Let the polynomial defining w implicitly as a function of z be of irreducible degree n in w. Then, if nodes are counted according to their multiplicity, the z-plane with introduced cuts and with poles excluded is imaged n times in the w-plane.  It is possible to get conformality between the z-plane and each of the branches of the implicit function by removing from the z-plane the cuts and also any other points where the derivative dw/dz is zero or fails to exist. To find these points simply calculate or attempt to calculate the derivative. What is left of the z-plane is an open set where the conditions for conformality are satisfied.
  As z tends to a singularity its images tend to points which we shall call range limits. We also extend the terminology so that z can have a singularity at infinity and w can have a range limit at infinity. Branch-points and nodes are cases of singularities and range limits which have been discussed in previous notes. Now going back to the z-plane complete with cuts a neighbourhood of infinity is mapped complete with cuts onto neighbourhoods of the correspondiig range limits.
  A simple example illustrating range limits is provided by the equation (w^2-1)z-2w=0. There are nodes at +i and -i and the corresponding branch-points are at -i and +i. There are range limits at +1 and -1. Cuts from +i up to infinity and from -i down to infinity are imaged by four arcs which join the nodes to the range limits forming an oval. The interior of the oval is the range of one branch, the exterior is the range of the other.
  Since infinity is not a branch-point it is possible alternatively to use just one cut from +i to -i. The image of this cut is the imaginary axis which separates the ranges of the two branches. Here the range limits +1 and -1 are interior points of the ranges. 0 is a singularity since there is only one corresponding w-value. For z very small non-zero w is either very small or very large. The range limits are 0 and infinity.

Copyright Oct06 conesetter.