Laplace Transform
Concepts and Applications


Contents
1 Fundamentals of Laplace Transform 1
1.1 Definition of Laplace Transform . . . . . . . . . . . . . . . . . 1
1.2 Linearity Property of Laplace Transform . . . . . . . . . . . . 1
1.3 Transforms for Mathematical Models . . . . . . . . . . . . . . 2
1.3.1 Laplace Transform of Constant Functions . . . . . . . . 2
1.3.2 Laplace Transform of Linear Models . . . . . . . . . . 3
1.3.3 Laplace Transform of Power Models . . . . . . . . . . . 4
1.3.4 Laplace Transform of Polynomials . . . . . . . . . . . . 5
1.3.5 Laplace Transform of Expo. & Log. Functions . . . . . 5
1.3.6 Laplace Transform of Trigonometric Functions . . . . . 9
1.3.7 Laplace Transform of Hyperbolic Functions . . . . . . . 12
1.3.8 Laplace Transform of Periodic Functions . . . . . . . . 13
1.4 Sufficient Conditions for the Existence of Laplace Transform . 14
1.4.1 Piecewise Continuous Functions . . . . . . . . . . . . . 14
1.4.2 Functions with an Exponential Order . . . . . . . . . . 15
1.5 Inverse of Laplace Transform . . . . . . . . . . . . . . . . . . . 16
2 Advance Topics and Applications 17
2.1 Shifting Properties of Laplace Transform . . . . . . . . . . . . 17
2.1.1 First Shifting Property . . . . . . . . . . . . . . . . . . 17
2.1.2 Second Shifting Property . . . . . . . . . . . . . . . . . 17
2.2 Transform of Derivatives and Integration . . . . . . . . . . . . 18
2.3 Differentiation and Integration of Laplace Transform . . . . . 20
2.4 Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Mathematical Application: Test of Linear Independence . . . . 23
3 Laplace-Stieltje’s Transform 25
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Obtaining Central Momen