Inverse Laplace transformation of $\sinh(ks)$

Question

What is the analytic solution to the following inverse Laplace transformation:

$$ f(t)=L^{-1}{\Big\{ {\sinh(ks)} \Big\}} $$

where $k$ is a constant.

Answer 1

Consider a function defined by $$ f(t) = c_{1} \, \delta(t + a) + c_{2} \, \delta(t - a). $$ Take the Laplace transform of this function to obtain $$ f(t) \doteqdot c_{1} \, e^{-a \, s} \, H(a) + c_{2} \, e^{a s} \, H(-a), $$ where $H(x)$ is the Heaviside step function. With care of the conditions of the Heaviside function one will find that $$\text{L}^{-1}\{ \sinh(a s) \} = \frac{\delta(t + a) - \delta(t -a)}{2}$$ and if $ a$ and $t$ are positive $$\text{L}^{-1}\{ \sinh(a s) \} = - \frac{\delta(t -a)}{2}.$$