ca.analysis and odes - How many ways can we characterize gamma function?

maybe I can give you some help.
Gamma function is also called the second Euler integral.

Here comes some characterizations.

a f(s)=

$$t(x)=int_{0}^{+infty}{t^(s-1)}{exp(-t)}dt$$ s>0

b f(s)=

$$lim n!n^s/[s(s+1)...(s+n)]$$ $$nrightarrow +infty$$

c

$$B（p，q）=Gamma(p)Gamma(q)/Gamma(pq）$$ p>0 q>0

d

$$Gamma(2s)=2^(2s-1）Gamma(s)Gamma(s+1/2)/sqrt(2pi)$$ s>0

e

$$Gamma(s)Gamma(1-s)=pi/sin(spi)$$ 0

May it help!

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