Simplifying Laplace Transform of Cosine with Angular Frequency and Phase Shift

[dRic2]

Homework Statement

I have to find the L-transform of $f(x) = cos(\omega t + \phi)$

Homework Equations

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The Attempt at a Solution

The straightforward approach is to write $cos(\omega t + \phi)$ as $cos(\omega t)cos(\phi) - sin(\omega t)sin(\phi)$ and it becomes: $$Lf(s) = \frac {s cos(\phi) - \omega sin(\phi)} {s^2 + \omega ^2}$$.

But can I try this other way ?
$cos ( \omega t + \phi ) = cos \left[ \omega \left( t - \left( - \frac { \phi} { \omega} \right) \right) \right]$ and now I can use the t-shift relation to get: $$ Lf(s) = e^{- \left( - \frac {\phi} {\omega} \right) s} L(cos(\omega t)) = e^{ \frac {\phi} {\omega} s} \frac {s} {s^2 + \omega ^2}$$

I don't know if there is a way to simplify my last result or if it is wrong...

 

[pasmith]

You are applying the time-shifting property incorrectly.

Going back to the definition, $$F(s) = \int_0^\infty e^{-st} \cos(\omega t + \phi)\,dt.$$ If I set $u = t + \frac{\phi}{\omega}$ I get $$F(s) = e^{s\phi/\omega}\int_{\phi/\omega}^\infty e^{-su} \cos(\omega u)\,du.$$ The expression on the right hand side is not $e^{s\phi/\omega}$ times the laplace transform of $\cos(\omega t)$ because the lower limit of the integral is no longer zero.

If $\phi/\omega > 0$ I can make the right hand side into $e^{s\phi/\omega}$ times the laplace transform of something by adding $$0 = e^{s\phi/\omega}\int_0^{\phi/\omega} 0e^{-su}\,du$$ to both sides, which leaves me with the laplace transform of $$g(t) = \begin{cases} 0 & 0 \leq t < \frac{\phi}{\omega} \\ \cos(\omega t) & t \geq \frac{\phi}{\omega}\end{cases}$$

 

[dRic2]

Oh, I see, Thank you! In fact now that I pay attention to it, in the book I'm reading the author takes care of the lower limit of the integral not being zero by writing the t-shift like this:
$$L(f(t-t_0)u(t-t_0)) (s) = e^{-t_0s} L(f)(s)$$
where $u(t)$ is the Heaviside step function. Of course here I can not use this property because I can not shift the "starting point" of my function with such a "trick".