• Complex Analysis
• complex number $\mathbb{C}$
  • complex plane
  • binary operation of complex number
  • complex number short syntax
  • Argand diagram
  • complex conjugation
  • Euler’s formula
    • de Moivre’s formula
  • complex root
  • standard topology ${\mathcal{O}}_s$
  • region
    • connectedness
    • bounded
  • complex infinity
  • Riemann sphere $\mathbb{S}$
• limit
  • connection between complex limit and real limit
• continuity
• differentiation
  • Cauchy-Riemann equations
    • Cauchy-Riemann equations in polar coordinate
    • differentiability from Cauchy-Riemann equations
  • holomorphicity
    • entire function
    • constancy from holomorphicity
• parametric curve
  • parametric derivative
  • parametric integral
  • reparametrization
    • parametric curve length
• contour
  • Jordan curve theorem
  • contour integral
    • upper bound theorem for contour integral (ML theorem)
    • path independence of contour integral
  • Cauchy-Goursat theorem
    • Cauchy’s theorem
  • Cauchy-Goursat theorem on simply connected domain
  • adoption of Cauchy-Goursat theorem on multiply connected domain
    • path deformation principle
  • Cauchy’s integral formula (Cauchy’s formula)
    • evaluating integral with Cauchy’s integral formula
  • Morera’s theorem
  • Liouville’s theorem
  • fundamental theorem of algebra
  • maximum modulus principle
• analyticity
• Laurent series
  • residue
    • Cauchy residue theorem
    • residue at infinity
• singularity
  • isolated singularity
    • essential isolated singularity
    • pole
      • residue at pole
    • zero of order $m$
      • zero and pole
• improper integral
  • Cauchy principal value $CPV$
  • improper integral for even rational function
  • Fourier integral
    • Jordan’s lemma
    • Jordan’s inequality
  • indented path
    • lemma for indented path
  • lemma for integral over branch point
• trigonometric integral
• Laplace transform
  • Bromwich formula
• meromorphic
  • theorem for meromorphic function
  • winding number
  • argument principle
  • Rouché’s theorem (dog on a leash theorem)
  • Brouwer’s fixed point theorem special version
  • Hopf’s theorem
  • vector field index theorem
    • vector field singularity
    • vector field index
• Möbius transformation
  • circle in Argand diagram
  • cross-ratio
• conformal transformation
  • inverse function theorem
  • conformal transformation preserve angle
  • conformal transformation preserve harmonic
• harmonic function
  • Dirichlet problem
  • maximum principle
  • uniqueness of solution to Dirichlet problem

Complex Analysis

complex number $\mathbb{C}$

$z=(x,y)$ in complex plane where $x,y\in \mathbb{R}$

• real part, $x=\text{Re}z$
• imaginary part, $y=\text{Im}z$

complex plane

• real axis/ imaginary axis
• $(0,y)$ pure imaginary number, $y\ne 0$

binary operation of complex number

• addition, $(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$
• multiplication, $(x_1,y_1)\cdot (x_2,y_2)=(x_1{x}_2-y_1{y}_2,x_1{y}_2+x_2{y}_1)$

complex number short syntax

$(x,y)=(x,0)+(0,y)=(x,0)+(0,1)\cdot (y,0)=:x+iy$

where $i:=(0,1)$

Argand diagram

diagram in ${\mathbb{R}}^2$ given by

$$x+iy\mapsto x\vec{i}+y\vec{j}$$

• modulus, $=$ Euclidean length in Argand diagram

  $$|z|=\sqrt{x^2+y^2}$$

• triangle inequality hold

complex conjugation

$\bar{z}=x-iy$

• $\overline{\stackrel{ˉ}{z}}=z$
• $|\bar{z}|=|z|$
• $\overline{z_1+z_2}={\bar{z}}_1+{\bar{z}}_2$
• $\overline{z_1{z}_2}={\bar{z}}_1{\bar{z}}_2$
• $x=\frac{z+\bar{z}}{2},y=\frac{z-\bar{z}}{2i}$
• $|z{|}^2=z\bar{z}$

