The local structure of a complex function can be measured in terms of Wirtinger derivatives. Using the real representations of mappings the results are rather unhandy. Wirtinger derivatives make life easy. Normally the number of solutions changes by two when crossing the bifurcation curve. 3.2 Beltrami Parameter and Ellipses Analytic or conformal mappings map small circles onto circles.
Arbitrary mappings map small circles onto ellipses.
Functions with for all z are analytic or conformal. The last ellipse field is of particular interest. The mathematical framework is then given by the Beltrami Equation. The complex theory of gravitational lensing, Beltrami equation and cluster lensing.