Editing Gamma function (section)

== Motivation ==
[[File:Generalized factorial function more infos.svg|thumb|250px|<math>\Gamma(x+1)</math> interpolates the factorial function to non-integer values.]]
<!--Can anyone change these into <math></math>?-->
The gamma function can be seen as a solution to the [[interpolation]] problem of finding a [[smooth curve]] <math>y=f(x)</math> that connects the points of the factorial sequence: <math>(x,y) = (n, n!) </math> for all positive integer values of <math>n</math>. The simple formula for the factorial, {{math|1=''x''! = 1 × 2 × ⋯ × ''x''}} is only valid when {{mvar|x}} is a positive integer, and no [[elementary function]] has this property, but a good solution is the gamma function <math>f(x) = \Gamma(x+1) </math>.<ref name="Davis" />

The gamma function is not only smooth but [[analytic function|analytic]] (except at the non-positive integers), and it can be defined in several explicit ways. However, it is not the only analytic function that extends the factorial, as one may add any analytic function that is zero on the positive integers, such as <math>k\sin(m\pi x)</math> for an integer <math>m</math>.<ref name="Davis" /> Such a function is known as a [[pseudogamma function]], the most famous being the [[Hadamard's gamma function|Hadamard]] function.<ref>{{Cite web |title=Is the Gamma function misdefined? Or: Hadamard versus Euler — Who found the better Gamma function? |url=https://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html}}</ref>

[[File:Gamma plus sin pi z.svg|thumb|250px|The gamma function, {{math|Γ(''z'')}} in blue, plotted along with {{math|Γ(''z'') + sin(π''z'')}} in green. Notice the intersection at positive integers. Both are valid extensions of the factorials to a meromorphic function on the complex plane.]]

A more restrictive requirement is the [[functional equation]] which interpolates the shifted factorial <math>f(n)  =  (n{-}1)!  </math> :<ref>{{cite book |title=Special Functions: A Graduate Text |first1=Richard |last1=Beals |first2=Roderick |last2=Wong |publisher=Cambridge University Press |year=2010 |isbn=978-1-139-49043-6 |page=28 |url=https://books.google.com/books?id=w87QUuTVIXYC}} [https://books.google.com/books?id=w87QUuTVIXYC&pg=PA28 Extract of page 28]</ref><ref>{{cite book |title=Differential Equations: An Introduction with Mathematica |edition=illustrated |first1=Clay C. |last1=Ross |publisher=Springer Science & Business Media |year=2013 |isbn=978-1-4757-3949-7 |page=293 |url=https://books.google.com/books?id=Z4bjBwAAQBAJ}} [https://books.google.com/books?id=Z4bjBwAAQBAJ&pg=PA293 Expression G.2 on page 293]</ref>
<math display="block">f(x+1) = x f(x)\ \text{ for all } x>0, \qquad f(1) = 1.</math><!-- please note that this is not the factorial function. It is the gamma function, which is a translated version of the factorial. See the cited sources -->

But this still does not give a unique solution, since it allows for multiplication by any periodic function <math>g(x)</math> with <math>g(x) = g(x+1)</math> and <math>g(0)=1</math>, such as  <math>g(x) = e^{k\sin(m\pi x)}</math>. 

One way to resolve the ambiguity is the [[Bohr–Mollerup theorem]], which shows that <math>f(x) = \Gamma(x)</math> is the unique interpolating function for the factorial, defined over the positive reals, which is [[Logarithmically convex function|logarithmically convex]],<ref name="Kingman1961">{{cite journal|last1=Kingman|first1=J. F. C.|title=A Convexity Property of Positive Matrices|journal=The Quarterly Journal of Mathematics|date=1961|volume=12|issue=1|pages=283–284|doi=10.1093/qmath/12.1.283|bibcode=1961QJMat..12..283K}}</ref> meaning that <math>y = \log f(x) </math> is [[Convex function|convex]].<ref>{{MathWorld | urlname=Bohr-MollerupTheorem | title=Bohr–Mollerup Theorem}}</ref>