Math 535 Homework #1
Winter 2008
You will find that linear fractional transformations are of considerable use in problems 1 and
2.
1. Suppose Cand Dare tangent circles, one inside the other. Find a circle C1which is tangent to
both Cand D, then for n≥2 find Cntangent to C,Dand Cn−1, with {Cj, j = 1,...,n}bounding
disjoint open discs. Prove that this process can be continued indefinitely and that the points of
tangency of the Cnall lie on a circle.
2. Suppose Cand Dare non-intersecting circles, one inside the other. Choose C1tangent to both
Cand D. Again choose Cntangent to C,Dand Cn−1, with {Cj, j = 1,...,n}bounding disjoint
open discs. Under some circumstances, the chain of circles may “close up” with Cntangent also
to C1. Show that for each n≥3, there is a choice of Cand Dso that the chain always closes up.
Given circles Cand D, show that the property of “closing up” does not depend on the choice of
C1.
It is known that given a collection of circles in a simply connected domain, there is collection
of circles in the unit disc with the same pattern of tangencies. If the circles are small enough, the
map from one set of discs to the other approximates the conformal map between the regions. See
Circle Packing by Stephenson for an introduction to the subject.
See Penrose and Rindler, Camb. Univ. Press 1984, Spinors and Space-Time for the connection
of LFTs and Lorentz transformations in the theory of relativity.
3.(a) Let fbe an analytic function on |z|<2 such that fis real-valued on a subarc of |z|= 1.
Prove that fis a constant.
(b) Suppose fis analytic on D. Suppose also that fextends to be continuous on D∪Jwhere Jis
a open subarc of ∂Dand has values on Jwhich lie on a circle C. Prove that fhas a meromorphic
extension to C\(∂D\J).
Ahlfors, Complex Analysis, page 173, last sentence before the exercises incorrectly says “ana-
lytic” instead. What’s wrong? Hint: For both a) and b) Consider f(1/z).
4. (Universal entire function) Show that there is an entire function fwith the property that for
every entire function g, for every ε > 0, and for every compact set Kin the plane, there is a c > 0
