18.04复变

complex algebra and plane

复数的模:magnitude,absolute value,norm,modulus
$z=x+yi=re^{i\theta}\text{ where r is the magnitude,theta is the args}$

• complexification(complex replacement)
  对于

$$\begin{aligned} I=\int e^xcos2xdx\\\\ I_c=\int (e^xcos2x+ie^xsin2x )dx\\\\ I=Re(I_c) \end{aligned}$$

• find nth root
  • 一个很方便的方法就是变成指数形式
  • geometry:
    
• inverse eular formulars

$$\begin{aligned} cost=\frac{e^{it}+e^{-it}}{2}\\\\ sint=\frac{e^{it}-e^{-it}}{2i} \end{aligned}$$

• de Moivre’s formula
  $(cos\theta +isin\theta)^n=cos(n\theta)+isin(n\theta)$
• 矩阵形式

$$\begin{aligned} z=Re(z)\times \begin{bmatrix} 1&0\\\\0&1 \end{bmatrix}+Im(z)\times \begin{bmatrix} 0&-1\\\\1&0 \end{bmatrix} \end{aligned}$$

这样复数乘法即可变为矩阵乘法

• polar form(stretch and rotation)

$$\begin{aligned} Z=\begin{bmatrix} r&0\\\\0&r \end{bmatrix} \begin{bmatrix} cos\theta&-sin\theta\\\\sin\theta&cos\theta \end{bmatrix} \end{aligned}$$

• complex functions as map
  • 不好可视化,这里视作映射
  • $w=f(z)\text{equal to }z\mapsto w$
• punctured plane:去除原点的复平面
• the function of arg(z)
  • the branch:我们选择了一个区间使得该函数变成单值的(指定函数取我们列出的值中)
    
    • 2pi处的不连续是无法避免的(橙线称为branch cut想连续就得除开该cut)
  • the principle branch of arg:-pi~pi,denoted by Arg(z)
• the function log
• we want to define it as the inverse of exponential
  $log(z)=log(|z|)+iarg(z)$
  • log 0 未定义,principle branch 仍然是-pi~pi
    
    • map circle to verticle line segment,ray to horizontal line
• power:$z^a=e^{alog(z)}$

analytic function

• a function is analytic if it has complex derivative
• open disk of r around z:all the points within the radius r of z
• open deleted disk=punctured disk(扣掉原点)
• 极限趋于某复数<=>实部和虚部分别趋近某值
• 只要它实部和虚部都连续它本身就连续
• the point at infinity:
  • extended complex plane=$\mathbb{C}\cup\{\infty\}$
  • one point at infinity:任何模趋近于无穷的序列都趋近于该无穷
  • $\frac{1}{\infty}=0$
  • the neighborhood of infinity:除开原点为圆心的圆

Stereographic projection from the Riemann sphere

(黎曼球面的立体投影)

• 复平面上放个球，从北极点(0,0,1)进行投影，球上的点与平面的点一一对应
  $P(a,b,c)\mapsto z=\frac{a+bi}{1-c}$

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

• cauchy-Riemann equation

$$\begin{aligned} \text{for analytic function } f(z)=u(x,y)+iv(x,y) \\\\f^{\prime}(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y} \end{aligned}$$

逆定理也成立

矩阵视角的复数求导(就是雅可比矩阵)

$$\begin{aligned} f^{\prime}(x)= \begin{bmatrix} u_x&u_y\\\\v_x&v_y \end{bmatrix} \text{ where }f(x)=u(x,y)+iv(x,y) \end{aligned}$$

• u,v二阶导存在且连续,则若f=u+iv可导,其导数也可导

• entire function:analytic on the complex plane

• 导数可能比原函数在更大的范围上analytic,但是反过来是不可能的
  

review calculus

• curl(旋度):$F(x,y)=(M(x,y),N(x,y))=>curlF=N_x-M_y$
• divergence(散度)$M_x+N_y$
• F为f的梯度,则f为potential funcion(势函数)
• 路径无关:保守向量场(在联通区域保守场和梯度场等价)
• 联通区域上保守向量场等价于旋度为0(simply connected:每一条内部曲线的内部都在区域中)
• 通量(flux)=所围区域内部的散度的积分($\int_C \mathbb{F}\cdot \mathbb{n}ds=\int\int_R div\mathbb{F}dxdy$)

Line integrals and Cauchy’s theorem

• complex line integral:$\int_\gamma f(z)dz=\int (u+iv)(dx+idy)=\int f(\gamma(t)\gamma^{\prime}(t))dt$
• fundamental theorem of complex line integral:

$$\begin{aligned} \text{if f(z) is a complex analytic function on an open region,then}\\\\ \int_{\gamma}f^{\prime}(z)dz=f(z_1)-f(z_0) \end{aligned}$$

• path independence:if the function f(z) has antiderivative on region A,then it’s integral is path independent on A
• cauchy’s theorem
  
