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

2. MATHEMATICS - II LECTURE - 11 CONCEPT OF PERIODIC FUNCTION S AND TRIGONOMETRIC SERIES [Chapter – 10.1] DEPARTMENT OF MATHEMATICS, CVRCE TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8 th EDITION]

3. DEPARTMENT OF MATHEMATICS, CVRCE OUTLINES  Periodic functions  Fundamental periods  Trigonometric functions  Some problem based on periodic functions  Assignments OUTLINES  Periodic functions  Fundamental periods  Trigonometric functions  Some problem based on periodic functions  Assignments

4. A function is said to be periodic if it is defined for all real x and if there is some positive real number p such that, for all x. (1) This number p is called a period of. Definition of Periodic Functions DEPARTMENT OF MATHEMATICS, CVRCE

5. From (1), we may write and for any integer n, we can write Hence are also periods of. Definition Contd… DEPARTMENT OF MATHEMATICS, CVRCE

6. Example - 1: Determine the smallest positive period p of cosx. Solutions: PROBLEMS INVOLVING PERIODIC FUNCTIONS Let a period of cosx be p By definition of period we have for every real number x

7. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING PERIODIC FUNCTIONS

8. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING PERIODIC FUNCTIONS So a period of the function cosx is 2n . Smallest positive period happens when n = 1,i.e. the smallest positive period of cosx is 2 . Example - 2: Determine the smallest positive period p of sinx. Solutions: Let a period of sinx be p By definition of period we have for every real number x

9. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING PERIODIC FUNCTIONS

10. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING PERIODIC FUNCTIONS So a period of the function sinx is 2n . Smallest positive period happens when n = 1,i.e. the smallest positive period of sinx is 2 .

11. DEPARTMENT OF MATHEMATICVRCE PROBLEMS INVOLVING PERIODIC FUNCTIONS Now, h(x + p) = af(x + p)+bg(x + p) = a f(x)+b g(x) = h(x) Example - 3: If f(x) and g(x) have period p then show that the function h(x) = af(x)+bg(x), where a and b are constants has the period p. Since f(x) has period p then we have f(x+p) = f(x). Solution: Since g(x) has period p then we have g(x+p) = g(x). So the function h(x) has period p.

12. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING PERIODIC FUNCTIONS Example 4: If p is a period of a periodic function f(x), then show that f(ax), a  0, is a periodic function of x of period p/a, and f(x/b), b  0, is a periodic function of x of period bp. Solution: Given that f(x) is a periodic function with period p, i.e. f(x+p)=f(x). Let g(x)= f(ax), a  0

13. DEPARTMENT OF MATHEMATICS, CVRCE PROBLEMS INVOLVING PERIODIC FUNCTIONS

14. DEPARTMENT OF MATHEMATICS, CVRCE DEPARTMENT OF MATHEMATICVRCE Some Remarks:-  If have period p, then the function also has the period p, where  If a periodic function has a smallest period this is called fundamental period of Example: The fundamental period of and is and that for  A constant function is a periodic function with an arbitrary period but no fundamental period. Some Remarks:-  If have period p, then the function also has the period p, where  If a periodic function has a smallest period this is called fundamental period of Example: The fundamental period of and is and that for  A constant function is a periodic function with an arbitrary period but no fundamental period.

15. Trigonometric Series Trigonometric series is a series of the form where are real constants also called as the coefficients of the series. DEPARTMENT OF MATHEMATICS, CVRCE Some Remarks:  Trigonometric series has a period of  If the series converges, its sum will be a function of  Trigonometric series can be used for representing periodic function f, simple or complicated of period p.  This series will then be called the Fourier series of f.

16. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS ON PERIODIC FUNCTIONS (1) Find the smallest positive period of the function sin2x. Solution: Let a period of sin2x be p. By definition of period we have for every real number x

17. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS ON PERIODIC FUNCTIONS So a period of the function sin2x is n . Smallest positive period happens when n = 1,i.e. the smallest positive period of sin2x is .

18. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS ON PERIODIC FUNCTIONS (2) Find the smallest positive period of the function cos2  x. Solution: Let a period of cos2  x be p. By definition of period we have for every real number x

19. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS ON PERIODIC FUNCTIONS So a period of the function sinpx is n. Smallest positive period happens when n = 1,i.e. the smallest positive period of sin2  x is 1.

20. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS ON PERIODIC FUNCTIONS (3) Find the smallest positive period of the function. Solution: Let a period of be p. By definition of period we have for every real number x

21. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS ON PERIODIC FUNCTIONS

22. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS ON PERIODIC FUNCTIONS So a period of the function is Smallest positive period happens when m = 1,i.e. the smallest positive period of is

23. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS ON PERIODIC FUNCTIONS (4) Find the smallest positive period of the function. Solution: Let a period of be p. By definition of period we have for every real number x

24. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS ON PERIODIC FUNCTIONS

25. DEPARTMENT OF MATHEMATICS, CVRCE SOME MORE PROBLEMS ON PERIODIC FUNCTIONS So a period of the function is Smallest positive period happens when m = 1, i.e. the smallest positive period of is

26. DEPARTMENT OF MATHEMATICS, CVRCE (4) Sketch or plot the following functions which are assumed to be periodic with period and for are given by the formulae: Solutions: (i) (4) Sketch or plot the following functions which are assumed to be periodic with period and for are given by the formulae: Solutions: (i)

27. DEPARTMENT OF MATHEMATICS, CVRCE (ii) (iii) (ii) (iii)

28. DEPARTMENT OF MATHEMATICS, CVRCE (iv) (v) (iv) (v)

29. DEPARTMENT OF MATHEMATICS, CVRCE Assignments (1) Find the smallest positive period p of the following: (2) If is a periodic function of x having period p, then show that is a periodic function of x having period Verify the results for (3) Sketch or plot the following functions which are assumed to be periodic with period and for are given by the formulae: Assignments (1) Find the smallest positive period p of the following: (2) If is a periodic function of x having period p, then show that is a periodic function of x having period Verify the results for (3) Sketch or plot the following functions which are assumed to be periodic with period and for are given by the formulae: