Complex Analysis - APS

1 Complex Number

$z=x+iy​$

• $x={\rm Re}\ z$ : real part of z
• $y={\rm Im}\ z$ : imaginary part of z
• equal : $x_1=x_2,y_1=y_2$

1.1 Arithmetic

• +
• -
• *

• /

• $\bar{z}=x-iy$ : conjugate of z

  

  ${\rm Re}\ z=\dfrac{z+\bar z}{2}\quad {\rm Im}\ z=\dfrac{z-\bar z}{2}$

1.2 Geometry of Complex Numbers

$z=x+iy\Rightarrow (x,y)$

$C\Rightarrow R^2$

• Rectangular coordinate (x,y)

• Polar coordinate $(r,\theta)$

  

• Modulus : $|z|==\sqrt{x^2+y^2}=r$
• Argument : ${\rm Arg}\ z ==\theta\in (-\pi,\pi) $, not unique, $+2k\pi$
  • principle value =${\rm arg} \ z$

1.2.1 Arithmetic

• + -
• * : $z_1z_2=r_1r_2(\cos\theta_{12}+i \sin\theta_{12})$
• power : $z^n=r^n(\cos n\theta+i\sin n\theta)$
• n-th root $=\sqrt[n]{z}(\cos \dfrac{\theta+2k\pi}{2}+i\sin \dfrac{\theta+2k\pi}{2})$

1.2.2 Eular Formular

$e^{i\theta}=\cos\theta+i\sin\theta$

$z=re^{i\theta}$

• $e^x=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dots$
• $e^{i\theta}=1+i\theta+\dfrac{(i\theta)^2}{2!}+\dfrac{(i\theta)^3}{3!}+\dots\=(1-\dfrac{\theta^2}{2!}+\dots)+i(\theta-\dfrac{\theta^3}{2}+\dots)\=\cos\theta+i\sin \theta$

• * : $z_1z_2=r_1e^{i\theta_1}r_2e^{i\theta_2}=r_1r_2e^{i(\theta_1+\theta_2)}$

• power : $z^n=(re^{i\theta})^n=r^ne^{in\theta}$

1.2.3 Stereographic Projection

• P : $C\rightarrow S^2$

• A : $\rightarrow P(A)$

  1. P is continuous
  2. P is a bijection between C and $S^2/{N}$
  3. $\lim\limits_{|A|\rightarrow \infty}P(A)=N\qquad P(\infty)=N$
     P is a bijection between $C\cup {\infty}$ and $S^2$
     • $C\cup {\infty}$ : the extended complex plane $\bar{C}$
     • $S^2$: Riemann Sphare

  $C\cup {\infty}$

1.3 The sets contained in C

1.3.1 the curves in C

curve C : $f(x,y)=0$ or $\begin{cases}x& =x(t)\y&=y(t)\end{cases}$(parametric representation)

Line

• A : $Ax\pm By\pm C=0$

  $\Downarrow$ let $x=\dfrac{z+\bar z}{2},\ y=\dfrac{z-\bar z}{2i}$

  $\bar bz+b\bar z+c=0\quad b\in C,c\in R$

• B : $\begin{cases}x&=x_0+v_1t\y&=y_0+v_2t\end{cases}$

  $\Downarrow$ add two equation together

  $z=z_0+v_0t\quad z_0,v_0\in C,t\in R$

  Circle

• A : $(x-x_0)^2+(y-y_0)^2=r^2$

  $\Downarrow$ $z_0=x_0+iy_0$

  $z\bar z+\bar b\bar z+b\bar z+c=0\quad b\in C,c\in R,|b|^2-c>0$

• B : $\begin{cases}x&=x_0+r\cos\theta\y&=y_0+r\sin\theta\end{cases}$

  $\Downarrow$ $Z=Z_0+re^{i\theta}$

  $z’=-r\sin\theta+ir\cos\theta=rie^{i\theta}$

  

