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?