Mapping of algebraic function to Riemann surface?

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When we map the algebraic function, [itex]w(z)[/itex], to a Riemann surface we essentially create a new "Riemann" coordinate system over a surface that is called the "algebraic function's Riemann surface".

This mapping allows one to create single-valued functions, [itex]f(z,w),[/itex] of the coordinate points over this surface, including the underlying algebraic function [itex]f(z,w)=w[/itex], that are single-valued, analytic functions except at special points called singular points.

May I ask what exactly is this type of mapping called?
 
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