An inverse rational function w(z) is defined by an equation
(1) zf(w)-g(w)=0
where f and g are polynomials. An example may have the maximum number
of singularities which are of the simplest kind or the minimum number
of singularities which are of the most complex kind or something in
between. This note is about the first case which means that a boundary
of a branch separates it from just one other and that all permutations
across cuts are transpositions. The relation
(2) z(w^(n+1)-w)-(w^n-c)=0
where c is near but not equal to 1 is the paradigm which is chosen for
easy calculation. The range-limits are 0 and the nth roots of unity.
Let the initial value for z be very large then the initial values of
the n+1 branches are very near the n+1 range-limits. So here a set of
branches can equally well be identified by the range-limits.
A slight digression may be useful here. What we are really
studying is the relations-theory of two complex variables. If w were
taken to be the independent variable then what we have called a
range-limit is not something novel but just a pole for z as a function
of w and might equally well be called an inverse pole.
In the general case at least one branch takes arbitrarily large
values and in this case just one does. This may be seen using Newton's
polygon or directly since (2) for large values of w approximates to
zw-1=0 or w=1/z. This shows incidentally that there is no branch-point
at infinity.
The nodes and branch-points are found by solving (2) and its
first partial derivative with respect to w for w and z. In what follows
W and Z are singularities while w and z remain ordinary points. The
equation for the nodes is
(3) W^2n+(n-1-(n+1)c)W^n+c=0.
The solutions can be grouped into n pairs, both of each pair
tending to the same nth root of unity as c tends to 1. Roughly speaking
|c-1| is a measure of the size of the ranges of the bounded branches
and if c were allowed to be 1 all but the unbounded branch would
degenerate to points. The corresponding branch-points are paired
similarly.
Now to obtain a suitable system of cuts. Start with radial cuts
from each branch-point to infinity. These are 2n in number but only n
cuts are needed to define the required number n of boundaries. Take a
great circle, discard the parts of the cuts outside the circle, and of
the circle itself retain only those arcs which join paired cuts. What
is retained is the required set of n cuts. The derived boundaries have
the shape of wedges pointed at the origin but with the points cut off a
little short of the origin. Since c is small each wedge contains just
one of the nth roots of unity and so by our assumption above just one
initial value. The boundaries of each bounded branch are boundaries
with the one unbounded branch. So the permutations across cuts are just
transpositions of the unbounded branch with the appropriate bounded
branch.
The cuts can of course be distorted so long as they retain their
end-points and are not distorted over a singularity. This is where a
topological theory goes beyond local analysis which may just obtain a
circle of convergence valid for a power series expansion.
Copyright Feb 08 conesetter.