Gamma function

1. In class we talked about the Gamma function T(p + 1) := / exx dx which is defined for p > -1. (a) (10 points) Show that for positive p, I(p + 1) = pl(p). (b) (10 points) Show that I(1) = 1. 2. Below are given the Laplace transforms F(s) for different functions. Find f(t) such that C[f(t)] = F(s). You may need to use partial fractions or complete the square. (10 points each) (a) s( $2 + 4) (b) S $2 + 2s +6 3. (15 points) Let F (s) = est f (t ) at = [If (t ) ]. Show that F' (s) = [[-tf(t)]. For a bonus five points, also show that for any positive integer n, we have that F() (s) = [[(-t)" f(t)]. You may assume without proof that differentiation across the integral is allowed. 4. (10 points) Using the results of the previous problem, calculate [[theat]. Do NOT use the integral definition of the Laplace transform. 5. (15 points) Solve the initial-value problem Sy" + 2y' + 5y =3e-20 y(0) = 1, y'(0) = 1 using the Laplace transform