Logarithm of Factorial/Historical Note
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Historical Note on Logarithm of Factorial
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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials