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## Due March 13th

### Problem  1

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

### Problem 2

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

### Problem 3

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

### Problem 5

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