If you are given a rectified sine wave, $$f(t)=\left|\sin\frac{t}{2}\right|$$ how do you find the Laplace transform of this?
I tried using the equation $$L\{ f(t)\} = \frac{1}{1-e^{-sT}} \int_0^T f(t) e^{-sT}\, dt$$ with period $T= 2\pi$ but I am not sure if that's correct.
If you're not sure go back to first principles:
$$L[f](s) = \int_0^\infty f(t)e^{-st} \ dt = \sum_{n=0}^\infty \int_{2n\pi}^{2(n+1)\pi} |\sin(t/2)| e^{-st} \ dt \ = \ ...$$
Can you take it from here?
Added: The last expression is equal to
$$\sum_{n=0}^\infty \int_0^{2\pi} \sin(t/2) e^{-s(t+2n\pi)} \ dt = \sum_{n=0}^\infty e^{-2n\pi s} \int_0^{2\pi} \sin(t/2) e^{-st} \ dt $$
$$ = \frac{1}{1 - e^{-2\pi s}} \int_0^{2\pi} \sin(t/2) e^{-st} \ dt$$
How about now?