Editing Factorial (section)

===Continuous interpolation and non-integer generalization===
[[File:Generalized factorial function more infos.svg|thumb|upright=1.6|The gamma function (shifted one unit left to match the factorials) continuously interpolates the factorial to non-integer values]]
[[File:Gamma abs 3D.png|thumb|Absolute values of the complex gamma function, showing poles at non-positive integers]]
{{Main|Gamma function}}
There are infinitely many ways to extend the factorials to a [[continuous function]].<ref name=davis/> The most widely used of these<ref name=borwein-corless/> uses the [[gamma function]], which can be defined for positive real numbers as the [[integral]]
<math display=block> \Gamma(z) = \int_0^\infty x^{z-1} e^{-x}\,dx.</math>
The resulting function is related to the factorial of a non-negative integer <math>n</math> by the equation
<math display=block> n!=\Gamma(n+1),</math>
which can be used as a definition of the factorial for non-integer arguments.
At all values <math>x</math> for which both <math>\Gamma(x)</math> and <math>\Gamma(x-1)</math> are defined, the gamma function obeys the [[functional equation]]
<math display=block> \Gamma(n)=(n-1)\Gamma(n-1),</math>
generalizing the [[recurrence relation]] for the factorials.<ref name=davis>{{cite journal | last = Davis | first = Philip J. | author-link = Philip J. Davis | doi = 10.1080/00029890.1959.11989422 | journal = [[The American Mathematical Monthly]] | jstor = 2309786 | mr = 106810 | pages = 849–869 | title = Leonhard Euler's integral: A historical profile of the gamma function | url = https://www.maa.org/programs/maa-awards/writing-awards/leonhard-eulers-integral-an-historical-profile-of-the-gamma-function | volume = 66 | year = 1959 | issue = 10 | access-date = 2021-12-20  | archive-date = 2023-01-01  | archive-url = https://web.archive.org/web/20230101190952/https://www.maa.org/programs/maa-awards/writing-awards/leonhard-eulers-integral-an-historical-profile-of-the-gamma-function | url-status = dead }}</ref>

The same integral converges more generally for any [[complex number]] <math>z</math> whose real part is positive. It can be extended to the non-integer points in the rest of the [[complex plane]] by solving for Euler's [[reflection formula]]
<math display=block>\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin\pi z}.</math>
However, this formula cannot be used at integers because, for them, the <math>\sin\pi z</math> term would produce a [[division by zero]]. The result of this extension process is an [[analytic function]], the [[analytic continuation]] of the integral formula for the gamma function. It has a nonzero value at all complex numbers, except for the non-positive integers where it has [[Zeros and poles|simple poles]]. Correspondingly, this provides a definition for the factorial at all complex numbers other than the negative integers.<ref name=borwein-corless>{{cite journal | last1 = Borwein | first1 = Jonathan M. | author1-link = Jonathan Borwein | last2 = Corless | first2 = Robert M. | doi = 10.1080/00029890.2018.1420983 | issue = 5 | journal = [[The American Mathematical Monthly]] | mr = 3785875 | pages = 400–424 | title = Gamma and factorial in the ''Monthly'' | volume = 125 | year = 2018| arxiv = 1703.05349 | s2cid = 119324101 }}</ref>
One property of the gamma function, distinguishing it from other continuous interpolations of the factorials, is given by the [[Bohr–Mollerup theorem]], which states that the gamma function (offset by one) is the only [[log-convex]] function on the positive real numbers that interpolates the factorials and obeys the same functional equation. A related uniqueness theorem of [[Helmut Wielandt]] states that the complex gamma function and its scalar multiples are the only [[holomorphic function]]s on the positive complex half-plane that obey the functional equation and remain bounded for complex numbers with real part between 1 and 2.<ref>{{cite journal | last = Remmert | first = Reinhold | author-link = Reinhold Remmert | doi = 10.1080/00029890.1996.12004726 | issue = 3 | journal = [[The American Mathematical Monthly]] | jstor = 2975370 | mr = 1376175 | pages = 214–220 | title = Wielandt's theorem about the {{nowrap|<math>\Gamma</math>-function}} | volume = 103 | year = 1996}}</ref>

Other complex functions that interpolate the factorial values include [[Hadamard's gamma function]], which is an [[entire function]] over all the complex numbers, including the non-positive integers.<ref>{{cite book|first=J.|last=Hadamard|author-link=Jacques Hadamard|chapter=Sur l'expression du produit {{math|1·2·3· · · · ·(''n''−1)}} par une fonction entière|title=Œuvres de Jacques Hadamard|publisher=Centre National de la Recherche Scientifiques|location=Paris|date=1968|chapter-url=http://www.luschny.de/math/factorial/hadamard/HadamardFactorial.pdf|orig-date=1894|language=fr}}
</ref><ref>{{cite journal | last = Alzer | first = Horst | doi = 10.1007/s12188-008-0009-5 | issue = 1 | journal = Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | mr = 2541340 | pages = 11–23 | title = A superadditive property of Hadamard's gamma function | volume = 79 | year = 2009| s2cid = 123691692 }}</ref> In the [[p-adic number|{{mvar|p}}-adic number]]s, it is not possible to continuously interpolate the factorial function directly, because the factorials of large integers (a dense subset of the {{mvar|p}}-adics) converge to zero according to Legendre's formula, forcing any continuous function that is close to their values to be zero everywhere. Instead, the [[p-adic gamma function|{{mvar|p}}-adic gamma function]] provides a continuous interpolation of a modified form of the factorial, omitting the factors in the factorial that are divisible by {{mvar|p}}.<ref>{{harvnb|Robert|2000}}. "7.1: The gamma function {{nowrap|<math>\Gamma_p</math>".}} pp. 366–385.</ref>

The [[digamma function]] is the [[logarithmic derivative]] of the gamma function. Just as the gamma function provides a continuous interpolation of the factorials, offset by one, the digamma function provides a continuous interpolation of the [[harmonic number]]s, offset by the [[Euler–Mascheroni constant]].<ref>{{cite journal | last = Ross | first = Bertram | doi = 10.1080/0025570X.1978.11976704 | issue = 3 | journal = [[Mathematics Magazine]] | jstor = 2689999 | mr = 1572267 | pages = 176–179 | title = The psi function | volume = 51 | year = 1978}}</ref>