Stirling's Formula for Gamma Function

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Theorem

Let $\Gamma$ denote the Gamma function.

Let $z \in \C$ with a strictly positive real part and $\size {\arg z} < \dfrac \pi 2$.

Then:

$\map \Gamma {z + 1} = \sqrt {2 \pi z} \, z^z e^{-z} \paren {1 + \dfrac 1 {12 z} + \dfrac 1 {288 z^2} - \dfrac {139} {51 \, 840 z^3} - \dfrac {571} {2\, 488 \, 320 z^4} + \dfrac {163 \, 879} {209\, 018 \, 880 z^5} + \dfrac {5\, 246\, 819} {75\, 246\, 796\, 800 z^6} + \cdots}$


Proof

From Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function, we have:

$\ds \map \Ln {\map \Gamma z} = \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \sum_{n \mathop = 1}^d \frac {B_{2 n} } {2 n \paren {2 n - 1} z^{2 n - 1} } + \OO \paren {\dfrac 1 {z^{2 d + 1} } }$

Taking the first $3$ terms in the sum, we get:

\(\ds \Ln \map \Gamma z\) \(=\) \(\ds \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \sum_{n \mathop = 1}^d \frac {B_{2 n} } {2 n \paren {2 n - 1} z^{2 n - 1} } + \OO \paren {\dfrac 1 {z^{2 d + 1} } }\) Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function
\(\ds \) \(=\) \(\ds \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \frac {B_2} {2 \times 1 \times z} + \frac {B_4} {4 \times 3 \times z^3}+ \frac {B_6} {6 \times 5 \times z^5}\) stopping after $3$ terms
\(\ds \) \(=\) \(\ds \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \dfrac {\dfrac 1 6} {2 \times 1 \times z} + \dfrac {-\dfrac 1 {30} } {4 \times 3 \times z^3}+ \dfrac {\dfrac 1 {42} } {6 \times 5 \times z^5}\) Definition of Bernoulli Numbers
\(\ds \) \(=\) \(\ds \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5}\)


Exponentiating both sides, we obtain:

