How many ways can we characterize gamma function? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T12:12:48Z http://mathoverflow.net/feeds/question/15367 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15367/how-many-ways-can-we-characterize-gamma-function How many ways can we characterize gamma function? unknown (google) 2010-02-15T21:25:16Z 2010-03-07T18:38:44Z <p>First let's state a well-known characterization of gamma function.</p> <p>If f is a positive function on positive real numbers such that: (1).f(x+1)=xf(x); (2).f(1)=1; (3).logf is convex, then f(x) is gamma function.</p> <p>Now here I'm wondering how many ways can we characterize gamma function like the above? Especially if we consider it as a function on complex plane with poles.</p> <p>ps: I'm not asking different ways to express gamma function explicitly, but the abstraction of it.</p> http://mathoverflow.net/questions/15367/how-many-ways-can-we-characterize-gamma-function/17390#17390 Answer by Gu Yejun for How many ways can we characterize gamma function? Gu Yejun 2010-03-07T16:19:49Z 2010-03-07T16:19:49Z <p>maybe I can give you some help. Gamma function is also called the second Euler integral.</p> <p>Here comes some characterizations.</p> <p>a f(s)= $$t(x)=\int_{0}^{+\infty}{t^(s-1)}{exp(-t)}dt$$ s>0</p> <p>b f(s)=$$\lim n!n^s/[s(s+1)...(s+n)]$$ $$n\rightarrow +\infty$$</p> <p>c $$B（p，q）=\Gamma(p)\Gamma(q)/\Gamma(pq）$$ p>0 q>0</p> <p>d $$\Gamma(2s)=2^(2s-1）\Gamma(s)\Gamma(s+1/2)/\sqrt(2\pi)$$ s>0</p> <p>e $$\Gamma(s)\Gamma(1-s)=\pi/sin(s\pi)$$ 0 <p>May it help!</p> http://mathoverflow.net/questions/15367/how-many-ways-can-we-characterize-gamma-function/17404#17404 Answer by vonjd for How many ways can we characterize gamma function? vonjd 2010-03-07T18:38:44Z 2010-03-07T18:38:44Z <p>Have a look here: <a href="http://dlmf.nist.gov/5/" rel="nofollow">http://dlmf.nist.gov/5/</a></p>