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Mostrando las entradas de marzo, 2020

### Homework 6: Complex Analysis

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 For a non-singular complex $2\times 2$ matrix $A$, let $\phi_A$ be the linear-fractional transformation seen in class. Verify directly that $\phi_A\circ\phi_B = \phi_{AB}$. For$A\in GL_2(\mathbb C)$, let $P_A$ be the mappi

### Homework 5: Complex Analysis

Due March 6th Problem 1 Suppose $f\in H(\mathbb C)$, $f(z+1) = - f(z)$ for all $z\in\mathbb C$, $f(0) = 0$ and $|f(z)| \le e^{\pi|\Im z|}$ for all $z\in\mathbb C$. Then $f(z) = c\sin\pi z$ for some constant $c$. ( Hint:  Use Problem 4 of Homework 2.) Problem 2 Suppose $f:\mathbb C\to\mathbb C$ is continuous, $f(z + 2) = f(z)$ and $|f(z)| \le e^{\pi|z|}$ for all $z\in\mathbb C$. Then there exists a constant $c$ such that $|f(z)| \le c e^{\pi|\Im z|}$ for all $z\in\mathbb C$. Problem 3 Find $\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}$ using the series for $\cot\pi z$. Problem 4 Prove that Theorem 7.5 in the text follows from Theorem 7.6. Problem 5 Suppose $f\in H(V)$, $W\subset \mathbb C$ is open, and $h:W\to V$ satisfies $f(h(z)) = z$ for all $z\in W$. Give an example where it is not true that $h\in H(W)$. If $h$ is