Texts for the course:
Ahlfors, Complex Analysis
Ahlfors, Conformal Invariants
Milnor, Dynamics in one complex variable.
Homework problems.
1. Verify the formula for the chordal spherical distance.
2. Prove that circles on the sphere correspond under the
stereographic projection to lines or circles in the plane.
3. Prove the formula relating the points in the complex
plane whose images on the sphere are diametrally opposite.
4*. Prove the formula for the spherical (intrinsic!)
distance in terms of projective coordinates:
distanse(z,w)=arccos|(z,w)|/|z||w|, where (z,w) is
the Hermitian dot product, and |z| is the corresponding
norm of the vector z.
5. Prove that the stereographic projection is conformal.
Reading: Ahlfors, Complex Analysis, Ch. 1, 2.4; Ch. 3, 1.4.
6. Prove that the area of a spherical triangle equals
R^2{(sum of the angles)-pi},
where R is the radius of the sphere. Hint: find the
area of a "diangle" first.
7. Solve all problems 1-4 on p. 227 of Ahlfors.
Reading: Ahlfors, Complex Analysis Ch. 5, section 5.
8*. Find a transcendental meromorphic function f in the
plane such that the family f(2^nz) is normal in 1<|z|<2.
Verify that this function has no Julia directions.
Hint: Let g be the canonical product of genus 0
with zeros at the points 2^n. Then set f(z)=g(z)/g(-z).
9. Using Zalcman's lemma,
give a complete proof that the family of
first derivatives of univalent functions in the unit disc
is normal.
10. Why the same proof does not work for second derivatives?
11. Prove that for every non-constant entire function
there exist inverse branches defined in discs of
arbitrarily large radii. (Valiron's theorem).
12. Solve 1,2 on p. 232.
Reading: Ahlfors, VI, 1.
13. Compute the hyperbolic (Riemannian) metric for the
upper half-plane and for the strip |Im z| less than pi/2.
(Starting from the known expression for the unit disc,
transplant it by a conformal map).
14. Find and prove a formula for
the hyperbolic distance between
two points z and w:
a. In the disc
b. In the upper half-plane.
Hint: for the upper half-plane, the formula
has the form ch(d/c)=1+|z-w|^2/(2 Im(z)Im(w)),
where z and w are two points in the upper half-plane,
d is the hyperbolic distance between them,
c is a constant (which you find yourself) and ch is
the hyperbolic cosine, ch(t)=(exp(t)+exp(-t))/2.
15. Find the length of a
hyperbolic circle of hyperbolic radius r.
16*.
Let A and B be two given disjoint circles in the
sphere.
Suppose that we have a sequence of circles
X_1,X_2,...,X_n such that each of them is tangent to A and B,
and X_{k+1} is tangent to X_k for k=1,...,n-1,
and in addition, X_n is tangent to X_1.
Now erase all these circles X_i. Take an arbitrary circle
Y_1 that is tangent to A and B, then draw a circle Y_2
that is tangent to A and B and Y_1, and continue this
construction (so that when Y_1,...Y_k are already
construcuted, Y_{k+1} is a circle tangent to A,B and Y_k).
Prove that Y_{n+1}=Y_1.
CORRECTION: This is not true as stated.
(Did you find a counterexample?)
I had to add the condition that all circles are disjoint
except for tangencies. That is they never cross.
Hint: Use something you learned in 530 to simplify
the problem.
17. Prove that every fractional-linear transformation
is a composition of some inversions.
(Inversion is a reflection
in a circle).
18*. (This exercise is not required for the rest
of the course; it is in 3-d geometry).
Let A,B and C be theree spheres in 3-space,
each tangent to the both others.
Let X_1,X_2,... be a sequence of distinct
spheres such that X_1 is tangent
to A,B,C, and X_k is tangent to A,B,C,X_{k-1}, for k=2,3,... .
Prove that X_7=X_1.
19. When two lattices in the plane define conformally
equivalent tori? A lattice is given by a pair of periods.
The answer to this question can be given in terms of the
ratios of these periods.
20. Describe all tori that have non-trivial conformal
automorphisms.
Read Ahlfors, Chap. 7.2 and 7.3, especially
3.4 and 3.5 which were not covered in 530.
21. Consider the group G of automorphisms of the upper
half-plane H generated by the map g(z)= kz, with some fixed
k>1. Find an explicit conformal map of the Riemann surface
H/G onto a ring in the plane.
22. Do the same for the group generated by g(z)=z+c,
where c is a real number.
Comment. To solve 21 and 22 you have to find a function f
analytic in H, wich the property that f(z)=f(w)
if and only if z and w belong to the same orbit of
the given group.
23. By definition, a round ring is the region of
the form r <|z| < R, where r is non-negative (possibly 0)
and R>r, possibly R=infinity. Classify the round rings,
that is tell when two round rings are conformally equivalent
and when they are not. (Problem in Ahlfors, p. 257, 1
gives a hint).
Problem 24 (pdf)
Problems on subharmonic functions:
25. Ahlfors, Complex Analysis: p. 247-248, 1-5; p. 251, 1.
26. For f in the class S (of normalized univalent functions
in the unit disc), compute the first 3 coefficients of
g(z)=sqrt{f(z^2)} in terms of the coefficients of f.
