Solved: The Gamma Function. The gamma function is denoted by (p) and is defined by
Chapter 6, Problem 30(choose chapter or problem)
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.
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