APPLICATIONS 225 To begin, we note that the gamma function is the solution to the difference equation Ak+1 = kak. (7.80) This follows directly from the results of Section 2.4. To determine the dom- inant or leading behavior of the solution to equation (7.80), we rewrite it in the form ak = eSk (7.81) Substitution of equation (7.81) into (7.80) gives Sk+1 - Sk = ln k, (7.82) whose exact solution is k-1 Sk = S1 + Σhr. (7.83) r=1 Since our interest is in k → ∞, the dominant behavior of Sk can be calculated by approximating the sum by an integral; therefore, In t dt ~ k In k. (7.84) Consequently, ar takes the form ak = k*bk, (7.85) where br is an unknown function whose dominant behavior will now be de- termined. To do this, we substitute equation (7.85) into equation (7.80) and obtain -(k+1) (1+ ) bk+1 = (7.86) Using the fact that lim (1+ h)'/h = e, (7.87) it follows that -(k+1) 1 (1+) lim (7.88) and for k → ∞, we have (). bk+1 ~ (7.89) Thus, the dominant behavior of br is given by the expression br ~ e-k (7.90) If we now set br = e¬kCk; (7.91)