Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment VIII.

1. Problem 1. a) If z_{1}, z_{2}, z_{3}, z_{4} are points on the circle, show that z_{1}, z_{3}, z_{4} and z_{2}, z_{3}, z_{4} determine the same orientation iff (z_{1}, z_{2}, z_{3}, z_{4}) > 0.

b) Prove that every reflection carries circles into circles.

c) Suppose circles C_{1}, C_{2} are both symmetric with respect to a line l, and C_{3} is the reflection image of C_{1} with respect to C_{2}. Then C_{3} is also symmetric with respect to the line l, compare with the symmetry principle! Hint: Let l be the real axis, and let the center of C_{2} be on the real axis. What is the relation between z and z^{∗}?

d) Reflect ℵ) the imaginary axis, ℶ) the line x = y, and ℷ) the circle |z| = 1, in the circle |z − 2| = 1. Hint: In ℵ) and ℷ) use the real axis as l. In ℶ) use x + y = 2.

2. Problem 2. a) Find the linear transformation which carries |z| = 1 and |z − 1/4| = 1/4 into concentric circles. What is the ratio of the radii? Hint: Use Problem 3 d) of Assignment VII with R = 1. If 0 → −β, 1/2 → β, what is β?

b) Same problem for |z| = 1 and x = 2. Hint: Use w = 1/z.

c) Find all circles which are orthogonal to |z| = 1 and |z − 1| = 4. Hint: Use the inversion that transforms the given circles into concentric ones.

3. Problem 3. a) Find the fixed points of the linear transformations

ℵ) w = \frac{z}{2z − 1},  ℶ) w = \frac{2z}{3z − 1},  ℷ) w = \frac{3z − 4}{z − 1},  ℸ) w = \frac{z}{2 − z} .

Is any of transformations elliptic, hyperbolic, parabolic?

b) Suppose that the coefficients of the transformation Sz = (az + b)/(cz + d) are normalized by ad − bc = 1. Show that S is elliptic iff −2 < a + d < 2, parabolic if a + d = ±2, hyperbolic if a + d < −2 or > 2.

4. Problem 4.

a) Show that a linear transformation which satisfies S^{n}z = z for some integer n is necessary elliptic.

Hint: Consider the number of fixed points. If there are two distinct finite fixed points a and b, say. Then,

\frac{z − a}{z − b} = \frac{S^{n}z − a}{S^{n}z − b} = k \frac{S^{n−1}z − a}{S^{n−1}z − b} = k^{2} \frac{S^{n−2}z − a}{S^{n−2}z − b} = ...

If S has only one fixed point, consider T z = 1/(z − a) + a and observe T ST^{−1} has only ∞ as a fixed point. Hence, T ST^{−1}z = cz + d, and c = 1.

b) If S is hyperbolic or loxodromic, show that S^{n}z converges to a fixed point as n → ∞, the same for all z, except when z coincides with the other fixed point, (the limit is the attractive, the other the repellent fixed point). What happens when n → −∞? What happens in the parabolic case?

Hint: Let at first two fixed points z_{1}, z_{2} be finite. Passing to ”new coordinates”, we can write v = L(ξ) = Kξ, where

v = \frac{w − z_{1}}{w − z_{2}} , ξ = \frac{z − z_{1}}{z − z_{2}} .

Hence, L(L(...L(ξ))) = K^{n}ξ...