Euler’s formula

$$\begin{gathered} e^{i\theta }=\cos \theta +i\sin \theta \\ \Rightarrow x+iy=r{e}^{i\theta } \end{gathered}$$

• $r=|z|$
• $\theta =\arg z$, argument of $z$
  • $-\pi <\text{Arg }z\le \pi$, principal argument of $z$
• $z_1{z}_2=r_1{r}_2{e}^{i({\theta }_1+{\theta }_2)}$
  • $\arg (z_1{z}_2)=\arg z_1+\arg z_2$
• $z^{-1}=\frac{1}{r}{e}^{-i\theta }$
• $\bar{z}=r{e}^{-i\theta }$

de Moivre’s formula

$$(\cos \theta +i\sin \theta {)}^n=e^{in\theta }=\cos n\theta +i\sin n\theta$$

complex root

$z_0=r_0{e}^{i{\theta }_0}$

$n$th root of $z_0$

$$\begin{gathered} z^n=z_0 \\ \Rightarrow z_k=\sqrt[n]{r_0}{e}^{i\left(\frac{{\theta }_0+2\pi k}{n}\right)},k\in \{0,1,\cdots ,n-1\} \end{gathered}$$

• $n$ distinct root
• principal root, when ${\theta }_0=\text{Arg }{z}_0,k=0$

standard topology ${\mathcal{O}}_s$

see context in Topology

$\epsilon >0$

open $\epsilon$-neighborhood

$$U_{\epsilon }(z_0)=\{z\in \mathbb{C}:|z-z_0|<\epsilon \}\in {\mathcal{O}}_s$$

region

nonempty connected set

• domain, open region

connectedness

$D\subseteq \mathbb{C}$ is connected $\iff$

$\forall z_1,z_2\in D,\exists$ polygonal line with finitely many segment between $z_1$, $z_2$

bounded

$\exists$ circle of finite radius containing the set

complex infinity

$x\to \infty$ or $y\to \infty$

Riemann sphere $\mathbb{S}$

$$x^2+y^2+z^2=1$$

intersection of the sphere with line through north poll and complex number

• north poll: $(0,0,1)$, correspond to complex infinity
• Argand plane $(x,y,0)$

limit

$$\underset{z\to z_0}{\lim }f(z)=w_0$$

$\iff \forall \epsilon >0,\exists \delta >0$ s.t.

$$|f(z)-w_0|<\epsilon \text{ whenever }|z-z_0|<\delta$$

• unique if exist
• independent of direction of approach

connection between complex limit and real limit

$f(z)=u(x,y)+iv(x,y)$

$$\underset{z\to z_0}{\lim }f(z)=\underset{(x,y)\to (x_0,y_0)}{\lim }u(x,y)+i\underset{(x,y)\to (x_0,y_0)}{\lim }v(x,y)$$

$\iff$ limit on the RHS exist

continuity

$f$ is continuous at $z_0\iff$

1. $f$ exist
2. ${\lim }_{z\to z_0}f(z)$ exist
3. $$\underset{z\to z_0}{\lim }f(z)=f(z_0)$$

differentiation

$f:D\to \mathbb{C},U_{\epsilon }(z_0)\subseteq D$

$f$ differentiable at $z_0\iff$

$$\underset{z\to z_0}{\lim }\frac{f(z)-f(z_0)}{z-z_0}$$

exist

• $f(z)$ depend on $\bar{z}\Rightarrow f$ not differentiable
• $u,v$ differentiable $\nRightarrow f$ differentiable

