Find the Laplace transform?

Question

Find the Laplace transform of f(t)=cos at, where a is a real constant.

Answer: F(s)=s/(s^2+a^2), s>0

 

Answer 1

 

Let's  Lf(t) be the Laplace transform of cos at (this is like cos-Furie trabsform of the fucntion e ^-st).

                                      Lf(t) = ∫₀,^∞ cos at e ^-stdt    (1)

If you integrate twice in parts, the same integral (1) will come out again. You can make sure that this integral satsfies to the equation

                                  Lf(t)= (1/s) - (a²/s²) Lf(t)                (2)

Solve the eqautio for Lf(t), and you will see that

                                                 Lf(t) = s/(s² + a²)

If you will have problems with integration, let me know.

                              

Answer 2

Hello Sun,

The Laplace transform of a function f(t) defined for t≥0 is

£{f(t)} = ∫₀^∞ e^-st f(t) dt,

provided of course that the above converges.

In your case, f(t) = cos(at), so you need to evaluate

£{cos(at)} = ∫₀^∞ e^-st cos(at) dt.

This is an integration-by-parts (twice) type integral, where you name the integral I, for instance, integrate by parts twice, and find I in the resulting second integration, whereafter you solve algebraically for I. 

The restriction on s (namely, s>0) comes from being able to evaluate to zero the limits of integration for the exponential function.

I hope this helps.  Someone else will probably come by and solve the integration for you later.

Regards,

Hassan H.