Editing Pi (section)

=== The gamma function and Stirling's approximation ===
[[File:Gamma plot points marked.svg|thumb|Plot of the gamma function on the real axis]]
The [[factorial]] function <math>n!</math> is the product of all of the positive integers through {{math|''n''}}. The [[gamma function]] extends the concept of [[factorial]] (normally defined only for non-negative integers) to all complex numbers, except the negative real integers, with the identity <math>\Gamma(n)=(n-1)!</math>. When the gamma function is evaluated at half-integers, the result contains {{pi}}. For example, <math> \Gamma\bigl(\tfrac12\bigr) = \sqrt{\pi} </math> and <math display="inline">\Gamma\bigl(\tfrac52\bigr) = \tfrac 34 \sqrt{\pi} </math>.<ref>{{harvnb|Bronshteĭn|Semendiaev|1971|pp=191–192}}.</ref>

The gamma function is defined by its [[Weierstrass product]] development:<ref>{{cite book |title=The Gamma Function |first=Emil |last=Artin |publisher=Holt, Rinehart and Winston |year=1964 |series=Athena series; selected topics in mathematics |edition=1st |author-link=Emil Artin}}</ref>

<math display=block>\Gamma(z) = \frac{e^{-\gamma z}}{z}\prod_{n=1}^\infty \frac{e^{z/n}}{1+z/n}</math>

where {{math|γ}} is the [[Euler–Mascheroni constant]].  Evaluated at {{tmath|1= z = \tfrac12 }} and squared, the equation {{tmath|1=\textstyle \gamma\bigl(\tfrac12\bigr)\vphantom)^2 = \pi}} reduces to the Wallis product formula.  The gamma function is also connected to the [[Riemann zeta function]] and identities for the [[functional determinant]], in which the constant {{pi}} [[#Number theory and Riemann zeta function|plays an important role]].

The gamma function is used to calculate the volume {{math|''V''<sub>''n''</sub>(''r'')}} of the [[n-ball|''n''-dimensional ball]] of radius ''r'' in Euclidean ''n''-dimensional space, and the surface area {{math|''S''<sub>''n''−1</sub>(''r'')}} of its boundary, the [[n-sphere|(''n''−1)-dimensional sphere]]:<ref>{{cite book |first=Lawrence |last=Evans |author-link=Lawrence C. Evans |title=Partial Differential Equations |publisher=AMS |year=1997 |page=615}}</ref>

<math display=block>V_n(r) = \frac{\pi^{n/2}}{\Gamma\bigl(\frac{n}{2}+1\bigr)}r^n,</math>

<math display=block>S_{n-1}(r) = \frac{n\pi^{n/2}}{\Gamma\bigl(\tfrac{n}{2}+1\bigr)}r^{n-1}.</math>

Further, it follows from the [[functional equation]] that

<math display=block>2\pi r = \frac{S_{n+1}(r)}{V_n(r)}.</math>

The gamma function can be used to create a simple approximation to the factorial function {{math|''n''!}} for large {{math|''n''}}: <math display="inline"> n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n</math> which is known as [[Stirling's approximation]].<ref>{{harvnb|Bronshteĭn|Semendiaev|1971|p=190}}.</ref>  Equivalently,

<math display=block>\pi = \lim_{n\to\infty} \frac{e^{2n}n!^2}{2 n^{2n+1}}.</math>

As a geometrical application of Stirling's approximation, let {{math|Δ<sub>''n''</sub>}} denote the [[simplex|standard simplex]] in ''n''-dimensional Euclidean space, and {{math|(''n''&nbsp;+&nbsp;1)Δ<sub>''n''</sub>}} denote the simplex having all of its sides scaled up by a factor of {{math|''n''&nbsp;+&nbsp;1}}. Then

<math display=block>\operatorname{Vol}((n+1)\Delta_n) = \frac{(n+1)^n}{n!} \sim \frac{e^{n+1}}{\sqrt{2\pi n}}.</math>

[[Ehrhart's volume conjecture]] is that this is the (optimal) upper bound on the volume of a [[convex body]] containing only one [[lattice point]].<ref>{{cite journal |author1=Benjamin Nill |author2=Andreas Paffenholz |title=On the equality case in Erhart's volume conjecture |year=2014 |arxiv=1205.1270 |journal=Advances in Geometry |volume=14 |issue=4 |pages=579–586 |issn=1615-7168 |doi=10.1515/advgeom-2014-0001 |s2cid=119125713}}</ref>