Functions from R 2 to R 2 : a study in nonlinearity

like f(z) = z 3 and g(z) = e z . Some teachers go further and introduce a few examples of conformal mappings. A picture is worth a thousand words, but more can be said on their favor: they provide a good exercise in combining theoretical facts in a consistent fashion. Indeed, to obtain the graph of a real function, a student considers its derivatives, asymptotic behavior and some special points, among other features. Something similar happens in the study of conformal mappings. In this text, we consider functions from R 2 to R 2 and along the way assemble a number of tools from undergraduate courses. We describe a graphical representation of such functions and, for functions which are visually too complicated, we