What is mathcal L^-1\(s e^-π s)/(s^2+1)\?

Exercise · ZW-UNIT-STEP-FUNCTION-010 · ★★★ Advanced

What is \(\mathcal L^{-1}\left\{\dfrac{s\,e^{-\pi s}}{s^2+1}\right\}\)?

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The result is \(u_\pi(t)\cos(t-\pi)\), and \(\cos(t-\pi)=-\cos t\). A common mistake is forgetting that the argument is shifted.

The unit step function \(u_c(t)\), with \(c\ge0\), equals 0 while \(t<c\) and equals 1 from \(t=c\) on; it is a switch that turns on at time \(c\).

\[u_c(t)=\begin{cases}0,&t