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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)

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)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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ences (World Scientific, Singapore, 2004), Chapter 7. 7.2.4 Approximating Factorials For integer k, the gamma function reduces to the factorial expression T(k) (k – 1)!. (7.77) Of great interest for many calculations in applied mathematics, the sciences, and engineering is the representation of the factorial for large values of k. This corresponds to k → , and such a representation is called an asymptotic expansion. For equation (7.77), the asymptotic expansion is 2T k T(k) = (k – 1)! ~ k A1 1+ k A1 k2 Аз +.. k3 (7.78) where the symbol “n" will mean "k → oo." In this equation, the constants A1, A2, and A3 are 1 Aj = 12' 1 A2 = 288 139 A3 = (7.79) 51, 840 The above expansion is called the Stirling series. We now derive equation (7.78) except for an overall multiplicative con- stant. 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 = In k, (7.82) whose exact solution is k-1 Sk = S1 +> Inr. (7.83) r=1 Since our interest is in k →x, the dominant behavior of Sk can be calculated by approximating the sum by an integral; therefore, Sk - In t dt ~ k In k. (7.84) Consequently, ak takes the form k* bk, (7.85) ak = 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 = 1+- br. (7.86) (†+)-- Using the fact that lim (1+ h)'/h = e, (7.87) it follows that -(k+1) 1 1 lim 1+ (7.88) e and for k → ∞, we have bk+1 ~ bk. (7.89) Thus, the dominant behavior of br is given by the expression bk ~e-k. (7.90) If we now set br = e-kck (7.91) Ck;
Transcribed Image Text:ences (World Scientific, Singapore, 2004), Chapter 7. 7.2.4 Approximating Factorials For integer k, the gamma function reduces to the factorial expression T(k) (k – 1)!. (7.77) Of great interest for many calculations in applied mathematics, the sciences, and engineering is the representation of the factorial for large values of k. This corresponds to k → , and such a representation is called an asymptotic expansion. For equation (7.77), the asymptotic expansion is 2T k T(k) = (k – 1)! ~ k A1 1+ k A1 k2 Аз +.. k3 (7.78) where the symbol “n" will mean "k → oo." In this equation, the constants A1, A2, and A3 are 1 Aj = 12' 1 A2 = 288 139 A3 = (7.79) 51, 840 The above expansion is called the Stirling series. We now derive equation (7.78) except for an overall multiplicative con- stant. 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 = In k, (7.82) whose exact solution is k-1 Sk = S1 +> Inr. (7.83) r=1 Since our interest is in k →x, the dominant behavior of Sk can be calculated by approximating the sum by an integral; therefore, Sk - In t dt ~ k In k. (7.84) Consequently, ak takes the form k* bk, (7.85) ak = 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 = 1+- br. (7.86) (†+)-- Using the fact that lim (1+ h)'/h = e, (7.87) it follows that -(k+1) 1 1 lim 1+ (7.88) e and for k → ∞, we have bk+1 ~ bk. (7.89) Thus, the dominant behavior of br is given by the expression bk ~e-k. (7.90) If we now set br = e-kck (7.91) Ck;
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