Complex Analysis, Spring 2011.

Instructor: Dmitry Ryabogin

Assignment I.

1. Problem 1. a) When does az + bz̄ + c = 0 represent a line?

b) Prove that the diagonals of a parallelogram bisect each other and that the diagonals of a rhombus are orthogonal.

c) Show that all circles that pass through a and 1/ā intersect the circle |z| = 1 at right angles. Hint: Observe that |a| = |1/ā|, and arg a = arg(1/ā). Then use your knowledge from the high school.

2. Problem 2. a) Let d(z, z′) be the distance between the stereographic projections of z and z′, and let Z and Z′ denote the stereographic projections of z, z′, and let N be a north pole. Show that the triangles NZZ′ and Nzz′ are similar, and use it to derive

d(z, z′) = \frac{2|z − z′|}{\sqrt{(1 + |z|^{2})(1 + |z′|^{2})}}.

b) Check that d is a metric on C.

c) What are stereographic projections of 1, −1, i, (1 − i)/\sqrt{2}?

d) Describe the stereographic projection of the pair of points that are symmetric with respect to: z = 0, real axis, unit circle (see Problem 1 c)).

3. Problem 3. a) Let the points z_{1}, z_{2}, ...z_{n} be located in the half-plane, with the boundary line passing through the origin. Prove that the points 1/z_{1}, 1/z_{2}, ..., 1/z_{n} have the same property (with respect to what line?).

b) Prove that

z_{1} + z_{2} + ... + z_{n} ≠ 0, \frac{1}{z_{1}} + \frac{1}{z_{2}} + ... + \frac{1}{z_{n}} ≠ 0.

4. Problem 4.

a) The functions

\frac{Rez}{z} ,  \frac{z}{|z|} ,  \frac{Re(z^{2})}{|z|^{2}} ,  \frac{zRez}{|z|}

are defined for z ≠ 0. Is it possible to define them at 0 in such a way that they would be continuous at this point? b) Are functions

\frac{1}{1 − |z|} , \frac{1}{1 + |z|^{2}}

continuous in (|z| < 1)? Are they uniformly continuous there?

c) Find constants a, b, c for which the function f(z) = x + ay + i(bx + cy) is analytic.

d) Let z = r(cos ϕ + i sin ϕ) and f(z) = u(r, ϕ) + iv(r, ϕ). Write down the Cauchy-Riemann equations in polar coordinates.

e) Prove that w = zRez is differentiable only at z = 0 and find w′(0).