\[F\left(a,1-a;\tfrac{1}{2};-z^{2}\right)=\frac{1}{2\sqrt{1+z^{2}}}\left(\left(%
\sqrt{1+z^{2}}+z\right)^{2a-1}+\left(\sqrt{1+z^{2}}-z\right)^{2a-1}\right),\]