Approximation to Stirling's Formula for Gamma Function
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Theorem
Let:
- $D_\epsilon = \set {z \in \C : \cmod {\Arg z} < \pi - \epsilon,\ \cmod z > 1}$
where:
- $\cmod {\Arg z}$ denotes the absolute value of the principal argument of $z$
- $\cmod z$ denotes the modulus of $z$
- $\epsilon \in \R_{>0}$.
Then for all $z \in D_\epsilon$, the gamma function of $z$ satisfies:
- $\map \Gamma z = \sqrt {\dfrac {2 \pi} z} \paren {\dfrac z e}^z \paren {1 + \map \OO {z^{-1} } }$
where $\map \OO {z^{-1} }$ denotes big-O of $z^{-1}$.
Proof
From Gamma Function is Unique Extension of Factorial:
| \(\text {(1)}: \quad\) | \(\ds \paren {y + n}^y n!\) | \(=\) | \(\ds \paren {y + n}^y \map \Gamma {n + 1}\) | Gamma Function Extends Factorial | ||||||||||
| \(\ds \) | \(\le\) | \(\ds \map \Gamma {y + n + 1}\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \paren {n + 1}^y \map \Gamma {n + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {n + 1}^y n!\) |
for $0 < y \le 1$ and $n \in \N$.
Let $x$ be given.
Let $n + 1$ be the largest natural number such that $n + 1 \le x$.
Let $x = y + n + 1$, and thus $0 < y \le 1$.
Then:
| \(\ds \frac {\map \Gamma x} {\sqrt {2 \pi} x^x x^{-1/2} e^{-x} }\) | \(\le\) | \(\ds \frac {\paren {n + 1}^y n!} {\sqrt {2 \pi} x^x x^{-1/2} e^{-x} }\) | from $(1)$ | |||||||||||
| \(\ds \) | \(\sim\) | \(\ds \frac {\paren {n + 1}^y n! \sqrt {2 \pi} n^n n^{1/2} e^{-n} } {\sqrt {2 \pi} x^x x^{-1/2} e^{-x} }\) | Stirling's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {n + 1}^y n! n^n n^{1/2} e^{-n} } {\paren {y + n + 1}^{y + n} \paren {y + n + 1}^{1/2} e^{-y - n - 1} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\frac {n + 1} {y + n + 1} }^y \paren {\frac n {y + n + 1} }^{1/2} \paren {1 + \frac {y + 1} n}^{-n} e^{y + 1}\) | ||||||||||||
| \(\ds \) | \(\to\) | \(\ds 1 \cdot 1 \cdot \frac 1 {e^{y + 1} } e^{y + 1} \text { as } n \to \infty\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
Similarly for the right hand side.
The result follows from Gamma Function Extends Factorial.
$\blacksquare$
Also see
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 17.7 \ (2)$