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Math 246a, Homework 3
Christoph Thiele
due Thursday April 27
Problem 1.
- Consider the standard model of the Riemann sphere in R^3. Show that all rotations of the sphere correspond do linear fractional transformations under the stereographic projection.
- Prove that stereographic projection provides a bijection between the set of circles on the Riemann Sphere and the set of circles and lines in the complex plane.
Problem 2.
(This would make a very typical qual problem.) Find a conformal map from the quarter circle {z : |z| < 1 , <(z) > 0 , =(z) > 0 } to the circle {z : |z| < 1 }. Hint: it can be constructed as a composition of linear fractional transformations and the function z → z^2 (used twice). Show that the map you construct is conformal. Why is z → z^4 not a solution to the problem?
Problem 3. Wirtinger Derivatives
- We use standard x, y coordinates for the complex plane. Consider the partial differential operators ∂ ∂z
( ∂ ∂x
− i
∂y
)
and ∂ ∂z
( ∂ ∂x
∂y
)
and prove that for any i, j ≥ 0
∂ ∂z
zj^ zk^ = jzj−^1 zk
and similarly for the other operator.
- Let N ≥ 0 be an integer and let aj,k, bj,k be complex numbers for j, k = 0,... , N. Prove that if ∑N
j,k=
aj,kzj^ zk^ =
∑^ N
j,k=
bj,kzj^ zk
then aj,k = bj,k for all j, k = 0,... , N. (For example use part 1)
Problem 4
The following is a qualifying exam problem from Fall 2003. “Let f (z) = u(x, y)+iv(x, y) denote a non-constant holomorphic function on some open domain D ⊂ C. Show that at each point of D, the “level curves” u(x, y) = const and v(x, y) = const intersect at right angles.” This problem was slightly ill posed. Give a counterexample to the problem. Then minimally modify the problem so that it becomes a good qual problem and solve it.
Problem 5
Let f : R^2 → R^2 be a holomorphic function. Assume |f (z)| = c is constant. Prove that f is constant.
Problem 6
Assume we have two circles in C such that one is inside the open disc surrounded by the other. Prove that there is a linear fractional transformation that moves these two circles into two concentric circles (we want inner circle to go to inner circle) Prove that the ratio of the two radii of the concentric circles is an invariant; i.e., if you have a different linear fractional transformation moving the two circles into concentric circles, then the ratio is the same. (again, inner circle goes to inner circle)
