# 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
f_{a}(t) = e^{-at}f(t) is desired. The Equation below show that
F_{a}(s) the transform of f_{a}(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.