Complex Analysis

Aleksandar Tuzlak ¹
ETH Zürich, D-PHYS

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Abstract

important thms., definitions, etc. (minimalistic)
based on Complex Analysis 401-2303-00S HS23

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Cauchy Riemann Equations (CR-eq.) : $f(z) = u(x, y) + iv(x, y)$

$$\partial_x u = \partial_y v$$

$$\partial_y u = -\partial_x v$$

corr. :
$f \in \mathcal{H(\Omega)}$, $\Omega$ open conn., $f' = 0 \implies f = const.$

$f \in \Omega \subset \mathbb{C}$ open, $u, v \in C^0(\Omega)$ $\wedge$ CR-eq. satisfied on $\Omega \implies f \in \mathcal{H}(\Omega)$

$$e^z := \sum_{n \in \mathbb{N}_0}^{} \frac{z^n}{n!}$$

$$\cos z := \frac{e^{iz} + e^{-iz}}{2} = \sum_{n \in \mathbb{N_0}}\frac{(-1)^n}{(2n!)}z^{2n} = cosh(iz)$$ $$\sin z := \frac{e^{iz} - e^{-iz}}{2i} = \sum_{n \in \mathbb{N_0}}\frac{(-1)^n}{(2n+1)!}z^{2n+1} = -isinh(iz)$$

Goursat Corollary $\leadsto$ same for rectangles (2 triangles)

$\Omega \subset \mathbb{C}$ open, $T \subset \Omega$ Triangle , $T^\circ \subset \Omega$ $$\implies \int_{T}^{} (f\in \mathcal{H}(\Omega)) = 0$$

locally every $f \in \mathcal{H}$ has a primitive $F$

$f \in \mathcal{H}(D)$, $D$ open disc $\in \mathbb{C}$ $\implies \exists F \in D: F'= f$

$D$ (from above), $f \in C^0(D) : \int_{\text{$\partial$\{closed rect.\}}}^{}f \stackrel{Goursat}{=} 0$ when sides $||$ coord. ax. $\implies \exists$ $F$ in $D$

Cauchy's thm. for a disc Corr. $\leadsto$ circles (proof with "keyhole"), generally $\leadsto$ toy contours
$\int_{\gamma}^{}\frac{1}{z - z_0} = 2\pi i$, $\gamma$: toy contour (circle, rect., ...)

$f \in \mathcal{H}(D \backslash \{z_0\}) \quad \wedge \quad f \in C^0(D)$ $$\implies \int_{\gamma}^{}f = 0$$ $\gamma$ closed $\in C^{\infty}_{pw}$

CIF corr. $\leadsto f \in \mathcal{H}(\Omega) \implies f \in C^{\infty}(\Omega)$

$f \in \mathcal{H}(\Omega)$, $\bar{D} \subset \Omega$ open $$\implies f(z) = \frac{1}{2\pi i}\int_{\gamma = \partial D}^{}\frac{f(w)}{w - z} dw$$ $\forall z \in D$ $$\stackrel{\text{*}}{\iff}\frac{1}{2\pi i}\int_{\gamma = \partial D}^{}\frac{f(w)}{w - z} dw = f(z)w_{\gamma}(z)$$ $\forall z \in \Omega \backslash \gamma$, $\Omega$ simply conn. $\Omega \subset \mathbb{C}$ open is simply conn. if it is conn. & any 2 curves with same endpts. are Homotopic (~one blob with no holes) , *: generalized res.thm applied to $\frac{f(w)}{w - z} = g(w) \in \mathcal{M}(\Omega\backslash z)$ $$\leadsto C\subset \Omega: C^{\circ}\subset \Omega$$ $$\implies f^{(n)}(z) = \frac{n!}{2\pi i}\int_{\gamma = \partial D = C}^{}\frac{f(w)}{(w - z)^{n+1}} dw$$ $\forall z \in C^{\circ}$

winding number $w_{\gamma}$ $\leadsto$ prop.:
$\implies w_{\gamma} \in C^{\circ}(\Omega, \mathbb{C}), w_{\gamma}(z) \in \mathbb{Z}$, hence const. on any conn. sset. of $\Omega$, $w_{\gamma}(z) = 0$ if $|z|$ large enough

$z_0 \in \mathbb{C}$, $\not\in \gamma \in C^{\infty}_{pw}$ closed, $w_{\gamma}(z_0) := \frac{1}{2\pi i}\int_{\gamma}^{}\frac{1}{z - z_0}dz$

