  The elegance and usefulness of a complex formulation of the basic lensing
equations is demonstrated with a number of applications. Using standard tools
of complex function theory, we present, for instance, a new proof of the fact
that the number of images produced by a regular lens is always odd, provided
that the source is not located on a caustic. Several differential and integral
relations between the mean curvature and the (reduced) shear are also derived.
These emerge almost automatically from complex differentiations of the
differential of the lens map, together with Stokes' theorem for complex valued
1-forms.
