Laplace transforms

GUJARAT
TECHNOLOGICAL
UNIVERSITY
Subject : Advanced
Engineering Mathematics
Sem-3 (Chemical)
Topic : Laplace Transforms

Topics
 Definition of Laplace Transform
 Linearity of the Laplace Transform
 Laplace Transform of some Elementary Functions
 First Shifting Theorem
 Inverse Laplace Transform
 Differentiation & Integration of Laplace Transform
 Evaluation of Integrals By Laplace Transform
 Convolution Theorem
 Application to Differential Equations
 Laplace Transform of Periodic Functions
 Unit Step Function
 Second Shifting Theorem
 Dirac Delta Function

Definition of Laplace Transform
 Let f(t) be a given function of t defined for all
then the Laplace Transform of f(t) denoted by L{f(t)}
or or F(s) or is defined as
provided the integral exists, where s is a parameter real
or complex.
0t
)(sf )(s
dttfessFsftfL st
)()()()()}({
0



 

Linearity of the Laplace Transform
 If L{f(t)}= and then for any
constants a and b
)(sf )()]([ sgtgL 
)]([)]([)]()([ tgbLtfaLtbgtafL 
)]([)]([)}()({
)()(
)]()([)}()({
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00
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Laplace Transform of some Elementary
Functions
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Inverse Laplace Transform
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Differentiation & Integration of Laplace
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Laplace Transform of Periodic Functions
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Unit Step Function
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Laplace transforms

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Laplace transforms