Cauchy-Riemann equations

$f^{\prime }(z)$ exist at $z_0\Rightarrow$

$$\begin{gathered} u_x=v_y,u_y=-v_x \\ {f}^{\prime }=u_x+i{v}_x \end{gathered}$$

Cauchy-Riemann equations in polar coordinate

$f(r,\theta )=u(r,\theta )+iv(r,\theta )$

$$\begin{gathered} r{u}_r=v_{\theta },u_{\theta }=-r{v}_r \\ {f}^{\prime }=e^{-i\theta }(u_r+i{v}_r) \end{gathered}$$

differentiability from Cauchy-Riemann equations

$f$ defined around $z_0$,
at $z_0$

1. $u_x,u_y,v_x,v_y$ are continuous
2. satisfy Cauchy-Riemann equations

$$\Rightarrow f^{\prime }=u_x+i{v}_x$$

exist

holomorphicity

$f:D\to \mathbb{C}$ holomorphic at $z_0\iff$

$f$ differentiable at each point in $U_{\epsilon }(z_0)\subseteq D$

• a.k.a. analytic/ regular
• $u,v\in C^{\infty }(D)$ and are harmonic function in $D$

entire function

holomorphic at every finite point

constancy from holomorphicity

$f$ holomorphic, in $D$

$f$ is constant

$\iff f^{\prime }=0$

$\iff \bar{f}$ holomorphic in $D$

$\iff |f|$ is constant

parametric curve

complex-valued function

$$\begin{gathered} C:[a,b]\subseteq \mathbb{R}\to \mathbb{C}, \\ t\mapsto z(t)=x(t)+iy(t) \end{gathered}$$

• simple $\iff$ does not intersect self $\iff$ injection
  • simple closed (Jordan curve) $\iff$ simple except two end meet
• oriented
  • positively oriented $\iff$ simple closed counterclockwise
  • negatively oriented

parametric derivative

$x^{\prime }(t),y^{\prime }(t)$ continuous

$\Rightarrow C$ differentiable,

$$z^{\prime }(t)=x^{\prime }(t)+i{y}^{\prime }(t)$$

• velocity
  • speed $|z^{\prime }|$
• smooth $\iff$ differentiable and $z^{\prime }\ne 0$

  • $C$ has unit tangent vector in Argand diagram

    $$T(t)=\frac{z^{\prime }(t)}{|z^{\prime }(t)|}$$

parametric integral

$$\int z(t)=\int x(t)+i\int y(t)$$

• both real part and imaginary part commute
• antiderivative
• mean value theorem of integral fail

reparametrization

$$\stackrel{\mapsto \varphi (t)=tC:z(t)=z(t)}{t}$$

• curve length persist

parametric curve length

$$L(C)={\int }_a^b\sqrt{x^{\prime }(t{)}^2+y^{\prime }(t{)}^2}dt={\int }_a^b|z^{\prime }(t)|dt$$

contour

piecewise smooth curve $C$ joined from finite smooth parametric curve

• $z$ continuous
• $z^{\prime }$ piecewise continuous
• vertex, point where $C$ not smooth
• length, orientation, simple closed like parametric curve

Jordan curve theorem

simple closed contour separate $\mathbb{C}$ into

1. a bounded interior
2. an unbounded exterior

contour integral

contour $C:z(t):[a,b]\to \mathbb{C}$,
function $f(z)$ piecewise continuous on $C$

$${\int }_{C}f(z)dz={\int }_a^{b}f(z(t))z^{\prime }(t)dt$$

• independent of parametrization

• linearity

  $${\int }_C(cf(z)+g(z))dz=c{\int }_{C}f(z)dz+{\int }_{C}g(z)dz$$

  for contour $C=C_1\cup C_2$

  $${\int }_{C}f(z)dz={\int }_{C_1}f(z)dz+{\int }_{C_2}f(z)dz$$

• traversed in opposite direction

  $$\begin{gathered} -C:t\mapsto z(-t):[-b,-a]\to \mathbb{C} \\ \Rightarrow {\int }_{-C}f(z)dz=-{\int }_{C}f(z)dz \end{gathered}$$

upper bound theorem for contour integral (ML theorem)

contour $C$ of length $L$,
$f(z)$ piecewise continuous on $C$,
$f(z)$ bounded by $M$

