SERIES OF A FUNCTION OF ARBITRARY PERIODS with the Fourier coefficients of is given by the Euler’s formula The fourier series of a periodic function f(x) of period T given in an interval is given by

7. DEPARTMENT OF MATHEMATICS, CVRCE Some special cases If we put T=2L in eqn.(1), then the Fourier series of a periodic function having period T=2L in the interval is given by where the Fourier coefficients

8. DEPARTMENT OF MATHEMATICS, CVRCE If we put in eqn.(2), then the Fourier series of a periodic function having period T=2L in the interval is given by where the Fourier coefficients

9. DEPARTMENT OF MATHEMATICS, CVRCE If we put in eqn.(2), then the Fourier series of a periodic function having period T=2L in the interval is given by where the Fourier coefficients The values of are called Euler’s formulae.

10. DEPARTMENT OF MATHEMATICS, CVRCE Example(1): Find the Fourier series of the function Example(1): Find the Fourier series of the function Solution:

11. DEPARTMENT OF MATHEMATICS, CVRCE The Fourier series of f(x) in the interval  < x <  + T is where

12. DEPARTMENT OF MATHEMATICS, CVRCE

13. Hence, the required Fourier series is

14. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS 1. Find the Fourier series of the following periodic functions having period where Solution: The Fourier series of f(x) in the interval  < x <  + T is

15. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

16. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

17. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

18. DEPARTMENT OF MATHEMATICS, CVRCE Finally, SOME MORE PROBLEMS

19. DEPARTMENT OF MATHEMATICS, CVRCE Substituting the values of in eqn.(2), we obtain the required Fourier series of the given function as SOME MORE PROBLEMS

20. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS 2. Find the Fourier series of the following periodic functions having period p. where The Fourier series of f(x) in the interval  < x <  + T is Solution:

21. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

22. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

23. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS Substituting the values of in eqn.(2), we obtain the required Fourier series of the given function as

24. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS 3. Find the Fourier series of the following periodic functions having period p where The Fourier series of f(x) in the interval  < x <  + T is Solution:

25. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

26. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

27. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

28. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS Hence, the required Fourier Series is

29. DEPARTMENT OF MATHEMATICS, CVRCE Solution: SOME MORE PROBLEMS 4. Find the Fourier series of the periodic function f(x) having period p, where where The Fourier series of f(x) in the interval  < x <  + T is

30. SOME MORE PROBLEMS

31. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

32. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

33. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS

34. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS Substituting the values of in eqn.(2), we obtain the desired Fourier series as

35. DEPARTMENT OF MATHEMATICS, CVRCE Assignments Find the Fourier series of the following functions: