Math 621 Homework Assignment 6 Spring 2006
Due: Monday, April 24
1. Ahlfors, page 136 Problem 6: If γis a path, piecewise of type C1, contained in the
open unit disc D, then the integral
Zγ
|dz|
1− |z|2
is called the non-euclidean length or hyperbolic length of γ. Let f:D→Dbe
an analytic function from the disc into itself. Show that fmaps every γon a
path with smaller or equal non-euclidean length. Deduce that a linear fractional
transformation from Donto itself preserves non-euclidean lengths.
2. Ahlfors, page 136 Problem 7: (Modified) It can be shown, that the path of smallest
non-euclidean length, joining the origin 0 to a point z∈D, is the straight line
segment between them.
(a) Use this fact to show that the path of smallest non-euclidean length, that
joins two given points in the unit disk, is the piece of the circle Cwhich is
orthogonal to the unit circle ∂D. The shortest non-euclidean length is called
the non-euclidean distance.
(b) Show that the non-euclidean distance between z1and z2is
1
2log
1 +
z1−z2
1−¯z1z2
1−
z1−z2
1−¯z1z2
3. (a) Show that single valued analytic branches of f(z) = zα,α∈C, and f(z) = zz
can be defined in any simply connected region, which does not contain the
origin.
(b) (Ahlfors, problem 5 page 148) Show that a single vlaued analytic branch of
√1−z2can be defined in any region such that the points 1 and −1 are in the
same connected component of the complement. What are the possible values
of Rdz
√1−z2over a closed curve in the region?
4. Laurent Serries: Lang page 164: 8, 12, 13
5. Problem 5 from the basic exam of August 99: Consider the Laurent series
tan(z) = ∞
X
n=−∞
anzn,which is valid in the annulus π
2<|z|<3π
2. Find the coeffi-
cients anwith index −∞ < n ≤ −1. Hint: Use integration.
6. Isolated Singularities: Lang page 170: 1a,c,e, 4
7. Find the value of the integral ZC
3z3+ 2
(z−1)(z2+ 9)dz for:
(a) Cthe circle |z−2|= 2.
(b) Cthe circle |z|= 4.
1
