Editing Gamma function (section)

=== 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>