1. Simple : 1 to 1, has no self-intersection point($z(t_1)=z(t_2)\quad t_1=a,t_2=b$)
2. Smooth : z(t) has a derivative for each t
3. Pieceweise smooth : z(t) has a derivative in $(a,t_1)\cup(t_1,t_2)\cup(t_n,b)$
4. Closed : $z(t_{start})=z(t_{end})$
5. Orientation : t $\uparrow$
   • positive orientation : counter-clockwise direction in simple closed curve
   • negative orientation : clockwise direction

1.3.2 The domains in C

Note :the disk of radius r centered at $Z_0$ is the set ${z||z-z_0|<r}$ denoted by $B_r(z_0)$

1. Interiior point of S
   
2. S is open: each $p\in S$ is a interior point
   • closure of S : $\bar S=S\cup\delta S$
3. S is connected : each pair $z_1,z_2\in S$, curve c can connect $z_1,z_2$ and containted is S
4. 

Jordan’s Curve Theorem

plane $\rightarrow$ each simple closed curve C

$\rightarrow$ interior domain of C(bounded and connected) and exterior domain of C(boundless and connected)

Simple-connected : the interior domain of each simple closed curve $C\in D$ is also contained in D
or Multiply-connected

Let P be a property of some points ,D be a connected set, and p satisfy

1. at least 1 point p = P holds at 1 point p at least
2. P holds at p = P holds in some neighborhood of p
3. P holds on ${Z_n}\quad n\in N$, and $\lim\limits_{n\rightarrow\infty}z_n=z_0\in D$ = P holds at $z_0$
   then P must hold at each point of D

1.4 Functions with one complex variable

$f:D\subset C\rightarrow C\quad z\rightarrow f(z)$

$u={\rm Re}f(z)=u(x,y)\v={\rm Im}f(z)=v(x,y)$

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

2 Calculus of analytic function

2.1 The limit and continuity

Limit of a sequence ${z_n}$

for all z>0,if there exist some $N\ge0$, such that for each every n>N, $|z_n-A|<\varepsilon$ holds, then $A\rightarrow limits\ of\ {z_n}$, denoted by

$\lim\limits_{n\rightarrow\infty}z_n=A$

• $z_n=x_n+iy_n$, then $\lim\limits_{n\rightarrow\infty}z_n=\lim\limits_{n\rightarrow\infty}x_n+i\lim\limits_{n\rightarrow\infty}y_n$

The continuity of a function

1. [limit]for all $\varepsilon>0$,if there exist some of $\delta >0$, such that for each $0<|z-z_0|<\delta$,$|f(z)-A|<\varepsilon$ holds, $A\rightarrow$ limit of f(z) as $z\rightarrow z_0$, denoted by $\lim\limits_{z\rightarrow z_0}=A​$
   • also $f(x)=u(x,y)+iv(x,y)$, then $\lim\limits_{z\rightarrow z_0}f(z)=\lim\limits_{(x,y)\rightarrow(x_0,y_0)}u(x,y)+i\lim\limits_{(x,y)\rightarrow(x_0,y_0)}v(x,y)$
   • if $\lim\limits_{z\rightarrow z_0}f(z)$ exists, then must bounded
   • if $\lim\limits_{z\rightarrow z_0}f(z)=A\neq0$, f(z) must be non zero in some punctured neighborhood of $z_0$
2. if $\lim\limits_{z\rightarrow z_0}f(z)=f(z_0)$, then $f(z_0)$ is continuous at $z_0$, and if continuous in each $z\in D$, continuous in D
   • f(z) continuous at $z_0$, iff u(x,y) and v(x,y) are continuous at $(x_0,y_0)$
3. $\lim\limits_{z\rightarrow \infty}f(z)=A$ : for all $\varepsilon>0$, exist M>0, for each $|z|>m$, $|f(z)-A|<\varepsilon$ holds
4. $\lim\limits_{z\rightarrow z_0}f(z)=\infty$, for all G>0, exists some $\delta>0$, for each $0<|z-z_0|<\delta$, $|f(z)|>G$

2.2 Derivative

Differentiable

$$\lim\limits_{\Delta z\rightarrow\infty}\dfrac{f(z+\Delta z)-f(z)}{\Delta z}$ exists $\Longleftrightarrow\lim\limits_{\Delta z\rightarrow 0^+}\dfrac{f(z+\Delta z)-f(z)}{\Delta z}=\lim\limits_{\Delta z\rightarrow 0^-}\dfrac{f(z+\Delta z)-f(z)}{\Delta z}$$

