Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment IX.

1. Problem 1. a) Map the common part of the discs |z| < 1 and |z − 1| < 1 on the inside of the unit circle. Hint: Use w = (z − a)(z − b).

b) Map the region between |z| = 1 and |z − 1/2| = 1/2 on the inside of the unit circle. Hint: Use w = 1/(z − a).

c) Map the complement of the arc |z| = 1, y ≥ 0 on the outside of the unit circle. Hint: Map the region onto the complement of the segment ℑz = 0, −1 ≤ ℜz ≤ 1, (the wedge).

2. Problem 2. a) Map the outside of the parabola y^{2} = 2px on the disc |w| < 1 so that z = 0 and z = −p/2 correspond to w = 1 and w = 0.

b) Map the inside of the right-hand branch of the hyperbola x^{2} − y^{2} = a^{2} on the unit disc |w| < 1 so that the focus corresponds to w = 0 and the vertex to w = −1.

c) Map the inside of the lemniscate |z^{2} − a^{2}| = ρ^{2}, ρ > a, on the disc |w| < 1.

d) Map the outside of the ellipse (x/a)^{2} + (y/b)^{2} = 1 onto |w| < 1. Hint: Find s such that r + 1/r = as, 1/r − r = bs.

3. Problem 3. a) Let a fractional linear transformation carrying a unit disc into itself be

w = e^{iα} \frac{z − a}{1 − āz} ,  |a| < 1,  a = |a|e^{iλ}.

Prove that it can be hyperbolic, parabolic and elliptic, but not loxodromic. Hint: Compute K and use cos α = (e^{iα} + e^{−iα})/2, sin α = (e^{iα} − e^{−iα})/(2i).

b) Describe a for which each of these cases occurs.

c) Find fixed points and bring the transformation to the normal form. Hint: Put |a| = sin(α/2) sin β for some β ∈ (0, π) when w is elliptic. If w is hyperbolic, put |a| sin β = sin(α/2) .

d) Find arg w(e^{iϕ}) = θ(ϕ).

e) Find w′(0), w′(a).

f) What part of the unit disc is dilated, what part is shrinking?

g) Find max |w′(z)|, min |w′(z)| for |z| ≤ 1.