Editing Gamma function (section)

== History ==
The gamma function has caught the interest of some of the most prominent mathematicians of all time. Its history, notably documented by [[Philip J. Davis]] in an article that won him the 1963 [[Chauvenet Prize]], reflects many of the major developments within mathematics since the 18th century. In the words of Davis, "each generation has found something of interest to say about the gamma function. Perhaps the next generation will also."<ref name="Davis">{{cite journal |last=Davis |first=P. J. |date=1959 |title=Leonhard Euler's Integral: A Historical Profile of the Gamma Function |journal=[[American Mathematical Monthly]] |volume=66 |issue=10 |pages=849–869 |url=http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3104 |access-date=3 December 2016 |doi=10.2307/2309786 |jstor=2309786 |archive-date=7 November 2012 |archive-url=https://web.archive.org/web/20121107190256/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3104 |url-status=dead }}</ref>

=== 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]].

=== 19th century: Gauss, Weierstrass and Legendre ===
[[File:Euler factorial paper.png|thumb|250px|alt=De progressionibus transcendentibus, seu quarum termini generales algebraicae dari nequeunt|The first page of Euler's paper]]

[[Carl Friedrich Gauss]] rewrote Euler's product as
<math display="block">\Gamma(z) = \lim_{m\to\infty}\frac{m^z m!}{z(z+1)(z+2)\cdots(z+m)}</math>
and used this formula to discover new properties of the gamma function. Although Euler was a pioneer in the theory of complex variables, he does not appear to have considered the factorial of a complex number, as instead Gauss first did.<ref name="Remmert">{{cite book |last=Remmert |first=R. |translator-last=Kay |translator-first=L. D. |title = Classical Topics in Complex Function Theory |publisher = Springer |year = 2006 |isbn = 978-0-387-98221-2 }}</ref> Gauss also proved the [[multiplication theorem]] of the gamma function and investigated the connection between the gamma function and [[elliptic integral]]s.

[[Karl Weierstrass]] further established the role of the gamma function in [[complex analysis]], starting from yet another product representation,
<math display="block">\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{k=1}^\infty \left(1 + \frac{z}{k}\right)^{-1} e^\frac{z}{k},</math>
where {{math|''γ''}} is the [[Euler–Mascheroni constant]]. Weierstrass originally wrote his product as one for {{math|{{sfrac|1|Γ}}}}, in which case it is taken over the function's zeros rather than its poles. Inspired by this result, he proved what is known as the [[Weierstrass factorization theorem]]—that any entire function can be written as a product over its zeros in the complex plane; a generalization of the [[fundamental theorem of algebra]].

The name gamma function and the symbol {{math|Γ}} were introduced by [[Adrien-Marie Legendre]] around 1811; Legendre also rewrote Euler's integral definition in its modern form. Although the symbol is an upper-case Greek gamma, there is no accepted standard for whether the function name should be written "gamma function" or "Gamma function" (some authors simply write "{{math|Γ}}-function"). The alternative "pi function" notation {{math|1=Π(''z'') = ''z''!}} due to Gauss is sometimes encountered in older literature, but Legendre's notation is dominant in modern works.

It is justified to ask why we distinguish between the "ordinary factorial" and the gamma function by using distinct symbols, and particularly why the gamma function should be normalized to {{math|1=Γ(''n'' + 1) = ''n''!}} instead of simply using "{{math|1=Γ(''n'') = ''n''!}}". Consider that the notation for exponents, {{math|''x<sup>n</sup>''}}, has been generalized from integers to complex numbers {{math|''x<sup>z</sup>''}} without any change. Legendre's motivation for the normalization is not known, and has been criticized as cumbersome by some (the 20th-century mathematician [[Cornelius Lanczos]], for example, called it "void of any rationality" and would instead use {{math|''z''!}}).<ref>{{cite journal|last=Lanczos |first=C. |date=1964 |title=A precision approximation of the gamma function |journal=  Journal of the Society for Industrial and Applied Mathematics, Series B: Numerical Analysis|volume=1|issue=1 |page=86 |doi=10.1137/0701008 |bibcode=1964SJNA....1...86L }}</ref> Legendre's normalization does simplify some formulae, but complicates others. From a modern point of view, the Legendre normalization of the gamma function is the integral of the additive [[character (mathematics)|character]] {{math|''e''<sup>−''x''</sup>}} against the multiplicative character {{math|''x<sup>z</sup>''}} with respect to the [[Haar measure]] <math display="inline">\frac{dx}{x}</math> on the [[Lie group]] {{math|'''R'''<sup>+</sup>}}. Thus this normalization makes it clearer that the gamma function is a continuous analogue of a [[Gauss sum]].<ref>{{cite book |title=Notes from the International Autumn School on Computational Number Theory |author1=Ilker Inam |author2=Engin Büyükaşşk |edition= |publisher=Springer |year=2019 |isbn=978-3-030-12558-5 |page=205 |url=https://books.google.com/books?id=khCTDwAAQBAJ}} [https://books.google.com/books?id=khCTDwAAQBAJ&pg=PA205 Extract of page 205]</ref>

=== 19th–20th centuries: characterizing the gamma function ===
It is somewhat problematic that a large number of definitions have been given for the gamma function. Although they describe the same function, it is not entirely straightforward to prove the equivalence. Stirling never proved that his extended formula corresponds exactly to Euler's gamma function; a proof was first given by [[Charles Hermite]] in 1900.<ref name="Knuth">{{cite book |last= Knuth |first=D. E. |title = The Art of Computer Programming |volume=1 (Fundamental Algorithms) |publisher = Addison-Wesley |year = 1997 |isbn=0-201-89683-4 }}</ref> Instead of finding a specialized proof for each formula, it would be desirable to have a general method of identifying the gamma function.