Cauchy -Riemann equation

If f(z) is differentiable at $z_0$, so $\dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y}\ \dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x}$ holds zt $z_0$

• f(z) Differentiable at $z_0$ $\Longleftrightarrow$ u(x,y),v(x,y) differentiable & C-R equation holds at $z_0$
• f differentiable at each ponit in D, called analytic in D
• ………………..in some neighborhood of $z_0$, called analytic at $z_0$

2.3 Elementary Functions

2.3.1 Exponential function

$e^z=e^{x+iy}=e^x(\cos y+i\sin y)$

• continuous, analytic
• $e^{z+i2\pi}=e^z$
• $e^{z_1}e^{z_2}=e^{z_1+z_2}$
• range $C^* [C/{0}]$
• $(e^z)’=e^z$

2.3.2 Logarithm function

${\rm Ln} z= \ln|z|+i{\rm Arg}z$

• principle value : $\ln z=\ln|z|+i {\rm arg}z$
• continuous, analytic in $C/{z\in R|z\le0}$, right half plane
• $(\ln z)’=\dfrac{1}{z}$
• $\ln(z_1z_2)\neq \ln z_1+\ln z_2$
• range
  

2.3.3 Trigonometric function

$\cos \theta=\dfrac{e^{i\theta}+e^{-i\theta}}{2}, \sin\theta=\dfrac{e^{i\theta}-e^{-i\theta}}{2i}$

• $(\cos z)’=-\sin z, (\sin z)’=\cos z$
• range = C

  2.3.4 Power function

  $z^\alpha=e^{\alpha {\rm Ln}z}$
• $z^{\dfrac{q}{p}}$ has p value
• $z^\frac{1}{n}=\sqrt[n]{r}e^{i\frac{\theta+2k\pi}{n}} \quad k=0,1,,,n-1$

2.4 Complex Integral

$\int^b_af(x)dx=\lim\limits_{\Delta x\rightarrow0}\sum\limits^n_{k=0}f(\zeta_k)(x_{k+1}-x_k)\\Delta x=\max {x_{k+1}-x_k},a=x_0<x_1\dots<x_{n+1}=b$

Evaluating $\int_cf(z)dz$

1. find a parameterization of C.$z(t),\alpha\le t\le\beta$
2. $\int_c f(z)dz=\int^\beta_\alpha f(z(t))z’(t)dt$

Independence of Path

$\int_c f(z)dz$ independence of path in D

$\Longleftrightarrow$ $F’(z)=f(z)$ exists

then $\int_c f(z)dz=F(z_2)-F(z_1)$

2.5 Cauchy’s Theorems

Cauchy-Goursat Theorem

f(z) analytic in D, continuous up to $\partial D$

$\int_{\partial D}f(z)dz=\int_{\partial D}(u+iv)d(x+iy)=\int_{\partial D}udx-vdy+i\int_{\partial D}vdx+udy$

$\Downarrow$ Green function : $\int_{\partial D}Pdx+Qdy=\iint\limits_D(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y})dxdy$

$\iint\limits_D(-\dfrac{\partial v}{\partial x}-\dfrac{\partial u}{\partial y})dxdy+i\iint\limits_D(\dfrac{\partial u}{\partial x}-\dfrac{\partial v}{\partial y})dxdy$

$\Downarrow$ C-R equation

0

$\int\limits_{\partial D}f(z)dz=0$

Cauchy’s Integral Formular

$\int\limits_{\partial D}\dfrac{f(z)}{z-z_0}dz=2\pi if(z_0)$

$\int\limits_{\partial D}\dfrac{f(z)}{(z-z_0)^n}dz=2\pi i\dfrac{f^{(n-1)}(z_0)}{(n-1)!}$

$=\sum\limits^k_{i=1}\sum\limits^{k}{j=1}\dfrac{A{i\alpha_j}}{(z-z_i)^{\alpha_j}}$

Mean Value Property

$B_r(z_0)\subset D$, and f(z) analytic in D, then $\dfrac{1}{2\pi}\int^{2\pi}_0f(z_0+re^{it})dz=f(z_0)$