  • the third : f has an antiderivative
• extension of cauchy’s:it demand the function to be analytic on the simply connected region
  
• definition:winding number is the number of times the curve goes around the origin zero counterclockwise
  • also equal to the times across the x-axis ,+1 from below
• 只有当函数在给定区域analytic才有推广到复数的微积分基本定理

cauthy’s integral formula

• C为简单闭曲线,函数在C及其内部解析,对C内任一点(c是线就不行)

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

  只要知道边界就能知道区域内的任一点
  导数

  $$f^{(n)}(z)=\frac{n!}{2\pi i}\int_C\frac{f(w)}{(w-z)^{n+1}}dw$$

triangle inequality

$$\begin{aligned} |z_0+z_1|\le|z_0|+|z_1| \\\\ |z_0-z_1|\ge|z_0|-|z_1|\\\\|\int_g f(z)dz|\le \int_g |f(z)|dz\\\\ |\int_g f(z)dz|\le \int_g |f(z)||dz|\\\\\text{where the z represent the complex variable} \end{aligned}$$

wonderful ex

$$\int_{-\infty}^{+\infty}\frac{1}{(x^2+1)^2}dx$$

• 思路:实数上的积分视作复平面上的实数轴的积分,先给出一个以R为半径的半圆计算积分然后令R趋近于无穷,这里注意计算曲线积分使用夹逼定理

proof

consequence

• analytic implies existence of all order derivatives
• inequality:
• entire+bounded implies the function is just a constant
• maximum modulus principle:非常数的解析函数模长在区域内不可能取极大值,最大值只能在边界取到
• mean value property:在A上解析:$f(z_0)=\frac{1}{2\pi}\int_0^{2\pi}f(z_0+re^{i\theta})d\theta$

problemset3

$f(z)=\bar{z}$这个函数不analytic,实际上在任意点处都不可微
the plane except the non-negative real axises can be simply connected(no whole)

harmonic functions

（调和函数）
definition:a function is harmonic if it’s twice differentiable and satisfy the Laplace’s Equation

$$\nabla^2u=u_{xx}+u_{yy}=0$$

the operator $\nabla^2$ is called Laplacian

• theorem:If a function of z=x+iy f(z)=u(x,y)+iv(x,y) is analytic on region A,then both u and v are harmonic on region A
  • note:analytic->infinite differentiable so twice derivative continuous,mix partial agree
• theorem:any harmonic function u(x,y) on an simply connected region is the real part of an analytic function

properties

definition:u and v are harmonic conjugates if they are real part and image part of an analytic function

(调和函数具有线性性,积分为连续线性组合的极限)
积分本质上是多个函数值按权重求和取极限
解析函数实际上=调和函数的共轭对=调和核的积分

• mean value property:$u(x_0,y_0)=\frac{1}{2\pi}\int_0^{2\pi}u(z_0+re^{i\theta})d\theta\text{ where }z_0=x_0+iy_0$
• maximum principle:有极大值或极小值则为常值函数,最大最小值只能在边界取到
• harmonic conjugates’ level curve(等值线) are orthogonal(梯度不为零的时候谈论才有意义)

exam1

• 欧拉公式在x为复数的情况下依然成立

• $$\text{show that the following function maps the upper plane to the unit circle}f(z)=\frac{z-i}{z+i}$$

solution:
first show that the function maps real axis to the circle
then show maps origin to the zero,it’s enough to say the conclusion

• Cauchy–Riemann中的x,y是针对的直角坐标,不能把极坐标直接拿来算
  • 极坐标下的关系式:$u_r=\frac{1}{r}v_{\theta}$ $v_r=-\frac{1}{r}u_{\theta}$
• cauchy’s integral formula理解:仅要求分子的g(z)在区域内解析,整个式子是允许$\frac{1}{z-a}$在a处有奇点的
  • 本身刻画的就是围绕一阶奇点的积分的关系

Two dimensional hydrodynamics and complex potentials

• hydrodynamics:流体动力学 potentials:势
• velocity field(速度场),如果和时间无关即可称为stationary flow incompressibility:div equals to zero irrotational flow:curl zero
• e.g:F=ay(shearing flow is rotational),the above flows faster ,negative sign indicates that the curl is clock-wise

potentials and streams

$\Phi=\phi+i\psi$:the complex potential function,real part the potential function,imag stream function
the derivative be complex velocity
$\mathbb{F}=\nabla\phi=(\phi_x,\phi_y):=(u,v)$:must be divergence and rotation free
incompress and irrotation always has the complex potential function

• e.g:

$$\begin{aligned} F=a(\frac{x}{r^2},\frac{y}{r^2})\\\\\Phi^{\prime}=u-iv=\frac{a\bar{z}}{z\bar{z}}=\frac{a}{z}\\\\\text{so we have}\Phi=alogz \end{aligned}$$

• stream function:velocity field F is tangent to(与其相切) the level curve of $\psi$ every where
• stagnation points:where velocity field equals to zero

