Ricardo A. Sáenz

Due May 22nd Problem 1 Let $latex a\in\mathbb C$ and $latex f,g\in\mathscr O_a$. Then $latex f = g$ if and only if $latex f^{(k)}(a) = g^{(k)}(a)$ for all $latex k\in\mathbb N$. Problem 2 Let $latex \gamma_0, \gamma_1$ paths from $latex a$ to $latex b$ in $latex V$. Then $latex \gamma_0$ and $latex \gamma_1$ are path-homotopic in $latex V$ if and only if $latex \gamma_0-\gamma_1$ is homotopic to a constant in $latex V$, where $latex \gamma_0-\gamma_1:[0,1]\to\mathbb C$ is defined by $latex (\gamma_0-\gamma_1)(t) = \begin{cases}\gamma_0(2t) & 0\le t\le 1/2\\ \gamma_1(2-2t) & 1/2\le t\le 1. \end{cases}$ Problem 3 Let $latex V\subset W\subset\mathbb C$ open connected, $latex u:W\to\mathbb R$ harmonic and $latex f\in H(V)$ such that $latex \Re f = u$ in $latex V$. Then $latex (f,V)$ admits unrestricted continuation in $latex W$.

Due May 15th Problem 1 Let $latex f:[0,1]\to(0,1)$ continuous and $latex V=\{x+iy:x\in(0,1), f(x) < y < 1\}$. Then every point $latex x + if(x), x\in(0,1)$ in $latex \partial V$ is simple. Problem 2 Define $latex f\in H(\mathbb D)$ by $latex f(z) = e^{i/(z-1)^2}$ and $latex \gamma:[0,1]\to\mathbb C$ by $latex \gamma(t) = 1 + (1-t)e^{3\pi i/4}$. $latex \gamma([0,1))\subset\mathbb D$ (draw a picture) $latex \displaystyle \lim_{t\to1} f(\gamma(t))$ exists $latex \displaystyle \lim_{r\to 1} f(r)$ does not exist Find $latex \gamma_1$ such that $latex |f(\gamma_1(t))|\to\infty$ as $latex t\to1$

Due May 8th Problem 1 If $latex P(z)$ is a polynomial, then there exists $latex z_0$ such that $latex |z_0|=1$ and $latex |P(z_0) - \dfrac{1}{z_0}| \ge 1$. Problem 2 There exists a sequence $latex P_n(z)$ of polynomials such that $latex P_n(0) =1$ for all $latex n$ and $latex P_n(z)\to 0$ for all $latex z\in\mathbb C$, $latex z\not=0$. Problem 3 Assume Runge's Theorem and prove the following theorem: Let $latex V\subset\mathbb C$ open and $latex \Gamma$ a cycle in $latex V$. If   $latex \displaystyle \int_\Gamma f(z) dz = 0$  for all $latex f\in H(V)$ of the form $latex f(z) = 1/(z-a)$, then  $latex \displaystyle \int_\Gamma f(z) dz = 0$  for all $latex f\in H(V)$. Problem 4 Let $latex V\subset\mathbb C$ be bounded, connected and open. There exists $latex f\in H(V)$ which cannot be extended to a function holomorphic in a strictly larger open set.