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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:

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。Example 3.2: Consider

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**Not every function has a Laplace transform**

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

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○ 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

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

exists

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

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

condition.

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＊ 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

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**○ 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

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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)

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Proof: Let

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

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

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**○ Laplace Transform of Integral**

From Eq. (3.1),

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**3.3. Shifting Theorems and Heaviside Function **

3.3.1.The First Shifting Theorem ◎ Theorem 3.7: ○ Example 3.6: Given

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○ Example 3.8:

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**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

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**。 Laplace transform of heaviside function**

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**3.3.3 The Second Shifting Theorem**

Proof:

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○ Example 3.11: Rewrite

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**◎ The inverse version of the second shifting theorem ○ Example 3.13: **

where rewritten as

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3.4. Convolution

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**◎ Theorem 3.9: Convolution theorem Proof: **

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

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○ Example 3.19:

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**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

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

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Proof:

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by Hospital’s rule ○ Example 3.20:

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**3.6 Laplace Transform Solution of Systems**

○ Example 3.22 Laplace transform Solve for

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**Partial fractions decomposition Inverse Laplace transform **

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**3.7. Differential Equations with Polynomial Coefficient**

◎ Theorem 3.13: Proof: ○ Corollary 3.1:

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**○ Example 3.25: Laplace transform **

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

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**Inverse Laplace transform**

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**○ Apply Laplace transform to algebraic expression for Y **

Differential equation for Y

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◎ Theorem 3.14: PC on [0, k],

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

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**Finding an integrating factor, Multiply (B) by , **

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In order to have

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

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**○Shifting Theorems: ○ Convolution: Convolution Theorem: ○ **

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○

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