>>11733267so I have a basic question about the gamma function G(a) for real positive arguments a>0.
G(a) := integral(0,inf, x^(a-1)*exp(-x) dx)
it's easy to show that
G(n+1)=n*G(n) (*)
and G(1)=1, yielding G(n+1)=n! for positive integers.
to show that G(a) exists for all a>0 it suffices to show that G(a)<inf for 0<a<1 because of (*), and that's what Im trying to do.
for 0<a<1 we have -1<a-1<0 and so I *think* I can split the integral into two parts (at x=1) and estimate those parts like this:
G(a) <= integral(0,1, x^(a-1)dx) + integral(1,inf, exp(-x)dx)
both terms are easily shown to be finite, the only question, is this step legit?
for the 1st term I use that 0<exp(-x)<1 for all positive x, so throwing that out should just make the integral larger.
similarly for the 2nd term I use that 0<x^(a-1)<1 for x>1 (remember that -1<a-1<0), so dropping it again should at most make the integral larger
is this correct? and is there an easier way of showing G(a) exists for all positive arguments?
