Laplace transform of the multiplication of two functions

[Debdut]

I have two functions $\phi(t)=\cos(\omega t)$ and $f(t)=u(t)−u(t−k)$ with $f(t)=f(t+T)$, $u(t)$ is the unit step function.
The problem is to find Laplace transform of $\phi(t) \cdot f(t)$.

I have tried convolution in frequency domain, but unable to solve it because of gamma functions. Also a doubt is arising about the limits of the convolution. Laplace transform of multiplication of simple functions and convolution of their individual transforms are not matching when taking limits like $0 \rightarrow s$ or $s \rightarrow \infty$ or $0 \rightarrow \infty$.

I have also thought about integration by parts of $\phi(t)f(t)e^{−st}$ as its limits are known: $0 \rightarrow \infty$. But value of a periodic function at infinity is undefined.

I am stuck, please help...

<Moderator's note: Member has been warned not to remove template.>

 

[Ray Vickson]

I have two functions $\phi(t)=\cos(\omega t)$ and $f(t)=u(t)−u(t−k)$ with $f(t)=f(t+T)$, $u(t)$ is the unit step function.
The problem is to find Laplace transform of $\phi(t) \cdot f(t)$.

I have tried convolution in frequency domain, but unable to solve it because of gamma functions. Also a doubt is arising about the limits of the convolution. Laplace transform of multiplication of simple functions and convolution of their individual transforms are not matching when taking limits like $0 \rightarrow s$ or $s \rightarrow \infty$ or $0 \rightarrow \infty$.

I have also thought about integration by parts of $\phi(t)f(t)e^{−st}$ as its limits are known: $0 \rightarrow \infty$. But value of a periodic function at infinity is undefined.

I am stuck, please help...

<Moderator's note: Member has been warned not to remove template.>

Assuming that $T > k$ the problem is do-able. Here is one way to do it.

Your transform has the form
$$ \sum_{n=0}^{\infty} \int_0^k e^{-s(t+nT)} \cos(wt+nT)) \, dt $$
Expand the $\cos(wt +n w T)$ and then do the integrations.

 

[Debdut]

Thank you very much sir.
I must say that I don't fully understand the above expression. The only formulae that I have got are from this site:
http://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf
Could you please offer some explanation or redirect me to any reference where I can learn it more?

 

[Ray Vickson]

Thank you very much sir.
I must say that I don't fully understand the above expression. The only formulae that I have got are from this site:
http://tutorial.math.lamar.edu/pdf/Laplace_Table.pdf
Could you please offer some explanation or redirect me to any reference where I can learn it more?

Just apply the definition of the Laplace transform: $\int_0^{\infty} e^{-st} F(t) \, dt$. Now use the fact that $F(t)$ is the product of a periodic (period $T$) "rectangular" function of width $k$ and $\cos(wt)$. Express the integrals over $T < t < 2T, 2T < t < 3T, \ldots$ as integrals over $0 < t < T$ with shifted values of $t$.

Reading about Laplace transforms won't help much; just go back to the start and apply the definition.

 

[Debdut]

OK, thank you again.