A holomorphic map between complex manifolds is a map whose expression in local complex coordinate
charts is given by holomorphic functions.
For maps between Riemann surfaces, a nonconstant
holomorphic map has local form 
after suitable choices of coordinates. Points with
are ramification
points.
Holomorphic Map
See also
Branched Covering Map, Complex Manifold, Holomorphic Function, Ramification Point, Riemann SurfaceExplore with Wolfram|Alpha
References
Miranda, R. Algebraic Curves and Riemann Surfaces. Providence, RI: Amer. Math. Soc., 1995.Referenced on Wolfram|Alpha
Holomorphic MapCite this as:
Weisstein, Eric W. "Holomorphic Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HolomorphicMap.html