Author:
Marichal Jean-Luc,Zenaïdi Naïm
Abstract
AbstractThe asymptotic behavior of the gamma function for large values of its argument can be summarized as follows: for any a ≥ 0, we have the following asymptotic equivalences (see Titchmarsh (The Theory of Functions, 2nd edn. Oxford University Press, Oxford, 1939, Section 1.87))
$$\displaystyle \begin{aligned} \Gamma (x+a) ~\sim ~ x^a\,\Gamma (x) \mbox{as }x\to \infty {\,},\end{aligned} $$
Γ
(
x
+
a
)
∼
x
a
Γ
(
x
)
as
x
→
∞
,
$$\displaystyle \begin{aligned}\Gamma (x) ~\sim ~ \sqrt {2\pi }{\,}e^{-x}x^{x-\frac {1}{2}} \mbox{as }x\to \infty {\,},\end{aligned} $$
Γ
(
x
)
∼
2
π
e
−
x
x
x
−
1
2
as
x
→
∞
,
$$\displaystyle \begin{aligned}\Gamma (x+1) ~\sim ~ \sqrt {2\pi x}{\,}e^{-x}x^x \mbox{as }x\to \infty {\,}, \end{aligned} $$
Γ
(
x
+
1
)
∼
2
π
x
e
−
x
x
x
as
x
→
∞
,
where both formulas (6.2) and (6.3) are known by the name Stirling’s formula.
Publisher
Springer International Publishing
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