One way to prove equivalence would be to find a [[differential equation]] that characterizes the gamma function. Most special functions in applied mathematics arise as solutions to differential equations, whose solutions are unique. However, the gamma function does not appear to satisfy any simple differential equation. [[Otto Hölder]] proved in 1887 that the gamma function at least does not satisfy any [[algebraic differential equation|''algebraic'' differential equation]] by showing that a solution to such an equation could not satisfy the gamma function's recurrence formula, making it a [[transcendentally transcendental function]]. This result is known as [[Hölder's theorem]].

A definite and generally applicable characterization of the gamma function was not given until 1922. [[Harald Bohr]] and [[Johannes Mollerup]] then proved what is known as the [[Bohr–Mollerup theorem]]: that the gamma function is the unique solution to the factorial recurrence relation that is positive and ''[[logarithmic convexity|logarithmically convex]]'' for positive {{mvar|z}} and whose value at 1 is 1 (a function is logarithmically convex if its logarithm is convex). Another characterisation is given by the [[Wielandt theorem]].

The Bohr–Mollerup theorem is useful because it is relatively easy to prove logarithmic convexity for any of the different formulas used to define the gamma function. Taking things further, instead of defining the gamma function by any particular formula, we can choose the conditions of the Bohr–Mollerup theorem as the definition, and then pick any formula we like that satisfies the conditions as a starting point for studying the gamma function. This approach was used by the [[Bourbaki group]].

[[Jonathan Borwein|Borwein]] & Corless review three centuries of work on the gamma function.<ref>{{cite journal
| last1                 = Borwein
| first1                = Jonathan M. 
| author-link1          = Jonathan Borwein
| last2                 = Corless
| first2                = Robert M. 
| title                 = Gamma and Factorial in the Monthly
| journal               = American Mathematical Monthly
| language              = en
| publisher             = Mathematical Association of America
| date      = 2017
| volume                = 125
| issue = 5 
| pages                 = 400–24
| arxiv                 = 1703.05349
| bibcode= 2017arXiv170305349B
| doi                   = 10.1080/00029890.2018.1420983
| s2cid = 119324101 
}}</ref>

=== Reference tables and software ===

Although the gamma function can be calculated virtually as easily as any mathematically simpler function with a modern computer—even with a programmable pocket calculator—this was of course not always the case. Until the mid-20th century, mathematicians relied on hand-made tables; in the case of the gamma function, notably a table computed by Gauss in 1813 and one computed by Legendre in 1825.<ref>{{Cite web |title=What's the history of Gamma_function? |url=https://yearis.com/gamma_function/ |access-date=2022-11-05 |website=yearis.com}}</ref>

[[File:Jahnke gamma function.png|thumb|300px|A hand-drawn graph of the absolute value of the complex gamma function, from ''Tables of Higher Functions'' by [[Eugen Jahnke|Jahnke]] and {{ill|Fritz Emde|de|lt=Emde}}.]]

Tables of complex values of the gamma function, as well as hand-drawn graphs, were given in ''[[Tables of Functions With Formulas and Curves]]'' by [[Eugen Jahnke|Jahnke]] and {{ill|Fritz Emde|de|lt=Emde}}, first published in Germany in 1909. According to [[Michael Berry (physicist)|Michael Berry]], "the publication in J&E of a three-dimensional graph showing the poles of the gamma function in the complex plane acquired an almost iconic status."<ref>{{cite news|last=Berry |first=M. |url=http://scitation.aip.org/journals/doc/PHTOAD-ft/vol_54/iss_4/11_1.shtml?bypassSSO=1 |title=Why are special functions special? |newspaper=Physics Today |date=April 2001}}</ref>

There was in fact little practical need for anything but real values of the gamma function until the 1930s, when applications for the complex gamma function were discovered in theoretical physics. As electronic computers became available for the production of tables in the 1950s, several extensive tables for the complex gamma function were published to meet the demand, including a table accurate to 12 decimal places from the U.S. [[National Bureau of Standards]].<ref name=Davis />
[[File:Famous complex plot by Janhke and Emde (Tables of Functions with Formulas and Curves, 4th ed., Dover, 1945) gamma function from -4.5-2.5i to 4.5+2.5i.svg|alt=Reproduction of a famous complex plot by Janhke and Emde (Tables of Functions with Formulas and Curves, 4th ed., Dover, 1945) of the gamma function from −4.5 − 2.5i to 4.5 + 2.5i|thumb|Reproduction of a famous complex plot by Janhke and Emde (Tables of Functions with Formulas and Curves, 4th ed., Dover, 1945) of the gamma function from −4.5 − 2.5i to 4.5 + 2.5i]]
Double-precision floating-point implementations of the gamma function and its logarithm are now available in most scientific computing software and special functions libraries, for example [[TK Solver]], [[Matlab]], [[GNU Octave]], and the [[GNU Scientific Library]]. The gamma function was also added to the [[C (programming language)|C]] standard library ([[math.h]]). Arbitrary-precision implementations are available in most [[computer algebra system]]s, such as [[Mathematica]] and [[Maple (software)|Maple]]. [[PARI/GP]], [[MPFR]] and [[MPFUN]] contain free arbitrary-precision implementations. In some [[software calculator]]s, e.g. [[Windows Calculator]] and [[GNOME]] Calculator, the factorial function returns Γ(''x'' + 1) when the input ''x'' is a non-integer value.<ref>{{Cite web|title=microsoft/calculator|url=https://github.com/microsoft/calculator|access-date=2020-12-25|website=GitHub|language=en}}</ref><ref>{{Cite web|title=gnome-calculator|url=https://gitlab.gnome.org/GNOME/gnome-calculator|access-date=2023-03-03|website=GNOME.org|language=en}}</ref>

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