$$\Rightarrow \left|{\int }_{C}f(z)dz\right|\le ML$$

• proof using lemma

  $$\left|{\int }_a^{b}z(t)dt\right|\le {\int }_a^b|z(t)|dt$$

  proof of lemma

  $$I:={\int }_a^{b}w(t)dt=r_I{e}^{i{\theta }_I}$$

  lemma hold for $I=0$. for $I\ne 0$

  $$\begin{gathered} r_I>0 \\ \Rightarrow r_I=\text{Re }\left(e^{-i{\theta }_I}{\int }_a^{b}w(t)dt\right) \\ ={\int }_a^b\text{Re }\left(e^{-i{\theta }_I}w(t)\right)dt\le {\int }_a^b\left|e^{-i{\theta }_I}w(t)\right|dt \\ ={\int }_a^b|z(t)|dt \end{gathered}$$

path independence of contour integral

$f:D\subseteq \mathbb{C}\to \mathbb{C}$ continuous in $D$

$\exists$ antiderivative $F$ s.t. $\forall C$ in $D$ from $z_1$ to $z_2$

$${\int }_{C}f(z)dz=F(z_2)-F(z_1)$$

$\iff \forall C$ closed,

$${\oint }_{C}f(z)dz=0$$

Cauchy-Goursat theorem

on $D\subseteq \mathbb{C}$,
simple closed contour $C$ in $D$,
$f:D\to \mathbb{C}$ holomorphic on and within $C$

$$\Rightarrow {\int }_{C}f(z)dz=0$$

Cauchy’s theorem

same as Cauchy-Goursat theorem, except $f^{\prime }(z)$ continuous

Cauchy-Goursat theorem on simply connected domain

on simply connected domain $D\subseteq \mathbb{C}$,
closed contour $C$ in $D$,
$f:D\to \mathbb{C}$ holomorphic on $D$

$$\Rightarrow {\int }_{C}f(z)dz=0$$

• simply connected domain: every simple closed contour enclose only point in $D$
  • multiply connected domain
• holomorphic function on simply connected domain has antiderivative
  • entire function has antiderivative

adoption of Cauchy-Goursat theorem on multiply connected domain

on multiply connected domain $D\subseteq \mathbb{C}$,
positively oriented contour $C$, negatively oriented simple closed contour $C_i$,
$C_i$ inside of and disjoint from $C$,
$f:D\to \mathbb{C}$ holomorphic on $D$ and on the region between $C,C_i$

$$\Rightarrow {\int }_{C}f(z)dz+{\int }_{C_i}f(z)dz=0$$

path deformation principle

for the following case, deformation of contour integral persist its value

positively oriented contour $C_1,C_2$,
$f$ holomorphic between $C_1,C_2$

$$\Rightarrow {\int }_{C_1}f(z)dz={\int }_{C_2}f(z)dz$$

Cauchy’s integral formula (Cauchy’s formula)

$C$ positively oriented contour,
$f$ holomorphic inside and on $C$

$\forall z_0$ inside $C$

$$f(z_0)=\frac{1}{2\pi i}{\int }_C\frac{f(z)dz}{z-z_0}$$

• derivative

  $$f^{\prime }(z_0)=\frac{1}{2\pi i}{\int }_C\frac{f(z)dz}{(z-z_0{)}^2}$$

• at $z_0$, $f$ holomorphic $\Rightarrow \forall n\in \mathbb{N},f^{(n)}$ exist and holomorphic

  $$f^{(n)}(z_0)=\frac{n!}{2\pi i}{\int }_C\frac{f(z)dz}{(z-z_0{)}^{(n+1)}}$$

• $f$ holomorphic $\Rightarrow u,v$ have continuous partial derivative at all order

evaluating integral with Cauchy’s integral formula

on positively oriented contour $C$, evaluate integral

$$I={\int }_{C}g(z)dz$$

1. find $f(z),z_0$ s.t. $z_0$ inside $C$ and

   $$g(z)=\frac{f(z)}{(z-z_0{)}^n}$$

2. calculate $f^{(n-1)}(z_0)$

3. apply Cauchy’s integral formula

   $$I=\frac{2\pi i}{(n-1)!}{f}^{(n-1)}(z_0)$$

Morera’s theorem

$f$ continuous on $D$

$\forall$ closed contour $C$

$${\int }_{C}f(z)dz=0$$

$\Rightarrow f$ holomorphic throughout $D$

Liouville’s theorem

bounded entire function is constant

fundamental theorem of algebra

non-constant polynomial of degree $n$ has $n$ root

maximum modulus principle

non-constant holomorphic function $f$ on open $D$
$\Rightarrow |f|$ has no maximum on $D$

