Editing Gamma function (section)

==== Euler's definition as an infinite product ====
<!-- Linked to from [[Binomial coefficient]] -->For a fixed integer <math>m</math>, as the integer <math>n</math> increases, we have that<ref>{{Cite journal |last=Davis |first=Philip |title=Leonhard Euler's Integral: A Historical Profile of the Gamma Function |url=https://ia800108.us.archive.org/view_archive.php?archive=/24/items/wikipedia-scholarly-sources-corpus/10.2307%252F2287541.zip&file=10.2307%252F2309786.pdf |website=maa.org}}</ref>
<math display="block">\lim_{n \to \infty} \frac{n! \, \left(n+1\right)^m}{(n+m)!} = 1\,.</math>

If <math>m</math> is not an integer, then this equation is meaningless, since in this section the factorial of a non-integer has not been defined yet. However, let us assume that this equation continues to hold when <math>m</math> is replaced by an arbitrary complex number <math>z</math>, in order to define the Gamma function for non-integers:

<math display="block">\lim_{n \to \infty} \frac{n! \, \left(n+1\right)^z}{(n+z)!} = 1\,.</math>
Multiplying both sides by <math>(z-1)!</math> gives
<math display="block">\begin{align}
(z-1)!
      &= \frac{1}{z} \lim_{n \to \infty} n!\frac{z!}{(n+z)!} (n+1)^z \\[8pt]
      &= \frac{1}{z} \lim_{n \to \infty} (1 \cdot2\cdots n)\frac{1}{(1+z) \cdots (n+z)} \left(\frac{2}{1} \cdot \frac{3}{2} \cdots \frac{n+1}{n}\right)^z \\[8pt]
      &= \frac{1}{z} \prod_{n=1}^\infty \left[ \frac{1}{1+\frac{z}{n}} \left(1 + \frac{1}{n}\right)^z \right].
\end{align}</math>This [[infinite product]], which is due to Euler,<ref>{{Cite journal |last=Bonvini |first=Marco |date=October 9, 2010 |title=The Gamma function |url=https://www.roma1.infn.it/~bonvini/math/Marco_Bonvini__Gamma_function.pdf |journal=Roma1.infn.it}}</ref> converges for all complex numbers <math>z</math> except the non-positive integers, which fail because of a division by zero. In fact, the above assumption produces a unique definition of <math>\Gamma(z)</math> as {{tmath|(z-1)!}}.

Intuitively, this formula indicates that <math>\Gamma(z)</math> is approximately the result of computing <math>\Gamma(n+1)=n!</math> for some large integer <math>n</math>, multiplying by <math>(n+1)^z</math> to approximate <math>\Gamma(n+z+1)</math>, and then using the relationship <math>\Gamma(x+1) = x \Gamma(x)</math> backwards <math>n+1</math> times to get an approximation for <math>\Gamma(z)</math>; and furthermore that this approximation becomes exact as <math>n</math> increases to infinity.

The infinite product for the [[reciprocal gamma function|reciprocal]]
<math display="block">\frac{1}{\Gamma(z)} = z \prod_{n=1}^\infty \left[ \left(1+\frac{z}{n}\right) / {\left(1 + \frac{1}{n}\right)^z} \right]</math>
is an [[entire function]], converging for every complex number {{mvar|z}}.