Hypergeometric function

Remarks

Existence of this analytic continuation follows from Euler's integral representation when $\operatorname{Re} c > \operatorname{Re} b > 0$. In general it follows from the hypergeometric differential equation, whose only finite singular points are $0$ and $1$. Uniqueness holds because $\C \setminus [1, \infty)$ is connected. This is the principal branch; continuation across the cut gives other branches of a multivalued function.

Example: ${}_2F_1(a, b; b; z) = (1 - z)^{-a}$ on the whole cut plane, with the principal power, since both sides are holomorphic there and agree for $\abs{z} < 1$.

Non-example: ${}_2F_1(1, 1; 2; z) = -\log(1 - z)/z$ has no holomorphic extension to a neighbourhood of $z = 1$ or to any annulus around $1$: going once around $z = 1$ changes $\log(1 - z)$ by $2\pi i$. So the cut, or a choice of branch, is needed.

Gauss's notation $F(a, b; c; z)$ is common, and the normalized function ${}_2F_1(a, b; c; z)/\Gamma(c)$ is entire in $c$.