The Fundamental Theorem of Algebra

Any polynomial with complex coefficients admits at least one root in C

Liouville’s Theorem

f analytic in C and bounded, then f is constant

Morera’s Theorem

$\int\limits_Cf(z)dz$ independent of Path in D, then f(z)
analytic in D

2.6 Harmonic function

h(x,y) a real-valued function in D, has continuous 2nd derivatives

$\dfrac{\partial^2 h}{\partial x^2}+\dfrac{\partial^2 h}{\partial y^2}=0$,called harmonic

• f analytic, u,v harmonic in D

Laplacian operators: $\Delta h==\dfrac{\partial^2 h}{\partial x^2}+\dfrac{\partial^2 h}{\partial y^2}$

harmonic conjugate pair : (u,v)
$\dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y}, \dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x}$

• v conjugate harmonic function of u

• (-v,u) harmonic,(v,u) not

• the harmonic conjugate function of u in D, if exists, $v=\int-u_ydx+u_xdy+C$

Mean Value Property

h(x,y) harmonic in D, $B_r(z_0)\subset D$,$z_0=x_0+iy_0$, then

$h(x_0,y_0)=\dfrac{1}{2\pi}\int^{2\pi}_0h(x_0+r\cos t),y_0+r\sin t)dt$

• proof, to set $h(x,y)={\rm Re}f(x,y)$

Maximum Principle

h harmonic in D, continuous up to $\partial D$, then

h cannot achieve maximum and minimum in D, unless constant

Maximum Modulus Principle

f(z) analytic in D, continuous up to $\partial D$, then

$|f(z)|$ cannot achieve maximum and minimum in D, unless constant

Liouville’s Theorem

h(x,y) harmonic in $R^2$, if upper/lower bounded, constant

a harmonic function must be smooth $\Rightarrow\ \dfrac{\partial^{\alpha+\beta}h}{\partial^\alpha x\partial^\beta y}$ exists

3 Series

3.1 Power Series

${z_n}$ a sequence, $S_n=\sum\limits^n_0z_k$, ${S_n}$ also a sequence
if $\lim\limits_{n\rightarrow\infty}S_n$ exists, called $\sum\limits^n_0z_k$ convergent / converges, define
$\sum\limits^n_0z_k=\lim\limits_{n\rightarrow\infty}S_n$
otherweise, called $\sum\limits^n_0z_k$ divergent / diverges

• [ Geometric Series ] $z_k=q^k$, $S_n=\dfrac{1-q^{n+1}}{1-q}$
  • $|q|<1, \lim\limits_{n\rightarrow\infty}S_n=\dfrac{1}{1-q}$, $\sum\limits^\infty_0q_k$ converges
  • $|q|\ge1, \lim\limits_{n\rightarrow\infty}S_n$ not exists, $\sum\limits^\infty_0q_k$ diverges
• [ Harmonic Series ] $\sum\limits^\infty_1\dfrac{1}{k}$ divergent
  
• $\sum\limits^n_0z_k$ converges$\Longleftrightarrow$ $\lim\limits_{k\rightarrow\infty}z_k=0$

Comparison Test

${b_n}$ a positive sequence

• $|z_n|\le b_n $ for each n, $\sum\limits^\infty_1 b_n$ converges, so is $\sum\limits^\infty_1 z_n$
• $|z_n|\ge b_n $ for each n, $\sum\limits^\infty_1 b_n$ diverges, so is $\sum\limits^\infty_1 z_n$

Ratio Test

• If $\lim\limits_{n\rightarrow\infty}|\dfrac{z_{n+1}}{z_n}|<1$, then $\sum\limits^\infty_1 z_n$ converges
• If $\lim\limits_{n\rightarrow\infty}|\dfrac{z_{n+1}}{z_n}|>1$, then $\sum\limits^\infty_1 z_n$ dinverges
• =1, can’t judge

Root Test

$\lim\limits_{n\rightarrow\infty}\sqrt[n]{|z_n|}=q$

• If q<1, then $\sum\limits^\infty_1 z_n$ converges
• If q>1, then $\sum\limits^\infty_1 z_n$ dinverges
• =1, can’t judge

