Gauss's Hypergeometric Theorem/Proof 2

Proof
From Euler's Integral Representation of Hypergeometric Function, we have:
 * $\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$

Where $a, b, c \in \C$.

and $\size x < 1$

and $\map \Re c > \map \Re b > 0$.

Since Euler's Integral Representation only applies where $\size x < 1$, we will determine the limit of the integral as $x \to 1$.

Therefore: