Pries: 619 Complex Variables II. Homework 2. Due Thursday 9/12
Computational
1. Let f (x, y) = x3 + y 3 + 1 − txy. Find the values of t for which Vf is not smooth.
P
2. Find the zeros and poles of f (z) = (z 2 +3)/(z 4 +3z 2 +2) and check P ∈C∞ ordP (f ) = 0.
3. Automorphisms of C∞ are of the form f (z) = (az + b)/(cz + d) where a, b, c, d ∈ R.
Prove that f (z) is invertible iff |ad − bc| = 1 and that f (z) stabilizes the upper half
plane iff ad − bc > 0.
Thoughtful
1. Miranda I.2 F. Show that the group law on a complex torus X is divisible: given P ∈ X
and an integer n ≥ 1, there is a point Q ∈ X such that nQ = P . In fact there are
exactly n2 such points.
2. If f (z) = p(z)/q(z)
is a rational function on C, show that ord∞ (f ) = deg(q) − deg(p).
P
Show that P ∈C∞ ordP (f ) = 0.
3. * Prove that the map f : H → D given by f (z) = z−i
is distance-preserving and a
z+i
diffeomorphism (bijective map such that both f and f −1 are differentiable).