Power Series

$\sum\limits^\infty_0 c_n(z_n-z_0)^n \quad z_o\in C$
$\Downarrow \sqrt[n]{c_n}|z-z_0|$

• If $(\lim\limits_{n\rightarrow\infty}\sqrt[n]{c_n})|z-z_0|<1$, $\sum\limits^\infty_0 c_n(z_n-z_0)^n$ converges
• If $(\lim\limits_{n\rightarrow\infty}\sqrt[n]{c_n})|z-z_0|>1$, $\sum\limits^\infty_0 c_n(z_n-z_0)^n$ diverges

Radius of convergence–R : $\sum\limits^\infty_0 c_n(z_n-z_0)^n$ converges in the disk $|z-z_0|<R$

• $0\le R\le \infty$
• in the disk, $(\sum\limits^\infty_0 c_n(z_n-z_0)^n)’=\sum\limits^\infty_0 c_n n(z_n-z_0)^{n-1},\int \sum\limits^\infty_0 c_n(z_n-z_0)^ndz=\sum\limits^\infty_0 \int c_n(z_n-z_0)^ndz$

3.2 Taylor Series

Calculus : $f(x)=a_0+a_1(x-x_0)+\dots+a_n(x-x_n)^n+Remainder$

• $a_k=\dfrac{f^{(k)}(x_0)}{k!}$

[ Taylor Series ] if f(z) analytic at $z_0$
$$f(z)=\sum\limits^\infty_0a_n(z-z_0)^n\qquad in\ |z-z_0|<r, a_n=\dfrac{f^{(n)}(z_n)}{n!}$$

• $e^z=\sum\limits^\infty_0\dfrac{z^n}{n!}\quad z\in C$
• $\cos z=\sum\limits^\infty_0\dfrac{(-1)^n}{(2n)!}z^{2n}\quad z\in C$
• $\sin z=\sum\limits^\infty_0\dfrac{(-1)^n}{(2n+1)!}z^{2n+1}\quad z\in C$
• $\ln(1-z)=-\sum\limits^\infty_1\dfrac{z^n}{n}\quad |z|<1$

Zeros of analytic function

if $f(z_0)=0$, then $z_0$ is a zero of f(z)

f(z) analytic and non constant, all its zeros isolate

Uniqueness of analytic function

f(z),g(z) are analytic in D, and $f(z)\equiv g(z)$ in some open set contained in D, then $f(z)\equiv g(z)$ in D

Fibonacci sequence

3.3 Laurent Series

f(z) analytic in annulus $r_1<|z-z_0|<r_2\ (0\le r_1<r_2\le+\infty)$, can be written
$$f(z)=\sum\limits^{+\infty}_{-\infty}c_n(z-z_0)^n$$

where $c_n=\dfrac{1}{2\pi}\int_\gamma\dfrac{f(z)}{(z-z_0)^{n+1}}dz$

Example

3.4 Isolated Singular points

$z_0$ a singular point : f(z) fails to be analytic at $z_0$ (valid / not defined), also in some neighborhood ${0<|z-z_0|<\delta}$

• [ removable ] singular point : $c_n=0$ for all n<0
  • $\lim\limits_{z\rightarrow z_0}f(z)$ exists
• [ pole ] $c_n=0$ for all n<-m, m>0, and $c_m\neq0$
  • $\lim\limits_{z\rightarrow z_0}f(z)=\infty$
• [ essential ] other case
  • $\lim\limits_{z\rightarrow z_0}f(z)$ not exists
  • $\lim\limits_{z\rightarrow z_0}f(z)\neq A / \infty$

Let $f(z)=\frac{g(z)}{h(z)}$, g,h analytic at $z_0$, $h(z_0)=0$ and $h(z) {\rm not} \equiv 0$, let $z_0$ be the zero with multiply m of h(z)

1. pole with order m of f(z) : $g(z_0)\neq 0$
2. removable : $z_0$ is the zero with multiply n of g(z), $n\ge m$
3. pole with order n-m : $z_0$ is the zero with multiply n of g(z), $n < m$

infinity $\infty$

f(z) analytic in ${|z|>R}$, then $\infty$ is a singular point of f(z)

