Editing Gamma function (section)

=== 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>