Math and stuff

Proposition Find a Laurent series for $\frac{z - 2}{z + 1}$ centered at $z = -1$ and the region in which it converges. Solution \[\begin{align*} \frac{z - 2}{z + 1} &= \frac{(z + 1) - 3}{z + 1} \\ &= 1 - \frac{3}{z + 1} \\ &= \frac{-3}{z + 1}...

Proposition Find a Laurent series for \[\begin{align*} \frac{1}{z(z - 2)^2} \end{align*}\] centered at $z = 2$ and specify the region in which it converges. Solution \[\begin{align*} \frac{1}{z(z - 2)^2} &= \frac{1}{(z - 2 + 2)(z - 2)^2} \\ &= \frac{1}{(z - 2)^3 + 2(z - 2)^2} \\ &= \frac{1}{2(z -...

Proposition Find a Laurent series for \[\begin{align*} \frac{1}{(z - 1)(z + 1)} \end{align*}\] centered at $z = 1$ and specify the region in which it converges. Solution \[\begin{align*} \frac{1}{(z - 1)(z + 1)} &= \frac{1}{(z - 1)(z - 1 + 2)} \\ &= \frac{1}{(z - 1)^2 + 2(z - 1)}...

Proposition Let $A \subset \mathbb{R}^n$ be an open set such that boundary $A$ is an $(n - 1)$-dimensional manifold. Show that $N = A \cup \text{boundary} A$ is an $n$-dimensional manifold-with-boundary. Prove a similar assertion for an open subset of an $n$-dimensional manifold. Solution 1 Let $x \in N$. Case...

Proposition Suppose $f$ is a nonconstant entire function. Prove that any complex number is arbitrarily close to a number in $f(\mathbb{C})$. Solution Let $z_0 \in \mathbb{C}$ and $r > 0$. Suppose that $\abs{f(z) - z_0)} \geq r$ for all $z \in \mathbb{C}$. Then the function $g(z) = \frac{1}{f(z) - z_0}$...