All the singular points of f(z) in the extended complex plane $\bar C$ iff f(z) has finite many singular points

• and not essential iff f(z) is a rational function $\dfrac{P(z)}{Q(z)}$

Let laurent series of f(z) in $|z|>R$ be $f(z)=\sum\limits^{+\infty}_{-\infty} c_nz^n$

• [ removable ] singular point : $c_n=0$ for all n>0
  • $\lim\limits_{z\rightarrow z_0}f(z)$ exists
• [ pole ] $c_n=0$ for all n>m, m>0, and $c_m\neq0$
  • $\lim\limits_{z\rightarrow z_0}f(z)=\infty$

• [ essential ] other case

  • $\lim\limits_{z\rightarrow z_0}f(z)$ not exists
  • $\lim\limits_{z\rightarrow z_0}f(z)\neq A / \infty$

• 0 is removable / pole / essential singular point of g(z) iff $\infty$ removable / pole / essential of $f(w)=g(\frac{1}{w})$

3.5 Residue Theorem

f(z) analytic in D except for finite many singular points,

C be a simple closed curve in D

no singular point on C

$$\int_cf(z)dz=\sum\limits^m_1\int\limits_{|z-z_k|=\varepsilon}f(z)dz=2\pi i \sum\limits^m_1 Res(f,z_k)$$

$\dfrac{1}{2\pi i}\int\limits_{|z-z_k|=\varepsilon}f(z)dz$ : the residue of f(z) at $z_c$, denoted as $Res(f,z_k)$

• $Res(f,z_k)=C_{-1}$ : $f(z)=\sum\limits^{+\infty}_{-\infty}c_n(z-z_0)^n$ be laurent series in punctured neighborhood of $z_0$

• [ Formula 1] $f(z)=\dfrac{g(z)}{(z-z_0)^{n+1}}$, g(z) analytic at $z_0$,

  $Res(f,z_0)=\dfrac{g^{(n)}(z_0)}{n!}$

• [ Formula 2 ] $f(z)=\dfrac{g(z)}{h(z)}$, $h(z_0)=0$ and $h’(z_0)\neq 0$,

  $Res(f,z_0)=\dfrac{g(z_0)}{h’(z_0)}$

Infinity $\infty$

f(z) analytic in ${|z|>R}$, then $\infty$ is a singular point of f(z)

$Res(f,\infty)=-\dfrac{1}{2\pi i}\int\limits_{|z|=\rho}f(z)dz\quad \rho>R$

• $Res(f,\infty)=C_{-1}$ : $f(z)=\sum\limits^{+\infty}_{-\infty}c_nz^n$ in ${|z|>R}$

• $\sum\limits^\infty_1Res(f,z_k)+Res(f,\infty)=0$ : f(z) analytic in $C/{z_1,\dots,z_n}$

• $Res(f(z),\infty)=-Res(f(\dfrac{1}{w})\cdot\dfrac{1}{w^2},0)$

3.6 Application of Residue Theorem

3.6.1 Improper Integral of Rational Function

$\int\limits^{+\infty}_{-\infty}\dfrac{P(x)}{Q(x)}dx$, $\int\limits^{+\infty}_0\dfrac{P(x)}{Q(x)}dx$

• P(x),Q(x) are polynomials
• ${\rm deg} P(x)+2\le {\rm deg} Q(x)$
• $Q(x)\neq 0$ in $(-\infty, +\infty)\ /\ [0,+\infty) $

$$\int\limits^{+\infty}_{-\infty}\dfrac{P(x)}{Q(x)}dx=2\pi i \sum\limits^k_1Res(\dfrac{P(z)}{Q(z)},z_j)$$

$$\int\limits^{+\infty}_{-\infty}\dfrac{P(x)}{Q(x)}dx=–\sum\limits^k_1Res(\dfrac{P(z)}{Q(z)}\log z,z_j)$$

