---
date: 2022-12-25 04:46
modification date: Sunday 25th December 2022 04:46:46
title: "Complex Analysis"
flashcard: "Quals::Complex Analysis"
---

# Complex Analysis

## Definitions and trivia

- [x] Types of isolated singularities ✅ 2022-12-26
	- Removable: $\abs{f(z)}$ is bounded in a punctured disc.
	  
	- Poles: $\lim_{z\to p} \abs{f(z)} = \infty$.
	  
	- Essential: neither removable nor a pole.
- [x] Dirichlet's Test ✅ 2022-12-25
	- If $\theset{a_n}, \theset{b_n}$ satisfy
		- $a_n \searrow 0$ 
		- There exists a uniform $M$ such that $\abs{\sum_{n=1}^N b_n} \leq M$ for every $N$,
	- Then $$ \sum_{n=1}^\infty a_n b_n < \infty .$$
- [x] Conformal Map ✅ 2022-12-25
	- A holomorphic map with nowhere vanishing derivative (locally injective).
- [x] Normal family ✅ 2022-12-25
	- A family of functions $\mcf \da \ts{f_j}_{j\in J}$ is **normal** iff every sequence $\ts{f_k}$ has a subsequence that converges locally uniformly, i.e. $\ts{f_{k_i}}$ converges uniformly on every compact subset.
- [x] Equicontinuity ✅ 2022-12-25
	- A family of functions $f_n$ is **equicontinuous** iff for every $\eps$ there exists a $\delta = \delta(\eps)$ (not depending on $n$ or $f_n$) such that  $$\abs{x-y}<\eps \implies \abs{f_n(x) - f_n(y)} < \eps\qquad \forall n.$$
- [x] Definition: An essential singularity ✅ 2022-12-25
	- An isolated singularity that is not a pole (or removable): $$\lim_{z\to z_0 } f(z) \text{ DNE }.$$
- [x] Definition: A pole $a$ of order $m$ ✅ 2022-12-25
	- The smallest $m$ such that $$\lim_{z\to a}(z-a)^{m+1}f(z) < \infty \text{ but } \lim_{z\to a}(z-a)^{k\leq m} f(z) = \infty.$$
- [x] Definition: A removable singularity ✅ 2022-12-25
	- A pole of order zero, so $$\lim_{z\to z_0}f(z) < \infty$$
	and $f$ is bounded on some neighborhood of $z_0$.

## Statements of theorems


- [x] Rouche ✅ 2022-12-25
	- If $f$ and $g$ are holomorphic on $\abs{z-z_0} \leq R$ with $f,g\neq 0$ on $\abs{z-z_0} = R$ and $$\abs{f-g} < \abs{f} + \abs{g},$$
	then $f$ and $g$ have the same number of zeros in this region.
- [x] Cauchy Inequalities ✅ 2022-12-25
	- The following: $$\abs{f^{(n)} (z_0) \over n!} \leq R^{-n} \sup_{\abs{z}=R} \abs{f(z)}.$$
- [x] Cauchy Integral Formula for $f(z)$ ✅ 2022-12-25
	- For $f$ holomorphic in $U\supseteq \bar D$, then for any $z\in D$, $$f(z) = {1 \over 2\pi i} \int _{\bd D} {f(\xi) \over \xi - z} \,d\xi.$$
- [x] Cauchy Integral Formula for $f^{(n)}(z)$ ✅ 2022-12-25
	- For $f$ holomorphic in $U\supseteq \bar D$ and $C$ is a circle such that $C^\circ \subset U$ then for any $z\in C^\circ$, $$f^{(n)}(z)=\frac{n !}{2 \pi i} \int_{C} \frac{f(\zeta)}{(\zeta-z)^{n+1}} d \zeta.$$
- [x] Maximum Length Lemma ✅ 2022-12-25
	- $$\abs{\int _\gamma f} \leq \sup_{z\in \gamma} \abs{f(z)} \cdot \ell(\gamma).$$
- [x] The generalized residue formula ✅ 2022-12-25
	- $$\Res_{z=z_0} f = \lim_{z\to z_0} {1 \over (n-1)!} \qty{\dd{}{z}}^{n-1} (z-z_0)^n f(z).$$
- [x] Cauchy-Goursat Theorem ✅ 2022-12-25
	- If $f$ is holomorphic on a simply connected region $\Omega$ containing a contour $\gamma$, then $$\int_\gamma f = 0.$$
- [x] Mean value theorem for holomorphic and harmonic functions ✅ 2022-12-25
	- [ ] $$f\left(z_0\right)=\frac{1}{2 \pi} \int_0^{2 \pi} f\left(z_0+r e^{i \theta}\right) d \theta=\frac{1}{\pi r^2} \iint_{D_r\left(z_0\right)} f(z) d A$$
- [x] Schwarz Lemma ✅ 2022-12-25
	- If $f: \mathbb{D} \rightarrow \mathbb{D}$ is holomorphic with $f(0)=0$, then
		- $|f(z)| \leq|z|$ for all $z \in \mathbb{D}$
		- $\left|f^{\prime}(0)\right| \leq 1$.
	- Moreover, if
		- $\left|f\left(z_0\right)\right|=\left|z_0\right|$ for any $z_0 \in \mathbb{D}$, or
		- $\left|f^{\prime}(0)\right|=1$, then $f$ is a rotation.
- [x] Morera's Theorem ✅ 2022-12-25
	- [ ] $$f\in C^0(\DD\interior, \CC),\,\int_T f = 0 \, \forall\text{ triangles } T \implies f\in \Hol(\DD, \CC)$$
- [x] Riemann's Removable Singularity Theorem ✅ 2022-12-25
	- [ ] Let $U\subset \CC$ be open, $a\in U$, and $f$ holomorphic on $U\setminus\theset{a}$. 
	Then TFAE
	- $f$ extends holomorphically to all of $U$
	- $f$ extends continuously to all of $U$
	- There exists a neighborhood of $a$ on which $f$ is bounded.
	- The limit characterization: $$\lim_{z\to a} (z-a)f(z) = 0.$$
- [x] Casorati-Weierstrass Theorem ✅ 2022-12-25
	- If $f$ is holomorphic on $\Omega\setminus\theset{z_0} \subseteq \CC$ where $z_0$ is an essential singularity and $V\subseteq \Omega$ then $f(V\setminus\theset{z_0}) \injects \CC$ is dense.
	- If $f$ is non-constant and entire then $f(\CC)\injects \CP^1$ is dense.
- [x] Maximum Modulus Principle ✅ 2022-12-25
	- If $f: \Omega \to \CC$ is holomorphic and not constant on $\Omega$, then $\abs{f}$ is unbounded in $\Omega^\circ$.
- [x] Open Mapping Theorem ✅ 2022-12-25
	- [ ] If $f: \Omega \to \CC$ is holomorphic and not constant on $\Omega$, then $f$ is an open map.

