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

2. TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8 th EDITION] MATHEMATICS - II LECTURE :18-19 FOURIER TRANSFORMS [Chapter – 10.10]

3. DEPARTMENT OF MATHEMATICS, CVRCE LAYOUT OF LECTURE PROPERTIES OF FOURIER TRANSFORMS FROM FOURIER INTEGRAL TO FOURIER TRANSFORM FORMAL DEFINITION OF FOURIER TRANSFORM AND INVERSE FOURIER TRANSFORM SOME PROBLEMS SOME PROBLEMS 1.Linearity Property 2. FOURIER TRANSFORM OF DERIVATIVES 3. CONVOLUTION THEOREM

4. FROM FOURIER INTEGRAL TO FOURIER TRANSFORM Fourier Integral of a function f(x) is given by 2(a) (1) 2(b)

5. Inserting 2(a) and 2(b) in (1), we get FROM FOURIER INTEGRAL TO FOURIER TRANSFORM

6. In (3), the integral in brackets is an even function of w, call it F(w) because cos(x – v)w is an even function of w, the function f does not depend on w, and we integrate with respect to v. Hence the integral of F(w) from w = 0 to ∞ is ½ times the integral of F(w ). Hence from (3) we have the following.

7. FROM FOURIER INTEGRAL TO FOURIER TRANSFORM (4) In (3), the integral in brackets with sine instead of cosine is an odd function of w, call it G(w) because sin(x – v)w is an odd function of w, the function f does not depend on w, and we integrate with respect to v. Hence the integral of G(w) from w = - ∞ to ∞ 0. Hence from (3) with sine instead of cosine gives the following.

8. FROM FOURIER INTEGRAL TO FOURIER TRANSFORM From(4) and (7) we have, Euler Formula: e ix = cosx + isinx ---------------------(7) From(4) and (5) we have,

9. FROM FOURIER INTEGRAL TO FOURIER TRANSFORM Equation (8) can be rewritten as In (9) we get

10. FROM FOURIER INTEGRAL TO FOURIER TRANSFORM Writing v = x in (10) we get Equation (12) is called the Fourier transform of f(x) and Equation (11) is called inverse Fourier transformation of

11. SOME PROBLEMS INVOLVING FOURIER TRANSFORMS Solution: Fourier transform of the given function is

12. SOME PROBLEMS INVOLVING FOURIER TRANSFORMS

14. Solution: Fourier transform of the given function is

15. SOME PROBLEMS INVOLVING FOURIER TRANSFORMS

17. LINEARITY PROPERTY OF FOURIER TRANSFORMS The Fourier Transform is a linear operation; that is, for any functions f(x) and g(x) whose Fourier transforms exist and any constants a and b, Proof:

18. Fourier Transform of the Derivative of a Function Proof:

19. Fourier Transform of the Derivative of a Function

21. Proof: By Fourier transform of derivative of a function, we have Fourier Transform of the Derivative of a Function

22. CONVOLUTION THEOREM OF FOURIER TRANSFORMS Definition of Convolution: The convolution of two functions f and g is denoted by the symbol f * g and is defined by

23. CONVOLUTION THEOREM OF FOURIER TRANSFORMS

26. SOME MORE PROBLEMS INVOLVING FOURIER TRANSFORMS Solution: Fourier transform of the given function is

27. SOME MORE PROBLEMS INVOLVING FOURIER TRANSFORMS

28. Solution: Fourier transform of the given function is

29. SOME MORE PROBLEMS INVOLVING FOURIER TRANSFORMS

31. TEST YOUR SKILLS