Ricardo A. Sáenz

Problem 1 Let $latex V\subset \mathbb C$ be open and connected and $latex \mathcal F\subset H(V)$. If $latex \mathcal F$ is a normal family, then $latex \mathcal F' = \{f': f\in\mathcal F\}$ is also a normal family. Problem 2 The converse to the result in Problem 1 is false. Find a "small" hypothesis to make the converse true. Problem 3 Let $latex V\subset \mathbb C$ be open and connected and $latex \mathcal F\subset H(V)$ a normal family. Let $latex f_n\in \mathcal F$ a sequence such that $latex f_n(z)\to f(z)$ for each $latex z\in S\subset V$, where $latex S$ has a limit point in $latex V$ and $latex f\in H(V)$. Then $latex f_n\to f$ in $latex H(V)$. Problem 4 Let $latex V\subset \mathbb C$ be open, $latex M>0$, and  $latex \displaystyle\mathcal F = \Big\{ f\in H(V): \iint_V |f(z)|^2 dxdy \le M \Big\}$. Then $latex \mathcal F$ is a normal family. Problem 5 Let $latex V\subset \mathbb C$ be open and connected and $latex \{f_n\}\subset H(

Problem 1 Consider the function on $latex [0,\infty)$ given by $latex \psi(t) = \dfrac{t}{1+t}$. $latex \psi$ is concave and increasing $latex \psi$ is continuous at $latex 0$ For any $latex \varepsilon > 0$ there exists $latex \delta > 0$ such that $latex \psi(t) < \delta$ implies $latex 0\le t < \varepsilon$ Problem 2 Let $latex d_j$ be a sequence of quasimetrics on $latex X$ that separates points, and $latex d$ the metric constructed from the $latex d_j$ as in class. Let, for each positive integer $latex N$, $latex x\in X$ and $latex \varepsilon>0$, $latex B_N(x,\varepsilon) = \{ y\in X: d_j(x,y) < \varepsilon, j = 1, 2, \ldots, N\}.$ Then $latex B_N(x,\varepsilon)$ is open with respect to the metric $latex d$. Problem 3 $latex S\in X$ is totally bounded in $latex (X,d)$ if and only if for any $latex \varepsilon > 0$ and any positive integer $latex N$ there exist $latex x_1, x_2, \ldots, x_m$ such that $latex \displaystyle S \subset \bigc

Problem 1 The map $latex \Phi(z) = i\dfrac{1+z}{1-z}$ is a biholomorphic mapping from the unit disk $latex \mathbb D$ onto the upper half-plane $latex \mathbb H$. $latex \Phi$ is called the Cayley transform . Problem 2 Let $latex A=\begin{pmatrix} a & b\\c & d \end{pmatrix}\in SL(2,\mathbb R)$ and $latex \phi_A$ the Möbius transformation $latex \phi_A(z) = \dfrac{az+b}{cz+d}$. Then $latex \phi_A\in\text{Aut}(\mathbb H)$. Problem 3 For $latex a\in\mathbb H$, give an explicit $latex A\in SL(2,\mathbb R)$ so that $latex \phi_A$ takes $latex a$ to $latex i$. Problem 4 If $latex \psi\in\text{Aut}(\mathbb H)$, then there exists $latex A\in SL(2,\mathbb R)$ so that  $latex \psi = \phi_A$. ( Hint:  Consider the matrix $latex A = \begin{pmatrix} \cos\theta & -\sin\theta\\\sin\theta & \cos\theta \end{pmatrix}$ and verify that $latex \Phi^{-1}\circ\phi_A\circ\Phi$ is a rotation in $latex \mathbb D$. Use the previous problem and the discussion in class, with

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 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 $latex \phi_A\circ\phi_B = \phi_{AB}$. For$latex A\in GL_2(\mathbb C)$, let $latex P_A$ be the mappi

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