Holomorphic implicit function theorem
= Holomorphic implicit function theorem
If a holomorphic function $F(z,w)$ satisfies $F(z_0,w_0)=0$ and $F_w(z_0,w_0)\ne0$, then near $(z_0,w_0)$ its zero set is the graph $w=g(z)$ of a unique holomorphic function. The analogous statement with $F_z\ne0$ uses $w$ as the local coordinate.