• for $f$ on $D\cup \partial D$, $|f|$ reach maximum always on $\partial D$, never on interior

analyticity

$f:U_{\epsilon }(z_0)\to \mathbb{C}$ analytic $\iff$

$$\forall z\in U_{\epsilon }(z_0),f(z)=\sum _{n=0}^{\infty }{a}_n(z-z_0{)}^n$$

Laurent series

annual domain $D$, $R_0<|z-z_0|<R_1$,
any closed contour $C$ in $D$,
function $f$ holomorphic in $D\Rightarrow$

$$\begin{gathered} f(z)=\sum _{n=-\infty }^{\infty }{c}_n(z-z_0{)}^n \\ {c}_n=\frac{1}{2\pi i}{\int }_C\frac{f(z)dz}{(z-z_0{)}^{n+1}} \end{gathered}$$

• punctured disk, $R_0=0$
• unique

residue

$$\begin{gathered} resf(z_0)=c_{-1}=\frac{1}{2\pi i}{\int }_{C}f(z)dz \\ \Rightarrow {\int }_{C}f(z)dz=2\pi iresf(z_0) \end{gathered}$$

Cauchy residue theorem

positively oriented contour $C$,
$f$ analytic on and within $C$ except on finite number of singularities $z_k,k\in \{1,2,\cdots ,n\}$

$${\int }_{C}f(z)dz=2\pi i\sum _{k=1}^{n}resf(z_k)$$

residue at infinity

all singularity are inside negatively oriented contour $C$

$${\int }_{C}f(z)dz=2\pi iresf(\infty )$$

• Cauchy residue theorem can be rewritten as

  $$\sum _{k=1}^{n}resf(z_k)+resf(\infty )=0$$

• by replacing $z$ with $\frac{1}{z}$

  $$res\left(\frac{1}{z^2}f\left(\frac{1}{z}\right)\right)(0)=-resf(\infty )$$

singularity

$z_0$ is singularity $\iff$

$f$ holomorphic in ${\hat{U}}_{\epsilon }(z_0)$ and not at $z_0$

isolated singularity

$f$ analytic in deleted neighborhood $\hat{U}(z_0)$

$z_0$ is isolated singularity

• $f$ has Laurent series about $z_0$
• not branch point
• isolated singular at infinity $\Leftarrow \exists R>0$ s.t. $f$ analytic for $R<|z|<\infty$

essential isolated singularity

$\nexists m\in {\mathbb{N}}^{+}$ s.t. $\forall n<-m,c_n=0$

$z_0$ is essential isolated singularity

• Casorati-Weierstrass theorem

  $\forall \epsilon ,R>0,w\in \mathbb{C},\exists z\in |z-z_0|<R$ s.t $|f(z)-w|<\epsilon$

pole

$\exists m\in {\mathbb{N}}^{+}$ s.t. $\forall n<-m,c_n=0$

pole of order $m$ at $z_0$

• simple pole $\Leftarrow m=1$

• removable singularity $\Leftarrow m=0$

  • removed by setting $f(z_0)=c_0$

residue at pole

$$resf(z_0)=\left\{\frac{1}{(m-1)!}\frac{d^{m-1}}{d{z}^{m-1}}\left[(z-z_0{)}^{m}f(z)\right]\right\}(z_0)$$

