In question 1) we get Laplace transform of $$ g(t) = t^a $$ is:
$$ \hat g(t)= {1/s^{a+1}}\int_0^\infty e^{-t}x^a $$
then I was stuck at question 2) which asks me to evaluate the inverse laplace transform of $ \hat g(p) $ which is
$$ {1/2\pi i}\int_0^\infty e^{pt}\hat g(p)dp $$
I know the answer should be $ t^a $ as the inverse transform comes back to itself, but I cannot figure out how to make the contour integration. I tried to apply Cauchy's residue theorem to eliminate the $ 1/2 \pi i $ but was stuck then. Thanks a lot for help!
