Confusion about Functional equation for $\Gamma$

Question

I have this theorem (the functional equation for $\Gamma$):

and this solution:

which uses the number $z=-1/2$ in the functional equation for $\Gamma$, which shouldn't be allowed.

What's missing?

Thanks

Answer 1

Sketch of argument discussed in my comment: We know that $\Gamma (z+1)=z\Gamma (z)$, valid for $\Re z>0$, and $\Gamma$ holomorphic for $\Re z>0.$ Consider $\Gamma (z+1)/z,$ for $\Re z>-1.$ By the properties of $\Gamma$, this function is meromorphic in this region, with a potential singularity at $z=0.$ Since $\Gamma(1)=1$, one can show that this is a simple pole. Also, if $\Re z>0,$ then $\Gamma (z+1)/z=\Gamma (z).$ Hence, $\Gamma$ uniquely extends meromorphically to $\Re z>-1,$ with a simple pole at $z=0.$ Now, one simply inducts, where you will get a function $$\frac{\Gamma (z+n)}{(z+n-1)(z+n-2)\cdots z}$$ meromorphic for $\Re z>-n$ that equals $\Gamma$ for $\Re z>0,$ with simple poles at $z=0,-1,\cdots, -n+1.$ By the uniqueness of analytic continuation, this extends all of the previous continuations. In the end, you get $\Gamma$ extended analytically to $\{z\in\mathbb{C}:\ z\neq 0,-1,-2,\cdots\}.$