Why Must the Real Part of \(c-a-b\) Be Positive in Gauss's Summation Formula?

[Ted123]

Why is the Gauss summation formula for complex parameters a,b,c: $$\displaystyle _2 F_1 (a,b;c;1) = \frac{\Gamma (c) \Gamma (c-a-b)}{\Gamma (c-a) \Gamma (c-b)}$$ only valid for $\text{Re}(c-a-b)>0,\;c\neq 0,-1,-2,-3,...$?

 

[Ted123]

It makes sense that $c\neq 0,-1,-2,-3,...$ so that $\Gamma (c)$ is defined but why does $\text{Re}(c-a-b)$ have to be positive?