The Gamma Function. The gamma function is denoted by ?(

The Gamma Function. The gamma function is denoted by ?( p) and defined by the integral ?( p + 1) = ? ? 0 e?x xp dx. (i) The integral converges for all p > ?1. (a) Show that, for p > 0, ?(p + 1) = p?(p). (b) Show that ?(1) = 1. (c) If p is a positive integer n, show that ?(n + 1) = n!. Since ?( p) is also defined when p is not an integer, this function provides an extension of the factorial function to nonintegral values of the independent variable. Note that it is also consistent to define 0! = 1. (d) Show that, for p > 0, p(p + 1) ? (p + n ? 1) = ?(p + n) ?(p) . Thus ?( p) can be determined for all positive values of p if ?(p) is known in a single interval of unit length, say, 0 < p ? 1. It is possible to show that ?(1?2) = ??. Find ?(3?2) and ?(11?2).