3.6.2 Fourier Type of Improper Integral

$\int\limits^{+\infty}{-\infty}\dfrac{P(x)}{Q(x)}\cos\alpha xdx$, $\int\limits^{+\infty}{-\infty}\dfrac{P(x)}{Q(x)}\sin\alpha xdx$, $a\in R$

• P(x),Q(x) are polynomials
• ${\rm deg}\ P(x)<{\rm deg}\ Q(x)$
• $Q(x)\neq 0$ for all $x\in R$

$${\rm Re}\ (\int\limits^{+\infty}{-\infty}\dfrac{P(x)}{Q(x)} e^{i\alpha x}dx)=\int\limits^{+\infty}{-\infty}\dfrac{P(x)}{Q(x)}\cos\alpha xdx\{\rm Im}\ (\int\limits^{+\infty}{-\infty}\dfrac{P(x)}{Q(x)} e^{i\alpha x}dx)=\int\limits^{+\infty}{-\infty}\dfrac{P(x)}{Q(x)}\sin\alpha xdx$$

when $\alpha<0$

$$(\int\limits^{+\infty}{-\infty}\dfrac{P(x)}{Q(x)} e^{i\alpha x}dx)=\overline{\int\limits^{+\infty}{-\infty}\dfrac{P(x)}{Q(x)} e^{i(-\alpha) x}dx}$$

3.6.3 $\int\limits^{2\pi}_0R(\cos \theta,\sin\theta)d\theta$

$$\int\limits^{2\pi}0R(\cos \theta,\sin\theta)d\theta=\int\limits{|z|=1}R\left(\dfrac{z+\frac{1}{z}}{2},\dfrac{z-\frac{1}{z}}{2i}\right)\dfrac{dz}{iz}$$

4 Integral Transform

4.1 Fourier Transform

f(x) a real-valued function in $(-\infty,+\infty)$

$$\hat{f}(w)=\int\limits^{+\infty}_{-\infty}f(x)e^{-iwx}dx\quad w\in R$$

• f(x) satisfies $\int\limits^{+\infty}_{-\infty}|f(x)|<+\infty\ \left(f\in L^1(-\infty,+\infty)\right)\\max{|{\rm Re}\ f(x)e^{-iwx}|,|{\rm Im}\ f(x)e^{-iwx}|}\le|f(x)|$, then $\hat f(x)$ is valid for all $w\in R$
• $f\in L^1(-\infty,+\infty)$, then $\hat f(w)$ continuous and $\lim\limits_{w\rightarrow\infty}\hat f(w)=0$

Properties

1. $l_1,l_2\in R,f_1f_2\in L^1$, then $(l_1f_1+l_2f_2)^\land=l_1\hat{f_1}+l_2\hat{f_2}$
2. Translation : $\alpha\in R, (f(x-\alpha))^\land=e^{-iw\alpha}\hat f(w)$
3. Similarity : $k\in R^+, (f(kx))^\land=\dfrac{1}{|k|}\hat f(\dfrac{w}{k})$
4. Convolution : $f_1,f_2\in L^1$, then $f_1\ast f_2=\int\limits^{+\infty}_{-\infty}f_1(x-y)f_2(y)dy$ also $\in L^1$, $(f_1\ast f_2)^\land=\hat{f_1}(w)\hat{f_2}(w)$
5. Derivative : $f,f’\in L^1$ $\Longrightarrow$ $(f’(x))^\land=iw\hat f(w)$

Inverse Fourier Trandsform

$f\in L^1$ and $f’$ is Pieceweise continuous
$$\check{f}(x)=\dfrac{1}{2\pi}\int\limits^{+\infty}_{-\infty}\hat f(x)e^{iwx}dw$$

• $(f(x))^\land=F(w)$ $\Longrightarrow$ $(f(w))^\vee=\dfrac{1}{2\pi}F(-x)$

4.2 Laplace Transform

f(t) in $(0,+\infty)$
$$F(s)\ or\ \mathscr{L}(f(x))=\int\limits^{+\infty}_0f(t)e^{-st}dt$$

