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 $\gamma(t) = 1 + (1-t)e^{3\pi i/4}$.

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