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

2. MATHEMATICS - II ● LAPLACE TRANSFORMS ● FOURIER SERIES
FOR BTECH SECOND SEMESTER COURSE [COMMON TO ALL BRANCHES OF ENGINEERING]
● LAPLACE TRANSFORMS
● FOURIER SERIES
● FOURIER TRANSFORMS
● VECTOR DIFFERENTIAL CALCULUS
● VECTOR INTEGRAL CALCULUS
● LINE, DOUBLE, SURFACE, VOLUME INTEGRALS
● BETA AND GAMMA FUNCTIONS
TEXT BOOK: ADVANCED ENGINEERING MATHEMATICS BY ERWIN KREYSZIG [8th EDITION]
DEPARTMENT OF MATHEMATICS, CVRCE

3. FOURIER INTEGRAL [chapter – 10.8] MATHEMATICS - II LECTURE :16
DEPARTMENT OF MATHEMATICS, CVRCE

4. LAYOUT OF LECTURE FROM FOURIER SERIES TO FOURIER INTEGRAL
INTRODUCTION & MOTIVATION
APPLICATIONS
EXISTENCE OF FOURIER INTEGRAL
SOME PROBLEMS
FOURIER SINE AND COSINE INTEGRALS
DEPARTMENT OF MATHEMATICS, CVRCE

5. INTRODUCTION & MOTIVATION
FOURIER SERIES ARE POWERFUL TOOLS IN TREATING VARIOUS PROBLLEMS INVOLVING PERIODIC FUNCTIONS. HOWEVER, FOURIER SERIES ARE NOT APPLICABLE TO MANY PRACTICAL PROBLEMS SUCH AS A SINGLE PULSE OF AN ELECTRICAL SIGNAL OR MECHANICAL FORCE VIBRATION WHICH INVOLVE NONPERIODIC FUNCTIONS. THIS SHOWS THAT METHOD OF FOURIER SERIES NEEDS TO BE EXTENDED. HERE WE START WITH THE FOURIER SERIES OF AN ARBITRARY PERIODIC FUNCTION fL OF PERIOD 2L AND THEN LET L SO AS TO DEVELOP FOURIER INTEGRAL OF A NON-PERIODIC FUNCTION.
Jean Baptiste Joseph Fourier (Mar21st 1768 –May16th 1830) French Mathematician & Physicist

6. FROM FOURIER SERIES TO FOURIER INTEGRAL
Let fL (x) be an arbitrary periodic function whose period is 2L which can be represented by its Fourier series as follows:
DEPARTMENT OF MATHEMATICS, CVRCE

7. FROM FOURIER SERIES TO FOURIER INTEGRAL
From (1), (2), (3), and (4), we get

8. FROM FOURIER SERIES TO FOURIER INTEGRAL
The representation (6) is valid for any fixed L, arbitrary large but finite. We now let L and assume that the resultant non-periodic function is absolutely integrable on x-axis , i.e. the resulting non-periodic function

9. FROM FOURIER SERIES TO FOURIER INTEGRAL
is absolutely integrable.
Set

10. FROM FOURIER SERIES TO FOURIER INTEGRAL
Applying (8) and (9) in (7), we get
Representation (10) with A(w) and B(w) given by (8) and (9), respectively, is called a Fourier integral of f(x).

11. FOURIER INTEGRAL

12. EXISTENCE OF FOURIER INTEGRAL
Theorem
If a function f(x) is piecewise continuous in every finite interval and has a right-hand derivative and left-hand derivative at every point and if the integral
exists, then f(x) can be represented by a Fourier integral. At a point where f(x) is discontinuous the value of the Fourier integral equals the average of the left and right hand limits of f(x) at that point.

13. PROBLEMS ON FOURIER INTEGRAL
EXAMPLE-1: Find the Fourier integral representation of the function
Solution:
The Fourier integral of the given function f(x) is

14. PROBLEMS ON FOURIER INTEGRAL

15. PROBLEMS ON FOURIER INTEGRAL

16. PROBLEMS ON FOURIER INTEGRAL

17. DIRICHLET’S DISCONTINUOUS FACTOR
From Example – 1 by Fourier integral representation, we find that
At x= 1, the function f(x) is discontinuous. Hence the value of the Fourier integral at 1 is ½ [f(1-0)+f(1+0)] = ½[0+1]=1/2.

18. DIRICHLET’S DISCONTINUOUS FACTOR
The above integral is called Dirichlet’s discontinuous factor.

19. Sine integral Dirichlet’s discontinuous factor is given by.
Putting x= 0 in the above expression we get .
The integral (1) is the limit of the integral as u  
The integral is called the sine integral and it is denoted by Si(u)

20. FOURIER COSINE INTEGRAL

21. FOURIER COSINE INTEGRAL
The Fourier integral of an even function is also known as Fourier cosine integral.

22. FOURIER SINE INTEGRAL

23. FOURIER SINE INTEGRAL
The Fourier integral of an odd function is also known as Fourier sine integral.

24. PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL
Solution:
The Fourier cosine integral of the given function f(x) is

25. PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL

26. PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL
The Fourier sine integral of the given function f(x) is

27. PROBLEMS INVOLVING FOURIER COSINE AND SINE INTEGRAL
Hence the Fourier sine integral of the given function is

28. LAPLACE INTEGRALS
From Example – 2 by Fourier cosine integral representation, we find that
(A)

29. The integrals (A) and (B) are called as Laplace integrals.
From Example – 2 by Fourier sine integral representation, we find that
(B)
The integrals (A) and (B) are called as Laplace integrals.

30. SOME MORE PROBLEMS
Solution:

31. SOME MORE PROBLEMS

32. SOME MORE PROBLEMS

33. SOME MORE PROBLEMS

34. SOME MORE PROBLEMS

35. SOME MORE PROBLEMS

36. SOME MORE PROBLEMS
Solution:

37. SOME MORE PROBLEMS

38. SOME MORE PROBLEMS

39. SOME MORE PROBLEMS
Solution:

40. SOME MORE PROBLEMS

41. SOME MORE PROBLEMS

42. The Fourier cosine integral of the function f(x) is given by
SOME MORE PROBLEMS
Solution:
The Fourier cosine integral of the function f(x) is given by

43. SOME MORE PROBLEMS

44. SOME MORE PROBLEMS

45. The Fourier cosine integral of f(x) is given by
SOME MORE PROBLEMS
Solution:
The Fourier cosine integral of f(x) is given by

46. SOME MORE PROBLEMS
Hence from (1) we obtain the required Fourier cosine integral as

47. The Fourier cosine integral of the given function f(x) is
SOME MORE PROBLEMS
The Fourier cosine integral of the given function f(x) is

48. SOME MORE PROBLEMS

49. Hence the Fourier cosine integral of the given function is
SOME MORE PROBLEMS
Hence the Fourier cosine integral of the given function is

50. The Fourier sine integral of the given function f(x) is
SOME MORE PROBLEMS
The Fourier sine integral of the given function f(x) is

51. Hence the Fourier sine integral of the given function is
SOME MORE PROBLEMS
Hence the Fourier sine integral of the given function is