Editing Gamma function (section)

=== 18th century: Euler and Stirling ===
[[File:DanielBernoulliLettreAGoldbach-1729-10-06.jpg|thumb|[[Daniel Bernoulli]]'s letter to [[Christian Goldbach]], October 6, 1729]]

The problem of extending the factorial to non-integer arguments was apparently first considered by [[Daniel Bernoulli]] and [[Christian Goldbach]] in the 1720s. In particular, in a letter from Bernoulli to Goldbach dated 6 October 1729 Bernoulli introduced the product representation<ref>{{cite web |url=https://www.luschny.de/math/factorial/history.html |title=Interpolating the natural factorial n! or The birth of the real factorial function (1729 - 1826) }}</ref>
<math display="block">x!=\lim_{n\to\infty}\left(n+1+\frac{x}{2}\right)^{x-1}\prod_{k=1}^{n}\frac{k+1}{k+x}</math>
which is well defined for real values of {{math|''x''}} other than the negative integers.

[[Leonhard Euler]] later gave two different definitions: the first was not his integral but an [[infinite product]] that is well defined for all complex numbers {{math|''n''}} other than the negative integers,
<math display="block">n! = \prod_{k=1}^\infty \frac{\left(1+\frac{1}{k}\right)^n}{1+\frac{n}{k}}\,,</math>
of which he informed Goldbach in a letter dated 13 October 1729. He wrote to Goldbach again on 8 January 1730, to announce his discovery of the integral representation
<math display="block">n!=\int_0^1 (-\log s)^n\, ds\,,</math>
which is valid when the real part of the complex number {{math|''n''}} is strictly greater than {{math|−1}} (i.e., <math>\Re (n) > -1</math>). By the change of variables {{math|1=''t'' = −ln ''s''}}, this becomes the familiar Euler integral. Euler published his results in the paper "De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt" ("On transcendental progressions, that is, those whose general terms cannot be given algebraically"), submitted to the [[St. Petersburg Academy]] on 28 November 1729.<ref>Euler's paper was published in ''Commentarii academiae scientiarum Petropolitanae'' 5, 1738, 36–57. See [http://math.dartmouth.edu/~euler/pages/E019.html E19 -- De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt], from The Euler Archive, which includes a scanned copy of the original article.</ref> Euler further discovered some of the gamma function's important functional properties, including the reflection formula.

[[James Stirling (mathematician)|James Stirling]], a contemporary of Euler, also attempted to find a continuous expression for the factorial and came up with what is now known as [[Stirling's formula]]. Although Stirling's formula gives a good estimate of {{math|''n''!}}, also for non-integers, it does not provide the exact value. Extensions of his formula that correct the error were given by Stirling himself and by [[Jacques Philippe Marie Binet]].