Math 621 Homework Assignment 3 Spring 2006
Due: Wednessday, March 8
1. (a) Find a function H(x, y) harmonic in the domain D:= {(x, y) : 0 <y<1}
and such that H≡0 on the line {y= 0}and H≡1 on the line y= 1.
(b) Find a function G(x, y) harmonic in the region inside the circle {|z|= 2}and
outside the circle {|z+ 1|= 1}and such that G≡0 on the inner circle and
G≡1 on the outer circle. Hint: See HW 2 problem 1(a).
2. Lang page 238-239 problems 12 part c (interpret the result geometrically), 14 parts
b,c, 13 part c (classify the fixed points according to Problem 14).
3. Ahlfors page 83 problem 2: Reflect the imaginary axis, the line x=y, and the
circle |z|= 1 in the circle |z−2|= 1.
4. Find the linear fractional transformation which carries the circle |z|= 3 into
|z−1|= 1, the point 3ito the origin, and the origin to i.
5. Ahlfors page 33 problem 4: What is the general form of a rational function (of
arbitrary degree) which has absolute value 1 on the circle |z|= 1? In particular,
how are the zeros and poles related to each other? (You may use the Fundamental
Theorem of Algebra stated in Corollary 7.6 page 130 in Lang. We will prove it later
in the course). Hint: Prove first that the rational function fsatisfies f=R◦f◦R,
where Ris the reflection with respect to the unit circle.
6. Ahlfors page 33 problem 5: If a rational function is real on |z|= 1, how are the
zeros and poles situated?
7. Let S3be the permutation group of the set {0,1,∞}. For each permutation σ,
denote by Tσthe fractional linear transformation taking 0,1,∞to σ(0), σ(1), σ(∞).
(a) Find the six l.f.t {Tσ:σ∈S3}.
(b) The orbit of λis the set {Tσ(λ) : σ∈S3}. Show that all but two S3orbits
in C\ {0,1}consist of six elements. One special S3orbit in C\ {0,1}consists
of two elements and the other special orbit consists of three elements.
8. We have seen that the cross ratios (z0
1:z0
2:z0
3:z0
4) and (z00
1:z00
2:z00
3:z00
4) are
equal, if and only if there exists a linear fractional transformation mapping the first
ordered 4-tuple to the second. We work out the analogous statement for unordered
sets of 4 distinct points. Let jbe the rational function (of degree 6)
j(λ) = 28(λ2−λ+ 1)3
λ2(λ−1)2
The composition j(z1:z2:z3:z4), of jand the cross ratio, is called the j-invariant
of the unordered set {z1, z2, z3, z4}. Use your answer to problem 7 to show
(a) The function j(z1:z2:z3:z4) is symmetric in the zi.
(b) There exists a linear fractional transformation mapping the unordered set
{z0
1, z0
2, z0
3, z0
4}onto {z00
1, z00
2, z00
3, z00
4}if and only if their j-invariants are equal.
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