1. Stating the problem: We want to evaluate the hypergeometric function $${}_2F_1(a,b,c,1)$$ under the conditions $$\mathrm{Re}(c-a-b)>0$$ and $$c$$ is neither zero nor a negative integer. 2. Explanation: The value of the Gauss hypergeometric function at 1 when $$\mathrm{Re}(c-a-b)>0$$ and $$c$$ not zero or negative integer is given by the formula $$ {}_2F_1(a,b,c,1) = \frac{\Gamma(c)\Gamma(c - a - b)}{\Gamma(c - a)\Gamma(c - b)} $$ 3. This corresponds exactly to option (a) in the question. Final answer: a. $${}_2F_1(a,b,c,1) = \frac{\Gamma(c)\Gamma(c - a - b)}{\Gamma(c - a)\Gamma(c - b)}$$