27. Prove by computation that the function g in Ex. 26 is odd.
28. Generalize the Exercises 26, 27 to the case g(z)=p-th root of f(z^p).
29. Prove (with any method) that the classes S and Sigma_0
of normalized univalent functions are normal families.
30. Problem 30
Problems on applications of the Ahlfors-Schwarz lemma:
31. Complete the construction of meromorphic functions
in the plane with three totally ramified values of
multiplicities (2,4,4) and (2,3,6).
32. Find all solutions of the equation
sum (1-1/m_k)=2 in integers m_k>1.
33*. Prove that the Landau constant
is at least 1/2.
Problems about extremal length:
34. Let G' and G" be two families of curves contained in
disjoint domains. Let G constsis of all curves g
such that g is the union of some curve g' of G' with
some curve g" of G". Prove that
extremal length of G is greater then or equal to
extremal length of G' plus extremal length of G".
("Curves" here are not necessarily connected;
they can be unions of finitely many connected pieces.)
35. Let a and b be extremal lengths of some families
of curves, and c the extremal length of the union
of these two families. Prove that
1/c is at least 1/a+1/b.
36. Let {D_k} be a sequence of disjoint nested doubly
connected regions in the plane. ("Nested" means
D_{k+1} is inside the bounded component of the complement
of D_k. Let a_k be the extremal length of the family
of all curves in D_k that connect the two
boundary components. Suppoose that the series
a_1+a_2+a_3+... is divergent.
Prove that the intersection of the bounded components
of the complements of D_k consists of one point.
Hint: use the result of problem 34.
37. Can a conformal map have radial limits everywhere
but not to have a continuous extension to the closed unit disc?
(Hint: yes. Use the first picture I made in the end of the
lecture 4.8, and the boundary point A in that picture).
38. Prove that every simply connected region has at least one
accessible boundary point. Furthermore, accessible boundary
points are dense on the boundary of this region.
39. Let f be a univalent function in the unit disc, continuous
in the closed unit disc. Suppose that a,b,c,d are four distinct
points of the unit circle, and f(a)=f(b), f(d)=f(c), but f(a) is different
from f(c).
Prove that the chords [a,b] and [c,d] are disjoint.
40* Suppose some disjoint letters T are drawn in the plane.
Prove that these letters is at most countable.
Dynamics problems
41. Prove that every polynomial of degree 2 is conjugate by an affine transformation (az+b) to a polynomial of the form z^2+c, and that such polynomials with different c are not conjugate to each other. (Thus from dynamical point of view there is a one-parametric family of quadratic polynomials).
42. Let P be the Weierstrass P-function with some periods. Prove that P(2z) is a rational function of P(z), and find this rational function explicitly. Prove that the Julia set of this function is the whole Riemann sphere. (So the Fatou set can be empty. This rational function is called a "Lattes example"). Can you find other examples of rational functions whose n-th iterates can be expressed by explicit formulas?
43. On an electronic calulator, enter the number 0.3 (or your other favorite small positive number). Then press the SIN button many times (until you get tired). Look at the results and make a conjecture, what happens. Then try to prove your conjecture.
44. For those who can handle a computer. Using your favorite programming tool write a program which will iterate z^2+c and represent the results graphically. Each pixel of your screen qill represent a point in the complex plane. Starting from this point, iterate the function up to 50 times. As soon as the result becomes large (say bigger than 4) stop iterating. Thus for each pixel, you compute an integer: how many iterates are needed for the trajectory to escape the disc of radius 4 centered at zero. Or perhaps it will not escape after 50 iterates. Then paint this pixel with some color depending on this integer. In a simpler version just use black and white: white if the point escapes after less than 50 iterates and black if it does not.
45. Prove that a rational function of degree at least 2 cannot have more than 2 disjoint completely invariant domains.
46. Let f_1,...f_n be non-constant analytic functions, and |f_k(0)|=1 for all k. Prove that in every neighborhood of 0 there exists a point z such that at least [n/2] of those functions satisfy |f_k(z)|<1.
47. Let g be a polynomial. Newton's method of finding the roots of g consists in iterating the rational function f(z)=z-g(z)/g'(z). Prove that every root of g is an attracting fixed point of f. What is the multiplier of this fixed point?
48. Study the Newton method for finding a square root, that if for solving the equation g(z)=z^2-a=0. Describe the regions in the plane where the Newton method converges to each root of this equation. What is the Julia set of f=id-g/g'?
49. Generalize the previous exercise to arbitrary quadratic equation.
50. Try the Newton method described in problems 47, 48 by finding the square root of 2 by hand. Then check with a calculator. Is this a good method of finding a square root by hand?
51. Let f(z) be a finite Blaschke product, that is f is a rational function that maps the unit disc into itself and the exterior of the unit disc into itself. What can the Julia set of f be? Give examples for all possibilities.
52. Derive from Ex. 51 that the only polynomials whose Julia set is the unit circle are cz^n, where |c|=1.