Math and stuff

Proposition Suppose that $f$ is holomorphic at $z_0$, $f(z_0) = 0$, and $f’(z_0) \ne 0$. Show that $f$ has a zero of multiplicity 1 at $z_0$. Solution By Theorem 8.14[A first course in complex analysis], $f$ is either identically 0 on some open disk around $z_0$ or $f(z) = (z...

Proposition Show that \[\begin{align*} \frac{z - 1}{z - 2} = \sum_{k \geq 0}\frac{1}{(z - 1)^k} \end{align*}\] for $\abs{z - 1} > 1$. Solution Suppose $\abs{z - 1} > 1$. \[\begin{align*} \frac{z - 1}{z - 2} &= \frac{z - 1}{(z - 1) - 1} \\ &= \frac{1}{1 - \frac{1}{z - 1}}...

Proposition Find the terms $c_nz^n$ in the Laurent series for $\frac{1}{\sin^2(z)}$ centered at $z = 0$ for $-4 \leq n \leq 4$. Solution Example 8.23 [A first course in complex analysis] has the Laurent series for $\frac{1}{\sin(z)}$. By simply squaring it, we obtain \[\begin{align*} c_i &= \begin{cases} 1 & (i...

Proposition Prove that if $f$ is entire and $\Im(f)$ is constant on the closed unit disk, then $f$ is constant. Solution Let $c$ be the value that $f$ takes on the closed unit disk. Let $g(z) = c$ for all $z$. $g$ is clearly entire. Then $f$ and $g$ agree...

Proposition If $f: A \rightarrow \mathbb{R}$ is non-negative and $\int_A f = 0$, show that $\{ x : f(x) \ne 0 \}$ has measure 0. Solution Let $n \in \mathbb{N}$ be given. We will show that $\{ x : f(x) > 1 / n \}$ has content 0. Let $\epsilon...