4. The Gamma function I(p) is defined by T(p) = ex-l dx. We will assume this improper integral converges for p > -1. (a) Show that, for p > 0, T(p + 1) = pl(p). (b) Show that I(1) = 1, and hence, that T(n + 1) = n! (c) Show that the Laplace transform of tP for p > -1 is given by [{tP } (s ) = [(p + 1) SP+1 for s > 0. (d) By using the fact that _ exp(-x?) dx = VT show that { {t- 1/ 2 } = 1/ and C{t1/21 = Vm 293/2 for s > 0