@user21820 Hi! I have a short question. I’m reading the book „The Gamma Function“ by J. Bonnar. On page ten he writes there ex. epsilon such that 0<=t^(x-1)e^(-t/2)<=1 for t=epsilon. Then he splits an integral at epsilon in two (1 to epsilon and epsilon to inf) and says the first is finite and for the second one for t >= epsilon he uses t^(x-1)e^(-t)=t^(x-1)e^(-t/2)e^(-t/2)<=e^(-t/2). This ist not really neat, or what do you think?
He splits \int_0^{\infty} t^(x-1)exp(-t) dt in \int_0^1 + \int_1^{\infty} and proves that the first one exists. That is clear. The his text is:
In the case of the second integral, first note that t^q exp(-t/2) \to 0 as t \to \infty for any q in \R. Hence for any x in \R there exits an \epsilon such that 0 \leq t^(x-1)e^(-t/2) \leq 1 for t = \epslion. So we further split the integral at \epsilon.
The first term is finite, being a finite integral. For the second term, when t \geq \epslion we have: t^(x-1)e^(-t)=(t^(x-1)e^(-t/2))e^(-t/2)<=e^(-t/2)
Ok I get the general idea the author is using, but I didn't read any of the formulae. I wouldn't do it that way only because it is obvious that my way works.
After all, he is still making an unjustified claim about a limit.
This is the Realm of Simply Beautiful…
This is the Realm of Simply Beautiful…