Riemann Surface

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Riemann surfaces, named after Bernhard Riemann, depict multi-valued functions by representing elements from a domain mapping to multiple places in a codomain. These surfaces resemble structures comprising infinite sheets separated by vertical distances and are configured in the complex plane utilizing vectors 1 and i. Complex numbers correspond to unique points in this plane. Branch cuts, lines or line-sections of multi-valued functions, offer another approach to represent such functions. They assign not a single point but infinite points from the domain. The sheets representing mappings from the origin are interconnected due to undefined status at the origin, making it non-part of the domain. Functions can be oriented by multiplying with complex numbers. In an example, f(z) = z^(1/3) is mapped using polar coordinates along with its branches or "branch cuts." These represent function discontinuity and non-differentiability. Increasing distance in polar coordinates results in crossing branch cuts where sheets meet, visibly depicted in 3D rendered images.