Taylor and Laurent series

geometric series:$f(x)=a\sum_{i=0}^\infty x^i$

• convergence of power series

$$\begin{aligned} f(z)=\sum_{n=0}^{\infty}a_n(z-z_0)^n\\\\ \text{there exist a number R such that}\\\\ \text{if the R>0,then the series converge to an analytic function on region}|z-z_0|<R\\\\ \text{the series diverge in}|z-z_0|>R,\text{R and the region:divergence radius\ disk}\\\\ \text{the derivative is given by term by term differentiation}\\\\ \text{the line integral along the divergence disk is given by term by term integration} \end{aligned}$$

• test:
  • radio test:就是比值绝对值判别收不收敛
  • root test:开n分之一次方的根

Taylor

• singularity
  • zero are isolated:if f is analytic and not identical to zero in region A,if f(z)=0,then we can draw small disk around z contains no zero
  • the first non-zero Taylor coefficient $a_k$ ,then k is the order of zero of f at that point
  • $\frac{1}{1+z^2}$加变减直接几何级数即可,negative power of z is called singular part,summation called regular\analytic part
  • a function is singular at point if it’s not analytic
  • isolated singularity:the deleted disk around point where analytic

laurent

• series
  suppose that f(z) is analytic on annulus $r_1<|z-z_0|<r_2$
  then $f(z)=\mathop{\sum}\limits_{n=1}\limits^{\infty}\frac{b_n}{(z-z_0)^n}+\mathop{\sum}\limits_{n=0}\limits^{\infty}a_n(z-z_0)^n$
  where $a_n=\frac{1}{2\pi i}\int_{\gamma} \frac{f(w)}{(w-z_0)^{n+1}}$ $b_n=\frac{1}{2\pi i}\int_{\gamma}f(w)(w-z_0)^{n-1}$
  前一项(analytic or regular part)在大于$r_1$的地方是收敛的，后一项(singular or principal part)在小于$r_2$的地方是收敛的
  级数的实际表达式取决于region
  sometimes we can use series to solve differential equation,just set the series and compare the coefficient
• poles
  
• residue
  consider the function f with isolated singularity at $z_0$,i.e at $0<|z-z_0|<r$,the residue at z0 is b1
  denoted by $Res(f,z_0)$ or $\mathop{Res}\limits_{z=z_0}f=b_1$
  if $\gamma$ is small,simple closed curve $\int_{\gamma}f(z)=2\pi ib_1$

problemset4&5

• 找函数满足边界都是0,axis of x,$0~\frac{\pi}{4}$四次方就能映射到整个上半平面(指数表达方法)

Residual theorem

pole(zero):分式(整式)的次数
holomorphic:equal to analytic
meromorphic:a function analytic except a set of poles of finite orders

• behavior near zero and pole
  $f(z)\approx a_n(z-z_0)^n\text{ (zero) }\frac{b_n}{(z-z_0)^n}\text{ (pole)}$
• Picards theorem
  if a function has essential singularity(不是可去奇点也不是极点的奇点) at $z_0$,then in the neighbor of it,the function take on all values infinite times possibly except one value
• quotient of function(函数的商)
  $h(z)=\frac{f(z)}{g(z)}$
  let f has zero of order m,g has zero of order n,then h(z) has zero of order m-n(negative means n-m pole),m=n mean analytic
• residue at simple pole
  suppose f has isolated singularity at $z_0$
  • if $g(z)=(z-z_0)f(z)$ is analytic around point,then this point is either simple pole or removable singularity,$Res(f,z_0)=g(z_0)$
  • if f(z) has simple pole,then $\mathop{lim}\limits_{z\rightarrow z_0}(z-z_0)f(z)=Res(f,z_0)$,limit exist also means simple pole or analytic
  • f has simple pole and g analytic,then $Res(fg,z_0)=g(z_0)Res(f,z_0)$ and if g not zero $Res(f/g,z_0)=\frac{1}{g(z_0)}Res(g,z_0)$
  • if g has simple zero,then 1/g has a simple pole and $Res(1/g,z_0)=\frac{1}{g^{\prime}(z_0)}$
• residue at finite pole
  
• cot(z)
  it’s fact that this function has simple pole at $n\pi$ and residue of 1 at these point
• cauchy residue theorem
  f在A上解析,有若干isolated singularities,C为不经过奇点的单连通区域边界,C上的线积分 $\int_C f(z)dz=2\pi i\sum \text{ residues of f inside C}$
• residue at $\infty$
  definition:
  when the curve is counter-clockwised,the interior is inside,when clockwised,the interior is outside
  the value of integral is defined as $Res(f,\infty)=-\frac{1}{2\pi i}\int_C f(z)dz$
  where C is big enough to contain all singularities
  $Res(f,\infty)=-Res(\frac{1}{\omega^2}f(\frac{1}{\omega}),0)$

Definite integrals using the residue theorem