## Proofs

- [ ] Prove Liouville's theorem
- [ ] Prove the FTA
- [ ] Prove Rouche's theorem
- [ ] Prove the Schwarz lemma.
	- In parts:
	- $f(z) = \sum_{k\geq 1}c_k z^k$ since $f(0) = 0$ implies $c_0 = 0$.
	- $g(z) \da f(z)/z = \sum_{k\geq 1}c_k z^{k-1}$ and $g(0) = c_1 = f'(0)$.
	- $\abs{f}\leq 1\implies \abs{g} \leq r\inv$ on $\abs{z} = r$, thus on $\abs{z} \leq r$ by MMP.
	- Take the limit $r\to 1$.
	- Part 2: extremum in interior implies $g(z) \equiv c$ is constant.
	- $\abs{f'(0)} = 1$ or $f(z) = z$ for some $z\neq 0$ implies $\abs{c} = 1$.
	- The actual source:
	- ![](attachments/2022-01-02_23-12-02.png)
- [ ] Prove the open mapping theorem
- [ ] Prove the Casorati-Weirstrass theorem.
- [ ] Prove the Riemann mapping theorem.
- [ ] Prove the argument principle.

## Conformal maps


- [x] Conformal map $\HH\to \DD$ ✅ 2022-12-25
	- ![](attachments/2021-12-28_04-52-21.png)
- [x] Conformal map: right-half-plane to $\DD$ ✅ 2022-12-25
	- ![](attachments/2021-12-28_04-52-47.png)
- [x] Conformal map: image of a region under the exponential map ✅ 2022-12-25
	- ![](attachments/2021-12-28_04-53-25.png)
- [x] Conformal map: strip to half-plane ✅ 2022-12-25
	- ![](attachments/2021-12-28_04-53-43.png)
- [x] Conformal map: $\HH \sm (-\infty ,0]$ to a horizontal strip ✅ 2022-12-25
	- ![](attachments/2021-12-28_04-55-00.png)
- [x] Conformal map: what does $z+a\inv$ do? ✅ 2022-12-25
	- ![](attachments/2021-12-28_04-55-40.png)
- [x] Conformal map: upper half-disc to $\HH$ ✅ 2022-12-25
	- ![](attachments/2021-12-28_04-56-02.png)
- [x] Example of a conformal map that is not injective. ✅ 2022-12-25
	- Example: $$z\mapsto e^z$$
	Not injective because it is periodic, not surjective because it's never zero.
- [x] Cross-ratio map ✅ 2022-12-25
	- $$R(z) \da (z, z_2, z_3, z_4) \da {z - z_3\over z-z_4}{z_2 - z_4 \over z_2 - z_3}.$$
	Sends:
	- $z_2 \to 1$
	- $z_3\to 0$
	- $z_4\to \infty$

## Convergence

- [x] An analytic function with convergence radius 1 which fails to converge at any point on $S^1$ ✅ 2022-12-25
	- Example: \( \sum_{n=1}^\infty nz^n \) 
- [x] An analytic function with convergence radius 1 which converges at every point on $S^1$ ✅ 2022-12-25
	- Example: $$\sum_{n=1}^\infty {z^n\over n^2}$$
- [x] An analytic function with convergence radius 1 that converges at every point on $S^1$ except $z=1$ ✅ 2022-12-25
	- Example: $$\sum_{n=1}^\infty {z^n\over n}$$