Question

Find Laplace transform F(p) for function f(t)=(t^n)sh(at)/n!

Answer 1

$$\begin{array}{l} \frac{t^n}{n!} sh(at) = \frac{1}{2} \left( \frac{t^n}{n!} e^{at} - \frac{t^n}{n!} e^{-at} \right) \rightarrow \\ \rightarrow \frac{1}{2} \left[ \frac{1}{(p - a)^{n+1}} - \frac{1}{(p + a)^{n+1}} \right] = \frac{(p + a)^{n+1} - (p - a)^{n+1}}{2(p^2 - a^2)^{n+1}} \end{array}$$