• only usable for $m=1,2$

• $m=1$

  $$resf(z_0)=[(z-z_0)f(z)](z_0)$$

• $m=2$

  $$resf(z_0)=\left\{\frac{d}{dz}[(z-z_0{)}^{2}f(z)]\right\}(z_0)$$

zero of order $m$

$f$ analytic at $z_0$,
$f(z_0)=0$,
$\exists m\in {\mathbb{N}}^{+}$ s.t.

$$\forall n<m,f^{(n)}(z_0)=0$$

• identically zero $\Leftarrow m=\infty$
  • otherwise, the zero is isolated

zero and pole

$p,q$ analytic,
$q$ has zero of order $m$ at at $z_0$,
$p(z_0)\ne 0$

$\Rightarrow \frac{p}{q}$ has pole of order $m$ at $z_0$

• $m=1\Rightarrow \frac{p}{q}$ has simple pole at $z_0$,

  $$res\left[\frac{p}{q}\right](z_0)=\frac{p(z_0)}{q^{\prime }(z_0)}$$

improper integral

Cauchy principal value $CPV$

$$CPV{\int }_{-\infty }^{\infty }f(x)dx=\underset{R\to \infty }{\lim }{\int }_{-R}^{R}f(x)dx$$

if RHS exist

• $f$ is even $\Rightarrow {\int }_{-\infty }^{\infty }f(x)dx=CPV{\int }_{-\infty }^{\infty }f(x)dx$

improper integral for even rational function

real, continuous, even, irreducible rational function $f=\frac{p}{q}$,
$q(x)$ has finitely many zero $z_k$ above the real axis

$\Rightarrow f$ has pole $z_k$ above the real axis,

let $C_R$ be upper semicircle with missing side $[-R,R]$, then

$${\int }_{-R}^{R}f(x)dx+{\int }_{C_R}f(x)dx=2\pi i\sum _{k}resf(z_k)$$

if ${\lim }_{R\to \infty }{\int }_{C_R}f(x)dx=0$, then

$${\int }_{-\infty }^{\infty }f(x)dx=2\pi i\sum _{k}resf(z_k)$$

Fourier integral

$k>0$

$${\int }_{-\infty }^{\infty }f(x)\sin (kx)dx\text{or}{\int }_{-\infty }^{\infty }f(x)\cos (kx)dx$$

integrate instead

$${\int }_{-R}^{R}f(x)e^{ikx}dx$$

Jordan’s lemma

$f(z)$ analytic above the imaginary axis outside $|z|<R_0$,
semicircle contour $C_R:|z|=R>R_0,0\le \theta \le \pi$

$$\begin{gathered} |f(z)|(C_R)\le M_R,\underset{R\to \infty }{\lim }{M}_R=0 \\ \Rightarrow \forall k>0,\underset{R\to \infty }{\lim }{\int }_{C_R}f(z)e^{ikz}dz=0 \end{gathered}$$

proof by parametric integral and Jordan’s inequality

Jordan’s inequality

$${\int }_0^{\pi }{e}^{-kR\sin \theta }d\theta <\frac{\pi }{kR}$$

indented path

improper integral with singularity on real axis

circumvent each singularity $x_i$ on real axis by upper semicircle $C_i:|z-x_i|=r_i$ with $r_i\to 0$

lemma for indented path

$f$ analytic on $0<|z-x_i|<r$,
clockwise upper semicircle $C_i:z=x_i+r_i{e}^{i\theta },\pi \ge \theta \ge 0,r_i<r$,
Laurent series of $f$ about $x_i$ contain no even negative power

$$\Rightarrow \underset{r_i\to 0}{\lim }{\int }_{C_i}f(z)dz=-i\pi resf(x_i)$$

proof by integrating Laurent series of parametric line integral term by term

• lemma hold at simple pole

• can prove

  $${\int }_{-\infty }^{\infty }\frac{\sin x}{x}dx=\pi$$

lemma for integral over branch point

circular segment of $C_r:z=r{e}^{i\theta },0\le \theta \le {\theta }_1$,
$f(z)$ continuous on $C_r\Rightarrow$

$$\underset{z\to 0}{\lim }zf(z)=0\Rightarrow \underset{r\to 0}{\lim }{\int }_{C_r}f(z)dz=0$$

proof by definition of limit and upper bound theorem

trigonometric integral

$${\int }_{{\theta }_0}^{{\theta }_1}f(\sin \theta ,\cos \theta )d\theta$$

define $C:z=e^{i\theta },{\theta }_0\le \theta \le {\theta }_1$

$$\begin{gathered} \sin \theta =\frac{z-z^{-1}}{2i},\cos \theta =\frac{z+z^{-1}}{2},d\theta =\frac{dz}{iz} \\ \Rightarrow {\int }_{{\theta }_0}^{{\theta }_1}f(\sin \theta ,\cos \theta )d\theta ={\int }_{C}f\left(\frac{z-z^{-1}}{2i},\frac{z+z^{-1}}{2}\right)\frac{dz}{iz} \end{gathered}$$

Laplace transform

Laplace transform of $f:{\mathbb{R}}_0^{+}\to \mathbb{R}$

$$\tilde{f}(s)={\int }_0^{\infty }f(t)e^{-st}dt:\mathbb{C}\to \mathbb{C}$$

Bromwich formula

inverse Laplace transform

$\gamma \in \mathbb{R}$ to the right of all sinularity of $\tilde{f}(s)$,
line contour ${\Gamma }_R$ from $\gamma -iR$ to $\gamma +iR$

$$f(t)=\frac{1}{2\pi i}\underset{R\to \infty }{\lim }{\int }_{{\Gamma }_R}\tilde{f}(s)e^{st}ds$$

meromorphic

$f$ is analytic except possibly for poles

theorem for meromorphic function

simple closed contour $C$,
$f$ non-zero and analytic on $C$, meromorphic within $C$

$$\frac{1}{2\pi i}{\int }_C\frac{f^{\prime }(z)}{f(z)}dz=Z-P$$

• $Z$, number of zero of $f$ within $C$ counted with order
• $P$, number of pole of $f$ within $C$ counted with order

winding number

closed contour $C:z=z(t),a\le t\le b,z(a)=z(b)=z_0$,
$f(z)=\rho e^{i\varphi }\ne 0$ on $C$,
$\Gamma :f(z(t))$

difference in argument

$${\Delta }_C\arg f=\varphi (b)-\varphi (a)=2\pi k$$

winding number of $\Gamma$ with respect to $0$

$$\nu (\Gamma ,0)=\frac{1}{2\pi }{\Delta }_C\arg f=k$$

argument principle

$$Z-P=\frac{1}{2\pi i}{\int }_C\frac{f^{\prime }(z)}{f(z)}dz=\frac{1}{2\pi }{\Delta }_C\arg f=\nu (\Gamma ,0)$$

Rouché’s theorem (dog on a leash theorem)

$f,g$ analytic on and within $C$,
$|g|<|f|$ on $C$

$\Rightarrow f,f+g$ have the same number of zero counted with order inside $C$

• constrain location of polynomial root using Rouché’s theorem
  1. split polynomial $P=f+g$ where $f(z)=z^n$
  2. $C:|z|=1+|a_{n-1}|+\cdots +|a_0|$
     $\Rightarrow |g|<|f|$ on $C$
  3. $\Rightarrow$ root of $P$ are within $C$

Brouwer’s fixed point theorem special version

$D:|z|\le 1,intD:|z|<1$,
$g:D\to intD$ analytic

$\Rightarrow \exists !z_0\in D$ s.t. $g(z_0)=z_0$ (fixed point)

• when changing $w$ continuously without crossing $\Gamma$, $\nu (\Gamma ,w)$ is topological invariant

Hopf’s theorem

closed curve $\Gamma ,\tilde{\Gamma }$,
one can be continuously deformed into another without crossing point $w$

