Laplace Transform of Half Wave Rectified Sine Curve

Theorem
Consider the half wave rectified sine curve:


 * $\map f t = \begin {cases} \sin t & : 2 n \pi \le t \le \paren {2 n + 1} \pi \\ 0 & : \paren {2 n + 1} \pi \le t \le \paren {2 n + 2} \pi \end {cases}$

The Laplace transform of $\map f t$ is given by:


 * $\laptrans {\map f t} = \dfrac 1 {\paren {1 - e^{-\pi s} } \paren {s^2 + 1} }$

Proof
We have that $\map f t$ is periodic with period $2 \pi$:


 * Half-wave-rectified-sine-curve.png

Hence: