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Mostrando las entradas de mayo, 2020

### Homework 14: Complex Analysis

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

### Homework 13: Complex Analysis

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

### Homework 12: Complex Analysis

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