Solved: The Gamma Function. The gamma function is denoted by (p) and is defined by

Chapter 6, Problem 30

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The Gamma Function. The gamma function is denoted by (p) and is defined by theintegral(p + 1) = 0exxp dx. (i)The integral converges as x for all p. For p < 0 it is also improper at x = 0,because the integrand becomes unbounded as x 0. However, the integral can be shownto converge at x = 0 for 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 extensionof the factorial function to nonintegral values of the independent variable. Note that it isalso consistent to define 0! = 1.(d) Show that, for p > 0,p(p + 1)(p + 2)(p + n 1) = (p + n)/ (p).Thus (p) can be determined for all positive values of p if (p) is known in a single intervalof unit lengthsay, 0 < p 1. It is possible to show that 12= . Find 32and 112.