Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment VI.

1. Problem 1. a) Find a linear transformation (function) that transforms a triangle with vertices at 0, 1, i onto a similar triangle with vertices at 0, 2, 1 + i.

b) Find a linear transformation with a fixed point 1 + 2i that transforms i into −i.

c) For w = 2z + 1 − 3i find a finite fixed point z_{0}, an angle θ of rotation about z_{0}, and a linear change of scale at z_{0}.

d) Find a general form of a linear transformation that transforms the upper-half plane onto itself.

e) Find a general form of a linear transformation that transforms the upper-half plane onto the lower-half plane.

2. Problem 2. a) Prove that the reflection z → z̄ is not a linear fractional transformation.

b) If

T_{1}z = \frac{z + 2}{z + 3}, T_{2}z = \frac{z}{z + 1},

find T_{1}T_{2}z, T_{2}T_{1}z and T_{1}^{−1}T_{2}z.

c) Show that any fractional linear transformation which transforms the real axis into itself can be written with real coefficients.

3. Problem 3. a) Give a precise definition of a single-valued branch of \sqrt{1 + z} + \sqrt{1 − z} in a suitable region, and prove that it is analytic.

b) Same problem for log log z.

c) Suppose that f(z) is analytic and satisfies the condition |f(z)^{2} − 1| < 1 in a region Ω. Show that either ℜf(z) > 0 or ℜf(z) < 0 throughout Ω.

4. Problem 4. a) Let E ⊂ C be a region with area A(E) that can be evaluated as a double Riemann integral

A(E) = ∫ ∫_{E} dxdy.

If f is a conformal mapping of an open set containing E, show that the area A(E′) of the image E′ = f(E) is given by

A(E′) = ∫ ∫_{E} |f′(z)|^{2}dxdy.

Hint: Use the change of variables formula in double integrals, then apply Cauchy-Riemann equations.

b) Let f(z) = z + ∑_{n=2}^{∞} a_{n}z^{n} be one-to-one and analytic in the disc D = {z : |z| < 1}.

Prove that A(f(D)) ≥ π, and A(f(D)) = π only provided f(z) = z.

Hint: Use a) together with |f′(z)|^{2} = f′(z)\overline{f′(z)} and

∫ ∫_{D} z^{m}z̄^{n} dxdy = ∫_{0}^{1} r^{m+n+1}dr ∫_{0}^{2π}e^{i(m−n)θ} dθ = 0 for m ≠ n.