Laplace Transform f(t)=tcos(t)

Use the complex form of the cosine.

 

I get [tex] \mathcal{L} [tcost] = \frac {1}{2} ( \frac {1}{(i-s)^{2}} + \frac {1}{(i+s)^{2}}) [/tex]

Hopefully that simplifies to [tex] \frac {s^{2}-1}{(s^{2}+1)^{2}} [/tex]

 

Even if you use the complex form of cos(t), you still have to integrate by parts (albeit it's rather simple).

[tex] \frac {1}{2} \int^{ \infty}_{0} e^{-st}t \frac {1}{2}(e^{it} + e^{-it})dt [/tex]

[tex] = \frac {1}{2} \int^{ \infty}_{0} te^{(i-s)t}dt + \frac {1}{2} \int^{ \infty}_{0} te^{-(i+s)t}dt [/tex]

 

You don't need to integrate by parts at all. Just compute the Laplace transform of cos(t). Differentiating with respect to s will then bring down a factor of minus t in the Laplace integral.

 

Cool shortcut. I've never seen it before.

[tex] \mathcal{L} [cos(t)] = \int^{ \infty}_{0} e^{-st}cos(t)dt [/tex]

[tex] \frac {d}{ds} \mathcal{L} [cos(t)] = \frac {d}{ds} \int^{ \infty}_{0} e^{-st}cos(t)dt [/tex]

Assuming it's okay to bring the derivative inside the integral,

[tex] = \int^{ \infty}_{0} \frac {d}{ds} e^{-st}tcos(t)dt [/tex]

[tex] = -\int^{ \infty}_{0}te^{-st}cos(t)dt [/tex]

[tex] = -\int^{ \infty}_{0}e^{-st}tcos(t)dt = - \mathcal{L} [tcos(t)] [/tex]

 

then it would appear that [tex] \frac {d^{2}}{ds^{2}} \mathcal{L} [cos(t)] = \mathcal{L} [t^{2}cos(t)] [/tex]

 

Another trick: To obtain the Laplace transform of, say, sin(t)/t you can compute the Laplace transform of sin(t) and then integrate w.r.t. s from p to infinity. The Laplace transform is then a function of the parameter p. If you put p = 0, you obtain the integral of sin(t)/t from zero to infinity. What's interesing about this is that the antiderivative of sin(t)/t cannot be evaluated in closed form.

 

I have to see for myself.

[tex] \mathcal{L} [sin(t)] = \int^{ \infty}_{0} e^{-st}sin(t)dt [/tex]

[tex] \int^{ \infty}_{p} \mathcal{L} [sin(t)] = \int^{ \infty}_{p} \int^{ \infty}_{0} e^{-st}sin(t)dtds [/tex]

Changing the order of integration (which I assume is allowed),

[tex] = \int^{ \infty}_{0} \int^{ \infty}_{p} e^{-st}sin(t)dsdt [/tex]

[tex] = \int^{ \infty}_{0} \frac {1}{t}e^{-pt}sin(t)dt [/tex]

[tex] = \int^{ \infty}_{0} e^{-pt} \frac {sin(t)}{t}dt = \mathcal{L} [ \frac {sin(t)}{t}] [/tex]

According to the table, [tex] \mathcal{L} [ \frac {sin(t)}{t}] = arctan( \frac {1}{p}) [/tex]

Should you then take the limit of [tex] arctan( \frac {1}{p}) [/tex] as p goes to zero?

[tex] \int^{ \infty}_{0} \frac {sin(t)}{t}dt = \lim_{p \to 0} arctan( \frac {1}{p}) = \frac {\pi}{2} [/tex] ?

 

You can derive that the Laplace transform of sin(t)/t is arctan(1/p) by integrating the Laplace transform of sin(t). If you integrate
1/(s^2+1) from p to infinity, you get pi/2 - arctan(p).