Meromorphic function $f \in \mathcal{M}$ $\mathcal{M}$ is a $\mathbb{C}$ vector space:
$f, g \in \mathcal{M}(\Omega):$
$af + bg, fg \in \mathcal{M}(\Omega)$
$0 \not\equiv f \in \mathcal{M}(\Omega)$ and zeroes of f don't have lim. pt. in $\Omega \implies \frac{1}{f} \in \mathcal{M}(\Omega)$

$f:\Omega\to\hat{\mathbb{C}}$, $\Omega$ open

$1)\quad S_f := \{z \in \Omega : f(z) = \infty\}$ has no lim. pt. in $\Omega$, i.e. $S_f$ si discrete in $\Omega$

$2)\quad f \in \mathcal{H}(\Omega \backslash S_f)$ $$\implies f \in \mathcal{M}(\Omega) \supseteq \mathcal{H}(\Omega)$$

combination of 2 thms. on zeroes and poles of functions sometimes useful but theoretically included in the thm. already:
$1) \quad f$ has pole of ord. $m$ at $z_0$ (i.e. $|\lim_{z \to z_0}(z - z_0)^m f(z)| < \infty$) and $m$ is the smallest such integer
$2) \quad \exists r>0: D_r(z_0) \subset \Omega \wedge h \in \mathcal{M}(D_r(z_0)): h(z) \not = 0 $
$ \forall z \in D^*_r(z_0)$
$h$ has a zero of ord. $m$ at $z_0$ and s.t.: $f(z) = \frac{1}{h(z)} \quad \forall z \in D^*_r(z_0)$
(ord. of zero $z_0$: $ord_{z_0}f := min(\{k \geq 0 : f^{k}(z_0) \not= 0\})$)

$f \in \mathcal{M}(\Omega), f \not= 0, z_0 \in \Omega$

$1)\quad ord_{z_0}f \iff \exists r>0, h \in \mathcal{H}(D_r(z_0)):$
$h(z_0) \not= 0, f(z) = (z - z_0)^{ord_{z_0}f}h(z) \quad \forall z \in D^*_r(z_0)$

$ord_{z_0}f < 0$ if $z_0$ pole

$ord_{z_0}f > 0$ if $z_0$ zero

$2)\quad ord_{z_0}(fg) = ord_{z_0}f + ord_{z_0}g, ord_{z_0}(\frac{f}{g})$ =
$ord_{z_0}f - ord_{z_0}g$

$3)\quad ord_{z_0}(f + g) \geq min({ord_{z_0}f, ord_{z_0}g})$

$f \in \mathcal{M}(\mathbb{\hat{C}})$ $$\implies \mathcal{M}(\mathbb{\hat{C}}) = \{\frac{P}{Q}: P, Q \in \mathbb{C}[z]\}$$ i.e. Meromorphic func. are rational func.

pwr. series exp. at $z_0$ f pwr. series, rad. of conv. = R $\implies f'$ obtained by termwise diff. has same rad. of conv. Laurent Series: $\sum_{n \in \mathbb{Z}}^{}a_n(z - z_0)^n$, Annulus $\mathcal{A}$...

$f \in \mathcal{H}(\Omega)$, $D \hat{=} D_R(z_0)$ from above $$\implies f(z) = \sum_{n \in \mathbb{N}_0}^{}a_n(z - z_0)^n, a_n := \frac{f^{n}(z_0)}{n!}$$ $\forall z \in D_R(z_0)$ $$\implies f \in \mathbb{C}^{\infty} \iff f \in \mathcal{H} \implies f' \in \mathcal{H}$$ radius of conv. of atleast $R = min\{|z - z_0|\}, R := (\limsup_{n \to \infty}|a_n|^{\frac{1}{n}})^{-1}$ $=$ $\lim_{n \to \infty}|\frac{c_n}{c_{n+1}}|$

Liouville's thm.

$f \in \mathcal{H}(\mathbb{C}) \wedge$ bdd. $$\implies f = const.$$

$\Uparrow$

Generalized Liouville's thm.

$f \in \mathcal{H}(\mathbb{C}) : |f(w)| \leq c|w|^n \quad \forall w \in \{z \in \mathbb{C} : |z| > C\}$ for $c, C > 0, n \geq 0$ $$\implies f \in \mathbb{C}[z]_n$$

$f \in \mathcal{H}(\Omega)$, $\Omega$ conn. open that vanishes on seq. of distinct pts. with lim in $\Omega$ $$\implies f \equiv 0$$

identity thm.

$f, g \in \mathcal{H}(\Omega)$, $\Omega$ open, conn. $\not= \emptyset$ TFAE:

$a)\quad f = g$

$b)\quad \exists a \in \Omega: f^{(n)}(a) = g^{(n)}(a)$ $\forall n \geq 0$

$c)\quad \{z \in \Omega: f(z) = g(z)\}$ has lim. in $\Omega$

Morera

$f \in C^0(D)$, $D$ open, $\forall T \subset D:$ $\int_{T}^{}f = 0$ $$\implies f \in \mathcal{H}(D)$$

$\{f_n\}_{n\geq1}$ seq.: $f_i \in \mathcal{H}(\Omega)$, that conv. unif. seq. $f_1, f_2,... : \Omega \to \mathbb{C}$ of func. def. on $\Omega \subset \mathbb{C}$ is unif. conv. to lim. $f:\Omega\to\mathbb{C}$ if:
$\forall\varepsilon>0 \exists N > 0 : |f(z) - f_n(z)| < \varepsilon$ $\forall n \geq N, z \in \Omega$ to a func. f in every comp. sset of $\Omega$ $$\implies f\in \mathcal{H}(\Omega)$$ $\stackrel{generally}{\leadsto} f_n^{(k)} \to f^{(k)}$

unif. conv. on compacta $\iff$

$1)\quad \forall a \in \Omega \exists \varepsilon > 0: B_{\epsilon}(a) \subset \Omega : (f_n|_{B_{\epsilon}(a)})$ conv. unif. $\wedge$

$2)\quad \forall $ comp. sset. $K \subset \Omega: (f_n|_K)$ conv. unif.

Weierstrass M-test

Riemann contin. thm.

$z_0 \in \mathbb{C}$, $f \in \mathcal{H}(\Omega\backslash\{z_0\})$, $\Omega \not= \emptyset$ TFAE:

$1)\quad f$ is holom. ext. to $\Omega$

$2)\quad f$ is cont. ext. to $\Omega$

$3)\quad f$ is bdd. in nbhd. of $z_0$ i.e. $\exists r>0 : |f(z)| \leq M \in \mathbb{R}$ $\forall z \in D^*_r(z_0)$

$4)\quad \lim_{z \to z_0}f(z) = 0$

Riemann's thm. on remov. sing. $3) \implies 1)$ from R-contin.thm

$f \in \mathcal{H}(\Omega\backslash\{z_0\})$, $f$ bdd. in $D^*_r(z_0)$ for some $D_r(z_0)\subset \Omega$ $$\implies z_0 \text{ is a remov. sing. of } f$$ i.e. $\exists F\in \mathcal{H}(\Omega, \mathbb{C}): F(z) = f(z) \quad \forall z\in \Omega\backslash\{z_0\}$

func. with poles special case of Laurent series:
$f(z) = \sum_{j = -n}^{\infty}a_j(z - z_0)^j$ same hypotheses as above

if $f$ has pole of ord. $n$ at $z_0$: $$f(z) = \sum_{j = 1}^{n} \frac{a_{-j}}{(z - z_0)^j} + G(z) = P(z) + G(z)$$

$G \in \mathcal{H}(D_r(z_0))$, $a_{-1} := res_{z_0}f$

residue $res_{z_0}f$

$f$ pole of ord. $n$ at $z_0$ $$res_{z_0}f := \lim_{z \to z_0}\frac{1}{(n - 1)!}(\frac{d}{dz})^{(n - 1)}(z - z_0)^nf(z)$$

$f, g \in \mathcal{H}(\{z_0\})$, $g(z)$ simple zero at $z_0$ $$\implies \frac{f}{g} \text{ has simple pole at $z_0$}$$ and $$res_{z_0}\frac{f(z)}{g(z)} = \frac{f(z_0)}{g'(z_0)}$$

General Residue Formula/thm.(using homotopy thm.) $\implies$ CIF

$\Omega \subset \mathbb{C}$ simply conn., $f\in \mathcal{M}(\Omega)$, $V:=\Omega\backslash S_f$, $\gamma \subset V$ closed curve $$\int_{\gamma}^{}f = 2\pi i \sum_{z_i \in S_f}w_{\gamma}(z_i)res_{z_i}(f)$$

Homotopy thm.

