Chapter 9 Laplace Transform §9.1 Definition of Laplace Transform §9.2 Properties of Laplace Transform §9.3 Convolution §9.4 Inverse Laplace Transform §9.5 Application of Laplace Transform

§9.1 Definition of Laplace Transform Definition

Unit Step: Unit Impulse

Ex:

TH Existence Theorem

Note: The conditions in the theorem are sufficient, not necessary.

Ex:9.1.5 Ex:9.1.6

§9.2 Properties of Laplace Transform 1.Linearity Ex.9.2.1

2.Derivation

Ex.9.2.2

Ex.

3.Integration

Ex Ex.

Homework P217:2.(1)(3)(5) 3 4 5(1)(2)(3)(4)

O t  f(t)f(t) f(t  ) 4.Delay Ex: 1 u(t)u(t)  tO

Ex:9.2.8

Ex:9.2.9

5.Displacement Ex:

6.Initial ＆ Terminal Value Theorems (1).Initial Value Theorem

(2).Terminal Value Theorem

Ex: Satisfying the conditions of the theorem, then you can use the theorem.

Ex: Ex: Table for properties on P201

§9.3 Convolution 1.Definition is called the convolution of,denoted as, i.e.. Note: Convolution in Fourier transform is same to that in Laplace transform.

Properties: 1.Commutative Law 2.Associative Law 3.Distributive Law 4.

Ex:

2.Convolution Theorem TH.9.3.1

Ex.9.3.2

Homework: P217:5.(5)-(13) 7.(1)(3)(5) 8

§9.4 Inverse Laplace Transform 1.Inverse Integral Formula From the inverse Fourier transform, we have the inverse Laplace transform formula.

R O Real axis Imaginary axis L CRCR  +jR jRjR singularities  analy TH.9.4.1

2.Evaluation (1).Using integral formula Ex:

(2).Using convolution theorem Ex:

(3).Using partial fraction Ex:

(4).Using properties Ex: (5).Using L-transform table

§9.5 Application of Laplace Transform 1.Evaluating the improper integral

Using Laplace transform solves the differential equation: The block diagram shows the details. Solution of Differential equation Algebra equation of 2.Solving Differential Equation

Ex:

Homework: P218: 9.(1)(3)(5) 10.(1)(3) 11.(1)(3)

1. The properties of Laplace Transform. 2. Application in solving differential equations. The key points and difficulties of the chapter.