\(\ds \map \exp {\Ln \map \Gamma z }\) \(=\) \(\ds \map \exp {\paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5} }\)
\(\ds \leadsto \ \ \) \(\ds \map \Gamma z\) \(=\) \(\ds \map \exp {\paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5} }\) Exponential Function is Inverse of Natural Logarithm
\(\ds \) \(=\) \(\ds \map \exp {\paren {\map \Ln z }^{\paren {z - \frac 1 2} } - z + \paren {\ln 2 \pi}^{\frac 1 2} + \dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5} }\) Logarithm of Power - Natural Logarithm
\(\ds \) \(=\) \(\ds z^{\paren {z - \frac 1 2} } \times e^{-z} \times \sqrt {2 \pi} \times \map \exp {\dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5} }\) Exponential Function is Inverse of Natural Logarithm, Product of Powers
\(\ds \) \(=\) \(\ds z^{\paren {z - \frac 1 2} } \times e^{-z} \times \sqrt {2 \pi} \times \paren {1 + \paren {\dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5} } + \dfrac {\paren {\dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5} }^2} {2!} }\) Power Series of Exponential Function
\(\ds \) \(\) \(\, \ds + \, \) \(\ds z^{\paren {z - \frac 1 2} } \times e^{-z} \times \sqrt {2 \pi} \times \paren {\dfrac {\paren {\dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5} }^3} {3!} + \dfrac {\paren {\dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5} }^4} {4!} }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds z^{\paren {z - \frac 1 2} } \times e^{-z} \times \sqrt {2 \pi} \times \paren {\dfrac {\paren {\dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5} }^5} {5!} + \dfrac {\paren {\dfrac 1 {12 z} - \dfrac 1 {360 z^3} + \dfrac 1 {1260 z^5} }^6} {6!} + \cdots}\)
\(\ds \) \(=\) \(\ds z^{\paren {z - \frac 1 2} } \times e^{-z} \times \sqrt {2 \pi} \times \leftparen {1 + \dfrac 1 {12 z} + \dfrac {\paren {\dfrac 1 {12 z} }^2} {2!} + \paren {-\dfrac 1 {360 z^3} + \dfrac {\paren {\dfrac 1 {12 z} }^3} {3!} } + }\) grouping terms
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {\dfrac {2 \paren {\dfrac 1 {12 z} } \paren {- \dfrac 1 {360 z^3} } } {2!} + \dfrac {\paren {\dfrac 1 {12 z} }^4} {4!} } + \paren {\dfrac 1 {1260 z^5} + \dfrac {3 \paren {\dfrac 1 {12 z} }^2 \paren {- \dfrac 1 {360 z^3} } } {3!} + \dfrac {\paren {\dfrac 1 {12 z} }^5} {5!} } +\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \rightparen {\paren {\dfrac {2 \paren {\dfrac 1 {12 z} } \paren {\dfrac 1 {1260 z^5} } + \paren {-\dfrac 1 {360 z^3} }^2 } {2!} + \dfrac {4 \paren {\dfrac 1 {12 z} }^3 \paren {-\dfrac 1 {360 z^3} } } {4!} + \dfrac {\paren {\dfrac 1 {12 z} }^6} {6!} } + \cdots }\)
\(\ds \) \(=\) \(\ds z^{\paren {z - \frac 1 2} } \times e^{-z} \times \sqrt {2 \pi} \times \leftparen {1 + \dfrac 1 {12 z} + \dfrac 1 {2! \times \paren {12 z}^2} + \dfrac {-12^2 + 5} {3! \times 5 \times \paren {12 z}^3} + \dfrac {-4 \times 12^2 + 5} {4! \times 5 \times \paren {12 z}^4} + }\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \rightparen {\dfrac {8 \times 12^4 -14 \times 12^2 + 7} {5! \times 7 \times \paren {12 z}^5} + \dfrac {254 \times 12^4 - 140 \times 12^2 + 35} {6! \times 5 \times 7 \times \paren {12 z}^6} + \cdots }\)
\(\ds \) \(=\) \(\ds z^{\paren {z - \frac 1 2} } \times e^{-z} \times \sqrt {2 \pi} \times \paren {1 + \dfrac 1 {12 z} + \dfrac 1 {288 z^2} - \dfrac {139} {51 \, 840 z^3} - \dfrac {571} {2\, 488 \, 320 z^4} + \dfrac {163 \, 879} {209\, 018 \, 880 z^5} + \dfrac {5\, 246\, 819} {75\, 246\, 796\, 800 z^6} + \cdots }\)
\(\ds \leadsto \ \ \) \(\ds z \map \Gamma z\) \(=\) \(\ds z \times z^{\paren {z - \frac 1 2} } \times e^{-z} \times \sqrt {2 \pi} \times \paren {1 + \dfrac 1 {12 z} + \dfrac 1 {288 z^2} - \dfrac {139} {51 \, 840 z^3} - \dfrac {571} {2\, 488 \, 320 z^4} + \dfrac {163 \, 879} {209\, 018 \, 880 z^5} + \dfrac {5\, 246\, 819} {75\, 246\, 796\, 800 z^6} + \cdots }\) multiplying both sides by z
\(\ds \leadsto \ \ \) \(\ds \map \Gamma {z + 1}\) \(=\) \(\ds z^{\paren {z + \frac 1 2} } \times e^{-z} \times \sqrt {2 \pi} \times \paren {1 + \dfrac 1 {12 z} + \dfrac 1 {288 z^2} - \dfrac {139} {51 \, 840 z^3} - \dfrac {571} {2\, 488 \, 320 z^4} + \dfrac {163 \, 879} {209\, 018 \, 880 z^5} + \dfrac {5\, 246\, 819} {75\, 246\, 796\, 800 z^6} + \cdots }\) Gamma Difference Equation, Product of Powers
\(\ds \) \(=\) \(\ds z^z \times e^{-z} \times \sqrt {2 \pi z} \times \paren {1 + \dfrac 1 {12 z} + \dfrac 1 {288 z^2} - \dfrac {139} {51 \, 840 z^3} - \dfrac {571} {2\, 488 \, 320 z^4} + \dfrac {163 \, 879} {209\, 018 \, 880 z^5} + \dfrac {5\, 246\, 819} {75\, 246\, 796\, 800 z^6} + \cdots }\)

$\blacksquare$


Also known as

Stirling's Formula for Gamma Function is also known as Stirling's asymptotic series.


Examples

First $10$ Integers

Stirling.png



Also see


Source of Name

This entry was named for James Stirling.


Sources

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