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

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