Homework 6: Complex Analysis

Due March 13th

Problem  1

Let $latex z\mapsto P(z)$ be the stereographical projecton. Then $latex P(z)$ and $latex P(w)$ are antipodal points if and only if $latex z\bar w = -1$.

Problem 2

Let $latex f(z)$ be an entire function such that $latex \lim_{z\to\infty} f(z) = \infty$. Prove, without any use of the Riemann sphere, that $latex f(z)$ is a polynomial.

Problem 3

Use the previous problem (and not the theorems seen in class) to prove that $latex \text{Aut}(\mathbb C)$ is equal to the set of nonconstant affine maps.

Problem 4

Use the previous problem (and not the theorems seen in class) to prove that $latex \text{Aut}(\mathbb C_\infty)$ is equal to the set of all linear-fractional transformations.

Problem 5

1. For a non-singular complex $latex 2\times 2$ matrix $latex A$, let $latex \phi_A$ be the linear-fractional transformation seen in class. Verify directly that $latex \phi_A\circ\phi_B = \phi_{AB}$.
2. For$latex A\in GL_2(\mathbb C)$, let $latex P_A$ be the mapping on the projective plane $latex \mathbb P_1$ and $latex L:\mathbb C_\infty\to\mathbb P_1$ as defined in class. If $latex M_A = L^{-1}\circ P_A\circ  L$, verify directly that $latex M_A = \phi_A$ on $latex \mathbb C_\infty$.

"Directly" means "work out all the calculations explicitly".