Riemann Surfaces, Branch Cuts and Charisma

The plotting of Riemann surfaces by computational software is discussed. The link between the branches of a multi-valued function g(z), which are defined on the range of g(z), and a Riemann surface, which is defined on the domain of g(z), is emphasized. The connection between the two is clarified by defining a property called charisma, which is used to plot the Riemann surface. The charisma is not uniquely defined and the choice can change the appearance of the surface. This approach is used to plot several surfaces using Maple.