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Homework 13: Complex Analysis

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 $latex \gamma(t) = 1 + (1-t)e^{3\pi i/4}$.
  1. $latex \gamma([0,1))\subset\mathbb D$ (draw a picture)
  2. $latex \displaystyle \lim_{t\to1} f(\gamma(t))$ exists
  3. $latex \displaystyle \lim_{r\to 1} f(r)$ does not exist
  4. Find $latex \gamma_1$ such that $latex |f(\gamma_1(t))|\to\infty$ as $latex t\to1$