$\Omega \subset\mathbb{C}$ open, $\gamma_{0, 1}:[a, b] \to \Omega$ s.t.:

$1)\quad \gamma_{0, 1}$ are closed and $\gamma_{0} \sim_{\Omega}\gamma_{1}$ (Homotopic) or

$2)\quad \gamma_{0, 1}$ have same endpts. and are homotopic with fixed endpts. $$\implies \int_{\gamma_0}^{}f = \int_{\gamma_1}^{}f$$ $f\in \mathcal{H}(\Omega)$

$\Downarrow$

$f \in \mathcal{H}(\Omega)$, $\Omega$ simply conn. $\implies \exists F: F' = f$, in part.: $\int_{\gamma_{closed}\in \Omega}^{}f \stackrel{\text{cauchy thm.}}{=} 0$, any 2 primitives differ by a const.

Casorati-Weierstrass ($\Leftarrow$ Picard thm. $f\in \mathcal{H}(D^*_r(z_0))$ with ess. sing. $z_0$ $\implies \#\{\mathbb{C}\backslash f(D^*_r(z_0))\} \leq 1$ )

$f \in \mathcal{H}(D^*_r(z_0))$ with ess. sing. $z_0$ $$\implies f(D^*_r(z_0)) \text{ is dense in $\mathbb{C}$}$$ dense in $\mathbb{C}$:
$\forall z \in \mathbb{C}, \varepsilon > 0 \exists w \in \mathbb{C}: |z - f(w)| < \varepsilon $

Arg. princ.

$\Omega \subset \mathbb{C}, \gamma \subset \Omega$ s.t. res. formula holds, $f\in \mathcal{M}(\Omega)$

if $f$ has no zeros or poles on $\gamma$ $$\implies \frac{1}{2\pi i} \int_{\gamma}^{}\frac{f'}{f} = \sum_{z_i \in Z_f \cap \gamma^{\circ}}^{}ord_{z_0}f - \sum_{z_i \in S_f \cap \gamma^{\circ}}^{}ord_{z_0}f$$ $$\hat{=} \quad \#(Z\cap\gamma^{\circ}) - \#(P\cap\gamma^{\circ})$$ $\frac{f'}{f} \in \mathcal{M}(\Omega) \quad \hat{=}$ $logf$, "logarithmic derivative of f"
has poles of ord. 1 at $z_0 \in \Omega$ i.e. $z_0$ is either pole or zero of $f$
$res_{z_0}(\frac{f'}{f}) = ord_{z_0}f$
$Z_f :=$ set of zeroes of f
$S_f :=$ set of poles of f

Rouché's thm. gives the nicest proof to fund. thm of algebra

$f, g \in \Omega \supset C, C^{\circ}$, $\Omega$ open

$|f(z)| > |g(z)| \quad \forall z \in C$ $$\implies f, f + g \text{ have same #zeros }\in C$$

open mapping thm.

$\Omega$ open conn., $f\in \mathcal{H}(\Omega)$, $f$ not a const (i.e. $f$ has iso. sing.) $$\implies f \text{ is open }$$ (map is open if it maps open sets to open sets)

$\implies f \not=$ const. if $f \in \mathcal{H}(D_r(0))$

$\implies$ it's not possible that: $f(z) \in \mathbb{R} \quad \forall z$ , since any sset of $\mathbb{R}$ is not open in $\mathbb{C}$

$\Downarrow$

max. mod. princ. special case of open mapping thm. since if $|f|$ attains local max. at $z \implies img(B_\varepsilon (z))$ cannot be open $\implies f =$ const.

$\Omega$ open, conn., $f \in \mathcal{H}(\Omega)$

$\exists z_0 \in \Omega: |f(z_0)| \geq |f(z)| \quad \forall z \in B_\varepsilon(z_0) $ $$\implies f = \text{ const. $\in \Omega$ }$$ alternatively: (contraposition of statement above)
$\Omega \subset \mathbb{C}$ open conn., $f \in \mathcal{H}(\Omega)$ not const.: $$\implies \not\exists z_0 \in \Omega : |f(z_0)| \geq |f(z)| \quad \forall z \in \Omega$$ ($f$ cannot attain a max. in $\Omega$)

in part. if $\bar{\Omega}$ bdd., $f \in C^0(\bar{\Omega})$ $$\implies max_{z \in \bar{\Omega}}|f(z)| = max_{z \in \partial\Omega}|f(z)|$$

Homotopy

branch of log, Log

existence of log

conformal map

$U, V \subset \mathbb{C}$ open

inj. holom. map : $f: U \to V$ is called conformal map

if $f$ is bij. : conformal equivalence conf. equiv. is an equivalence relation:
$\sim_C$ i.e.:
$U\sim_CU \quad \hat{=} \quad f: U\to U, u\mapsto u$ (id.)
as well as symmetry and transitivity (/biholom./holom. isomorph.) $\implies f\in \mathcal{H}$

$U$ and $V$ are conformally equivalent if $U = V \iff$ automorphism $\mathbb{C} \stackrel{Liouville}{\not\sim_C} \mathbb{D}$

Riemann thm. corr. $\leadsto$ any 2 proper, simply conn. open ssets of $\mathbb{C}$ are conf. equiv.

