Se p 20 03 Functions from R 2 to R 2 : a study in nonlinearity Nicolau C . Saldanha and Carlos Tomei

Calculus students learn how to draw graphs of functions from R to R and undergraduates studying complex variable learn about geometric properties of functions like f(z) = z and g(z) = e. 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 to R 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 still count preimages, in a manner reminiscent of Rouché ’s theorem. Why is it that such aspects of functions from the plane to the plane are not more familiar? A reason might be the following. Most of the information we compute about functions from the line to itself, or about holomorphic functions, concerns special points—typically critical points, where the derivative is zero. In the case of functions from the plane to the plane, we need to consider critical curves, where the Jacobian matrix is not invertible. Such curves are often impossible to describe in simple closed form. Enter the computer: we should think of the study of a given function from the plane to the plane as a description of certain relevant objects, in a way that these objects become amenable to numerics. In this sense, the time is ripe for this new case study in nonlinear theory, in the same way that we feel more at ease nowadays with showing students how to evaluate roots of polynomials of degree 6, or eigenvalues of 5× 5 matrices. The theory should operate on two levels: we should learn enough to get qualitative information about simple examples, and we should be able to derive numerical procedures to handle general cases. In particular, such procedures