Logarithm of Factorial

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Theorem

\(\ds \ln n!\) \(=\) \(\ds \sum_{k \mathop \ge 0} a_{k + 1} n^{\underline k}\)
\(\ds \) \(=\) \(\ds a_1 n + a_2 n \paren {n - 1} + a_3 n \paren {n - 1} \paren {n - 2} + \cdots\)

where:

$\ds a_m = \dfrac {\paren {-1}^m} {m!} \sum_{k \mathop \ge 1} \paren {-1}^k \binom {m - 1} {k - 1} \ln k$


Proof




Historical Note

The sequence of $a_1, a_2, \ldots$ was established by James Stirling during the course of his attempt extend the factorial to the real numbers.

However, although he established that $\paren {\dfrac 1 2}! = \dfrac {\sqrt \pi} 2$, he was not able to prove that this sum defined $n!$ for a general non-integer $n$.


It was Charles Hermite who finally proved in $1900$ that this formula does indeed define $n!$, by demonstrating that it is identical to the Euler form of the Gamma function.


Sources