Simplifying Laplace Transform of Cosine with Angular Frequency and Phase Shift

In summary, the task is to find the Laplace transform of ##f(x) = cos(\omega t + \phi)##. One approach is to use the t-shift relation, but it is applied incorrectly in this context. The correct approach is to use the definition of the Laplace transform and set a new variable, u, to account for the shift in the argument of the cosine function. This results in the Laplace transform of a piecewise function, which can be simplified if ##\phi/\omega > 0##.
  • #1
dRic2
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Homework Statement


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

Homework Equations


.

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...
 
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  • #2
You are applying the time-shifting property incorrectly.

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

If [itex]\phi/\omega > 0[/itex] I can make the right hand side into [itex]e^{s\phi/\omega}[/itex] times the laplace transform of something by adding [tex] 0 = e^{s\phi/\omega}\int_0^{\phi/\omega} 0e^{-su}\,du[/tex] to both sides, which leaves me with the laplace transform of [tex]g(t) = \begin{cases} 0 & 0 \leq t < \frac{\phi}{\omega} \\
\cos(\omega t) & t \geq \frac{\phi}{\omega}\end{cases}[/tex]
 
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  • #3
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".
 

Related to Simplifying Laplace Transform of Cosine with Angular Frequency and Phase Shift

What is the Laplace transform of cosine with angular frequency and phase shift?

The Laplace transform of cosine with angular frequency and phase shift is a mathematical operation that converts a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations and analyze systems.

How do you simplify the Laplace transform of cosine with angular frequency and phase shift?

To simplify the Laplace transform of cosine with angular frequency and phase shift, you can use the following formula:
L{cos(ωt + φ)} = s/(s^2 + ω^2) + (φs + ω)/(s^2 + ω^2). This formula can be derived from the definition of the Laplace transform and trigonometric identities.

What is the significance of the angular frequency and phase shift in the Laplace transform of cosine?

The angular frequency and phase shift in the Laplace transform of cosine represent the frequency and phase of the original function in the complex frequency domain. They provide important information about the behavior and characteristics of the system being analyzed.

Can the Laplace transform of cosine with angular frequency and phase shift be used to solve differential equations?

Yes, the Laplace transform of cosine with angular frequency and phase shift can be used to solve differential equations. By converting the differential equation into an algebraic equation in the complex frequency domain, it becomes easier to solve and analyze the system.

Are there any limitations to using the Laplace transform of cosine with angular frequency and phase shift?

One limitation of using the Laplace transform of cosine with angular frequency and phase shift is that it can only be applied to functions that are defined for all values of time. It also assumes that the system being analyzed is linear and time-invariant.

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