Properties

1. $l_1,l_2\in R$, then $\mathscr{L}(l_1f_1+l_2f_2)=l_1\mathscr{L}(f_1)+l_2\mathscr{L}(f_2)$
2. Translation : $t_0\in R, \mathscr{L}(f(t-t_0))=e^{-st_0}\mathscr{L}(f(t))$
3. Similarity : $k\in R^+, \mathscr{L}(f(kt))=\dfrac{1}{k}\mathscr{L} (f(t))(\dfrac{s}{k})$
4. $\mathscr{L}(e^{\alpha t}f(t))=\mathscr{L}(f(t))(s-\alpha)$
5. Convolution :$\mathscr{L}(f_1\ast f_2)=\mathscr{L}(f_1)\mathscr{L}(f_2)$
6. Derivative : $\mathscr{L}(f(t))=F(s)$ and $\lim\limits_{t\rightarrow 0^+}f(t)=f(0)$, $\mathscr{L}(f’(t))=sF(s)-f(0)$

Inverse Laplace Transform for Rational Function

P(s),Q(s) polynomial, and ${\rm deg}\ P(s)<{\rm deg}\ Q(s)$, then

$\dfrac{P(s)}{Q(s)}=\sum\limits^m_{j=1}\sum\limits^{n_j}{k=1}\dfrac{c{jk}}{(s-s_j)^k}$

$$\mathscr{L}^{-1}(\dfrac{P(s)}{Q(s)})=\sum\limits^m_{j=1}\sum\limits^{n_j}{k=1}\dfrac{c{jk}}{(k-1)!}t^{k-1}e^{s_jt}$$

4.3 Application

5. Conformal Mappings

5.1 Conformal Mappings

f(z) analytic in D, if$f’(z_0)\neq0\quad z_0\in D$, f is conformal at $z_0$
if conformal at each point in D, then conformal in D

• f conformal at $z_0$, f 1-1 locally
• f conformal in D, may not be 1-1 in D

• $f(z_0)=w$, $g(z)=\dfrac{1}{f(z)}$ conformal at $z_0$
• $f(\infty)=w$, $g(z)=f(\dfrac{1}{z})$ conformal at 0
• $f(\infty)=\infty$, $g(z)=\dfrac{1}{f(\frac{1}{z})}$ conformal at 0

5.2 Moebius Transformation

$f(z)=\dfrac{az+b}{cz+d}\quad (a,b,c,d\in C,\ \begin{vmatrix}a& b\c& d\end{vmatrix}\neq0)$

• $c\neq0$
  $f(\infty)=\frac{a}{c},f(-\frac{d}{c})=\infty$, $f^{-1}(z)=\dfrac{dz-b}{-cz+a}$, f bijection in C
• c=0
  $f(z)=az+b, f(\infty)=\infty$, $f^{-1}=\frac{1}{a}z-\frac{b}{a}$, f bijection in $\bar{C}$

Properties

1. Moebius Transformation are bijection in $\bar{C}$,inverse mapping also Moebius Transformation
   
2. Composition of 2 MT is a MT
3. MT are conformal in C

• f bijection $\bar{C}\rightarrow\bar{C}$, f conformal in $\bar{C}$, f must MT

Preservation of Circles

MT maps a line/circle to another line/circle in $\bar{C}$

Lemma : each MT is composition of :

1. Translation : $w=z+z_0\quad z_0\in C$
2. Rotation : $w=e^{i\theta}z\quad \theta\in R$
3. Scaling : $w=kz\quad k>0$
4. Inversion : $w=\frac{1}{z}\quad z_0\in C$

Crossing Ratio

$(z_0,z_1,z_2,z_3)=\dfrac{z_2-z_0}{z_2-z_1} : \dfrac{z_3-z_0}{z_3-z_1}$

1. Map unit circle to unit circle with $f(z_0)=0$
   
   $f(z)=e^{i\theta}\dfrac{z-z_0}{1-\overline{z_0}z}$

2. Map upper half plane to $|w|=1$, and $f(z_0)=0$
   
   $f(z)=e^{i\theta}\dfrac{z-z_0}{z-\overline{z_0}}$

5.3 Other Conformal Mapping

1. Power function $z^n, n=2,3,\cdots$
   
2. $e^z$ and $\ln z$
   

Riemann’s Theorem

Each simply connected domain other than C, can be conformal onto unit disk

Appendix