Mapping of algebraic function to Riemann surface?

[jackmell]

When we map the algebraic function, $w(z)$, 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, $f(z,w),$ of the coordinate points over this surface, including the underlying algebraic function $f(z,w)=w$, that are single-valued, analytic functions except at special points called singular points.

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

 

[fresh_42]

Holomorphic and meromorphic seem to be the properties which are normally used:
https://math.berkeley.edu/~teleman/math/Riemann.pdfhttps://link.springer.com/chapter/10.1007/978-3-319-16053-5_3