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.

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Copyright 1996 Cuthbert A. Nyack.