Branch cuts and Riemann surfaces

The plotting of Riemann surfaces by computational software is discussed. The link between the branches of a multi-valued function $g(z)$, defined on the range of $g(z)$, and a Riemann surface, defined on the domain of $g(z)$, is emphasized. The connection between the two is clarified by defining the $\mathit{\text{charisma}}$ of the argument $z$ to the function. This approach is used to plot several surfaces using Maple.