# Chapter 3: The Laplace Transform

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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:

。Example 3.2: Consider

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

○ 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

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

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

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

＊ 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

○ Theorem 3. 1: Laplace transform is linear Proof: ○ Definition 3. 3:
○ Theorem 3.1: Laplace transform is linear Proof: ○ Definition 3.3:. Inverse Laplace transform e.g., ＊ Inverse Laplace transform is linear

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)

Proof: Let

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

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

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

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

○ Example 3.8:

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

。 Laplace transform of heaviside function

3.3.3 The Second Shifting Theorem
Proof:

○ Example 3.11: Rewrite

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

3.4. Convolution

◎ Theorem 3.9: Convolution theorem Proof:

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

○ Example 3.19:

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

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

Proof:

by Hospital’s rule ○ Example 3.20:

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

Partial fractions decomposition Inverse Laplace transform

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

○ Example 3.25: Laplace transform

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

Inverse Laplace transform

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

◎ Theorem 3.14: PC on [0, k],

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

Finding an integrating factor, Multiply (B) by ,

In order to have

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

○Shifting Theorems: ○ Convolution: Convolution Theorem: ○