$\iff \nu (\Gamma ,w)=\nu (\tilde{\Gamma },w)$

vector field index theorem

vector field $V:z\mapsto f(z)$,
simple closed contour $C$ enclose singularity $z_i,i=1,2,\cdots$ of $V$ with index $I_V(C_i)\Rightarrow$

$$I_V(C_i)=\nu (\Gamma ,0)=Z-P=\sum _i{I}_V(C_i)$$

vector field singularity

zero or pole of $f$

vector field index

positively oriented contour $C_i$ enclose singularity $z_i$ of $V$

vector field index is integer number of rotation of $V$ traversing $C_i$

$$I_V(C_i)=\nu ({\Gamma }_i,0)$$

Möbius transformation

$$T(z)=\frac{az+b}{cz+d}:\bar{\mathbb{C}}\to \bar{\mathbb{C}},ad-bc\ne 0$$

• extended complex plane $\bar{\mathbb{C}}:=\mathbb{C}\cup \{\infty \}$

• only singularity is simple pole $-\frac{d}{c}$

• inverse

  $$T^{-1}(w)=\frac{b-dw}{cw-a}$$

• derivative non-zero

  $$T^{\prime }(z)=\frac{ad-bc}{(cz+d{)}^2}\ne 0$$

  • analytic except at pole
  • non-constant mapping

• is composition of linear transformation and inversion

  $$\begin{gathered} f_1(z)=cz+d,f_2(z)=\frac{1}{z},f_3(z)=\frac{a}{c}-\frac{ad-bc}{c}z \\ \Rightarrow T=f_1\circ f_2\circ f_3 \end{gathered}$$

• they are a group

• map circle to circle in $\bar{\mathbb{C}}$

  • line are circle of infinite radius through $\infty$

• need $3$ point to specify

circle in Argand diagram

$$\begin{gathered} Az\bar{z}+(B-iC)z+(B+iC)\bar{z}+D=0 \\ A,B,C,D\in \mathbb{R} \end{gathered}$$

cross-ratio

map $3$ distinct point $z_1,z_2,z_3$ to $w=T(z)$ by
$z_1\mapsto w_1,z_2\mapsto w_2,z_3\mapsto w_3$

$$\frac{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1)}=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}$$

• $\infty$ can be canceled

conformal transformation

$f:D\to \mathbb{C}$

1. analytic
2. $\forall z\in D,f^{\prime }(z)\ne 0$

• Möbius transformation

inverse function theorem

analytic $f$ at $z_0$,
$f^{\prime }(z_0)\ne 0$

$\Rightarrow \exists U(z_0),\exists !f^{-1}$ analytic

• derivative

  $$(f^{-1}{)}^{\prime }(w)=\frac{1}{f^{\prime }(z)}$$

conformal transformation preserve angle

$f$ conformal,
smooth $C_1,C_2$ at $z_0$ and intersect with acute angle $\psi$

$\Rightarrow f(C_1),f(C_2)$ intersect at $f(z_0)$ with acute angle $\psi$

proof by taking $f^{\prime }(z_0)$ through $C_1$ and $C_2$ and their equality

conformal transformation preserve harmonic

proof by constructing $f$ s.t. $\varphi =Ref$

• $T(C)=k$ on smooth $C\Rightarrow \Phi (T(C))=k$

harmonic function

$h:D\subseteq {\mathbb{R}}^2\to \mathbb{R}$ with continuous second partial derivative satisfy Laplace’s equation

$$h_{xx}+h_{yy}=0$$

Dirichlet problem

hormonic $\varphi :R\to \mathbb{R}$,
prescribed $\varphi (\partial R)$,
find $\varphi$

• Neumann problem, prescribed $\nabla \varphi (\partial R)$

maximum principle

simply connected $R$ bounded by simple closed $C$,
$\varphi$ harmonic on $R$

$\Rightarrow \varphi$ attain extremum on boundary

proof by constructing $f$ s.t. $\varphi =Ref$ and using maximum modulus principle

uniqueness of solution to Dirichlet problem

simply connected $R$ bounded by simple closed $C$

$\Rightarrow \varphi$ is unique

proof by assuming both ${\varphi }_1,{\varphi }_2$ are solutions and arguing their difference is $0$ using maximum principle