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The Gauss hypergeometric function is the normalized analytic solution at zero,initially for $|z|<1$ and $c\notin \{0,-1,-2,\dots \}$, followed by analytic continuation where possible.

The ordinary hypergeometric function is ${}_2{F}_1(a,b;c;z)={\sum }_{n\ge 0}(a{)}_n(b{)}_n{z}^n/((c{)}_{n}n!)$ near zero for admissible $c$. It solves the Gauss hypergeometric equation and extends by analytic continuation.

For the Euler integral to converge directly, assume $\mathrm{Re}b>0$ and $\mathrm{Re}(c-b)>0$. At zero it is the beta function $B(b,c-b)=\Gamma (b)\Gamma (c-b)/\Gamma (c)$. The normalization is thereforeOutside these initial parameter conditions the hypergeometric function is defined by analytic continuation wherever the parameters permit it.

The Möbius transformation $w=z/(z-1)$ sends $0\mapsto 0$, $\infty \mapsto 1$ and $1\mapsto \infty$. The hypergeometric function $F(a,c-b;c;w)$ has local exponent pairs $(0,1-c)$ at $w=0$, $(0,b-a)$ at $w=1$, and $(a,c-b)$ at $w=\infty$. Pulling back to $z$ and multiplying by $(z-1{)}^{-a}$ gives the pairsAt infinity the prefactor adds $a$ to both exponents, while at $z=1$ it subtracts $a$ from both pulled-back exponents. These are exactly the requested Riemann P-symbol exponents. Equivalently, the Pfaff transformation identifies this branch, up to a nonzero phase depending on the chosen branches, with $F(a,b;c;z)$.

When $c\notin \mathbb{Z}$, an independent Frobenius solution isIts exponent $1-c$ differs from the first branch's exponent zero, establishing linear independence. For resonant parameters, use a parameter limit giving the appropriate logarithmic branch. More directly, on a simply connected region avoiding the singularities and zeros of a nonzero first solution $y_1$, reduction of order givesThe Wronskian is a nonzero multiple of $z^{-c}(1-z{)}^{c-a-b-1}$, so this also produces an independent local solution in the resonant cases for which the first branch is defined.