For integer k, the gamma function reduces to the factorial expression Г(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 → o, and such a representation is called an asymptotic expansion. For equation (7.77), the asymptotic expansion is k k A1 I(k) = (k – 1)! ~ k (7.78) 1+ ... k2 e where the symbol "" will mean "k o." In this equation, the constants A1, A2, and A3 are 1 139 A1 A2 A3 : (7.79) 12 288 51, 840 The above expansion is called the Stirling series. We now derive equation (7.78) except for an overall multiplicative con- stant.
For integer k, the gamma function reduces to the factorial expression Г(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 → o, and such a representation is called an asymptotic expansion. For equation (7.77), the asymptotic expansion is k k A1 I(k) = (k – 1)! ~ k (7.78) 1+ ... k2 e where the symbol "" will mean "k o." In this equation, the constants A1, A2, and A3 are 1 139 A1 A2 A3 : (7.79) 12 288 51, 840 The above expansion is called the Stirling series. We now derive equation (7.78) except for an overall multiplicative con- stant.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Approximating Factorials
For integer k, the gamma function reduces to the factorial expression
Г(k) — (к — 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
expansion. For equation (7.77), the asymptotic expansion is
0, and such a representation is called an asymptotic
k
k
Г() — (k — 1)!.
A1
1+
A1
A3
k2
k3
(7.78)
k
e
where the symbol "" will mean "k –→ ." In this equation, the constants
А1, А2, and Аз аre
1
A1
1
139
A2
12'
Аз
51, 840
(7.79)
288
The above expansion is called the Stirling series.
We now derive equation (7.78) except for an overall multiplicative con-
stant.
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