Chapter 3: The Laplace Transform

1. Chapter 3: The Laplace Transform 3.1. Definition and Basic Properties 。 Objective of Laplace transform -- Convert differential into algebraic equations ○ Definition 3.1: Laplace transform s.t. converges s, t : independent variables ＊ Representation:

2. 。Example 3.2: Consider

4. * Not every function has a Laplace transform. In general, can not converge 。Example 3.1:

5. ○ Definition 3.2.: Piecewise continuity (PC) f is PC on if there are finite points s.t. and are finite i.e., f is continuous on [a, b] except at finite points, at each of which f has finite one-sided limits

6. If f is PC on [0, k], then so is and exists

7. ◎ Theorem 3.2: Existence of f is PC on If Proof:

8. * Theorem 3.2 is a sufficient but not a necessary condition.

9. ＊ There may be different functions whose Laplace transforms are the same e.g., and have the same Laplace transform ○ Theorem 3.3: Lerch’s Theorem ＊ Table 3.1 lists Laplace transforms of functions

10. ○ Theorem 3.1: Laplace transform is linear Proof: ○ Definition 3.3:. Inverse Laplace transform e.g., ＊ Inverse Laplace transform is linear

11. 3.2 Solution of Initial Value Problems Using Laplace Transform ○ Theorem 3.5: Laplace transform of f: continuous on : PC on [0, k] Then, ------(3.1)

12. Proof: Let

13. ○ Theorem 3.6: Laplace transform of : PC on [0, k] for s > 0, j = 1,2 … , n-1

14. 。 Example 3.3: From Table 3.1, entries (5) and (8)

16. ○ Laplace Transform of Integral From Eq. (3.1),

17. 3.3. Shifting Theorems and Heaviside Function 3.3.1.The First Shifting Theorem ◎ Theorem 3.7: ○ Example 3.6: Given

18. ○ Example 3.8:

19. 3.3.2. Heaviside Function and Pulses ○f has a jump discontinuity at a, if exist and are finite but unequal ○ Definition 3.4: Heaviside function 。 Shifting

20. 。 Laplace transform of heaviside function

21. 3.3.3 The Second Shifting Theorem ◎ Theorem 3.8: Proof:

22. ○ Example 3.11: Rewrite

23. ◎ The inverse version of the second shifting theorem ○ Example 3.13: where rewritten as

26. 3.4. Convolution

27. ◎ Theorem 3.9: Convolution theorem Proof:

28. ◎ Theorem 3.10: ○ Exmaple 3.18 ◎ Theorem 3.11: Proof :

29. ○ Example 3.19:

30. 3.5 Impulses and Dirac Delta Function ○ Definition 3.5: Pulse ○ Impulse: ○ Dirac delta function: A pulse of infinite magnitude over an infinitely short duration

31. ○ Laplace transform of the delta function ◎ Filtering (Sampling) ○ Theorem 3.12: f : integrable and continuous at a

32. Proof:

33. ○ Example 3.20: by Hospital’s rule

34. 3.6 Laplace Transform Solution of Systems ○ Example 3.22 Laplace transform Solve for

35. Partial fractions decomposition Inverse Laplace transform

36. 3.7. Differential Equations with Polynomial Coefficient ◎ Theorem 3.13: Proof: ○ Corollary 3.1:

37. ○ Example 3.25: Laplace transform

38. Find the integrating factor, Multiply (B) by the integrating factor

39. Inverse Laplace transform

40. ○ Apply Laplace transform to algebraic expression for Y Apply Laplace transform to Differential equation for Y

41. ◎ Theorem 3.14: PC on [0, k],

42. ○ Example 3.26: Laplace transform ------(A) ------(B)

43. Finding an integrating factor, Multiply (B) by ,

44. In order to have

45. Formulas: ○ Laplace Transform: ○ Laplace Transform of Derivatives: ○ Laplace Transform of Integral:

46. ○Shifting Theorems: ○ Convolution: Convolution Theorem: ○

47. ○