$\Omega$ proper $\not= \emptyset \wedge \Omega \subsetneq \mathbb{C}$ simply conn., $z_0\in \Omega$ $$\implies \exists ! F \text{ (conformal) } : \Omega \to \mathbb{D}:$$ $$F(z_0) = 0, F'(z_0) > 0$$

only autom. of $D_1$ that fix the origin are rotations

if $f:\mathbb{D} \to \mathbb{D}$ is an automorph. of $\mathbb{D}$ $$\implies \exists\theta\in \mathbb{R}, \alpha\in \mathbb{D} : f(z) = e^{i\theta}\psi_{\alpha}(z)$$ $$\psi_{\alpha}(z) = \frac{\alpha - z}{1 - \bar{\alpha}z}$$ conversely all maps of this form are automorph. of $\mathbb{D}$

Schwarz's Lemma ($\Leftarrow$ mod. princ.)

$f\in \mathcal{H}(\mathbb{D},\mathbb{D}): f(0) = 0$

$1) \quad |f(z)| \leq |z| \quad \forall z\in \mathbb{D}$

$2) \quad$ if for some $z_0 \not= 0$ we have $|f(z_0)| = |z_0| \implies f$ is a rot.

$3) \quad |f'(0)| \leq 1$,
$|f'(0)| = 1 \iff f$ is a rot. ($\exists\theta\in\mathbb{R}:f(z)= e^{i\theta}z$)

every automorph. $g\in \mathcal{H}(\mathbb{H}, \mathbb{H})$ is of the form: $$g(z) = \frac{az + b}{cz + d}$$

Montel thm.

Integrals

f even $\implies \int_{\mathbb{R^+}}^{}f = \frac{1}{2}\int_{\mathbb{R}}^{}f$

$1) \quad$ e.g.: $\int_{\mathbb{R}}^{}\frac{1}{(x^2 + a^2)^2}dx$ (sketch: path integral over $\mathbb{D} \cap \mathbb{H} \leadsto \gamma := \gamma_{R} \cup [-R, R]$, $\gamma_{R}\stackrel{R\to \infty}{\to}0, \lim_{R\to\infty}[-R, R] = (-\infty, \infty)$)
$f := \frac{P(x)}{Q(x)}$ $$I := \int_{\mathbb{R}}^{}fdx$$ $P, Q \in \mathbb{R}[x]: deg(Q) \geq deg(P) + 2$ s.t.: $|\int_{\gamma_R}^{} fdx| \leq C\frac{1}{R^{degQ - degP}}R \stackrel{R\to \infty}{\to}0$ , $\not\exists z_0 \in \mathbb{R}: Q(z_0) = 0$ $$\implies I = 2\pi i\sum_{\substack{z_i \in S_f\\ S_f \cap \gamma = \emptyset}}res_{z_i}f = 2\pi i \sum_{\substack{(imz_i > 0) \in S_f \\ S_f \cap \gamma = \emptyset}}res_{z_i}f$$ since our choice of path

$1.1) \quad$ e.g.: $\int_{\mathbb{R}}^{}\frac{1}{x^2 + 1} cos(ax)dx$ ($|e^{iz} = e^{i(x+iy)}| = e^{-y} \implies |e^{iz}| \leq 1 : imz > 0$) $$\int_{\mathbb{R}^{}}f cos(ax) dx$$ define $f_{new} := f(z)e^{iaz}$
sol. $\leadsto$ same as above

$2) \quad$ e.g.: $\int_{0}^{2\pi}\frac{1}{a + cos\theta}d\theta$ (sketch: contour integral around $C_1(0)$ with subst. $z := e^{i\theta}$) $P, Q \in \mathbb{R}[x], Q(x, y) \not= 0$ for $x^2 + y^2 = 1 \quad \forall x, y\in \mathbb{R}$ $$\int_{0}^{2\pi}\frac{P(cost, sint)}{Q(cost, sint)}dt$$ $\implies$ res. thm.

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