I am facing some troubles related to the Gamma function. As I do not want to add to much complex elements to my questions, imagine that in a part of my equation I have this: $\frac{\Gamma(-m-1/2)}{2}+\Gamma(m+1/2)\cdot 2$
I found online that it considers only one part or the other one according to the following condition: if $Rm>-1/2$ then the solution is $\Gamma(m+1/2)\cdot 2$ otherwise is $\frac{\Gamma(-m-1/2)}{2}$.
Could you kindly explain why? What happens if m isn't a complex number? I do not understand why the $\Gamma(x)$ when x is non-positive should be deleted and considered equal to zero. I cannot find any literature about that.
EDIT: I got it from here: Asymptotics of this HyperGeometric Function In point 2, you can obtain: $x^{2m}\left(x^1\frac{\Gamma(3/2)\Gamma(-m-1/2)}{\Gamma(-m)} + x^{-2m} \frac{\Gamma(1/2+m)}{2(\Gamma(3/2+m))}\right)$ I cannot find how they reach the result in point 2.
It seems that the solution provided in the link above is wrong according to the comments. May I ask you how can I work with the previous expression and how can I solve the limit when x tends to zero? (as in the link)