Advanced Calculus fi..

Advanced Calculus, Fifth EditionIn general, an integration by parts (Problem 10) shows thatAccordingly, we can apply induction to conclude thatFor s = 1, (6.81) gives00k! = L tke-' dt.This suggests a method of generalizing the factorial; that is, we could use Eq. (6.82)to define k! for k an arbitrary real number greater than - 1. It is customary to denotethis generalized factorial by r(k f 1); the Gamma function r(k) is then defined bythe equationwhen k is a positive integer or 0,The general integral (6.79) is expressible in terms of the Gamma function(Problem 14):Equation (6.80) then states thatthat is,This is thefunctional equation of the Gammafunction. The functional equation canbe used to define r(k) for negative k; thus we writein order to define r(- 4). r(- 3). . . . in terms of the known value of r( f ). Thisprocedure fails only for k = - 1, -2, . . . . In fact, we can show (Problem 13) that1-lim r (k) = +oo,k+O+