# Laplace Transform of Exp and Sine

Cuthbert Nyack
To find the Laplace transform F(s) of an exponential function f(t) = e -at for t >= 0. Substitute f(t) into the definition of the Laplace Transform to get. Integrating and evaluating the limits give:- For an exponential function F(s) has a simple pole on the negative real axis at s = -a.
To find the Laplace transform of a sine function f(t) = sin wt for t > 0. Substitute f(t) to give:- If we substitute the following expression for f(t) then we get:- Carrying out the integration gives And evaluating the limits we get which simplifies to:- For a sine F(s) has poles on the imaginary axis at -jw and at -jw. For the cosine the Laplace Transform is shown above and has poles at -jw and +jw and a zero at the origin.
It often happens that the transform of a function f(t) is known and the transform of fa(t) = e-atf(t) is desired. The Equation below show that Fa(s) the transform of fa(t) is obtained from the transform F(s) of f(t) by replacing s with s + a. Application of this result to sinwt and coswt is shown below.  Return to main page
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Copyright 1996 © Cuthbert A. Nyack.