Find the inverse Laplace transform of $L(s)= \frac{s}{s^2 + 25} e^{-\pi s}$

Question

$$L(s)= \frac{s}{s^2 + 25} e^{-\pi s}$$

I never seen such function. Can exponential function appear in Laplace transform?

Help required

Answer 1

I think you were close (in the comments), but in fact, with $\;\mathcal L\{f(t)\}=F(s)\;$:

$$\mathcal L^{-1}\left\{e^{-cs}F(s)\right\}= u_c(t)f(t-c)\implies\mathcal L^{-1}\left\{e^{-\pi s}\frac s{s^2+25}\right\}=u_\pi(t) \cos(5(t-\pi))=$$

$$=-u_\pi(t)\cos 5t$$