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^-atf(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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