Pries: 619 Complex Variables II. Homework 1. Due Thursday 9/5
Computational
1. Miranda I.1.F. Let U be an open subset of X = R2 . Show that these two coordinate
charts are not compatible: φ1 (x, y) = x + iy and
φ2 (x, y) =
x
y
p
p
+i
.
1 + x2 + y 2
1 + x2 + y 2
2. Miranda I.1.G. Show that the transition function for the two stereographic projection
maps is T by checking that φ0 = T ◦ φ∞ on U0 ∩ U∞ .
3. Find coordinate charts for the n-dimensional sphere:
n
S := {(x1 , . . . , xn+1 ) ∈ R
n+1
|
n+1
X
x2i = 1}.
i=1
Thoughtful
1. Show that the reflection ρy in S 2 over the plane y = 0 corresponds (via stereographic
projection φ0 ) to complex conjugation cc in C: cc ◦ φ0 = φ0 ◦ ρy .
2. Show that the set of points (z, w) ∈ C2 such that w2 = sin(z) is a Riemann surface.
3. * Show that the 1-point compactification of the real line is homeomorphic to the circle.