Analytic Continuation
In the complex plane $\mathbb C$, Euler’s formula and De Moivre’s theorem state that
- $e^{in\theta}=(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$ for any $n\in\mathbb Z$ and $\theta\in\mathbb R$.
- $z=re^{i\theta}$ for $z\in\mathbb C$ in polar form, where $r=|z|$ and $\theta=\arg(z)\in(-\pi,\pi]$.
- We denote $\arg(z)$ as the principal value of the multi-valued function $\mathrm{Arg}(z)$ as $$\arg(z)=\begin{cases}\arctan\dfrac {\Im(z)}{\Re(z)},&\text{if }\Re(z)>0,\\[10pt] \arctan\dfrac {\Im(z)}{\Re(z)}+\pi,&\text{if }\Re(z)<0,\Im(z)\ge 0,\\[10pt] \arctan\dfrac {\Im(z)}{\Re(z)}-\pi,&\text{if }\Re(z)<0,\Im(z)<0,\\[10pt] \dfrac \pi 2 \text{, or } -\dfrac \pi 2,&\text{if }z=i|z|, \text{ or } z=-i|z|.\end{cases},\quad \mathrm{Arg}(z)=\arg(z)+2k\pi,\ k\in\mathbb Z,$$ for $z\in\mathbb C^*$, and $\arg(0)$ is undefined.
- $\log (z)=\log |z|+i\arg(z)$ is the principal branch of $\mathrm{Log}(z)=\log|z|+i\mathrm{Arg}(z)$ for $z\in \mathbb C^*$.
- $\mathrm{Arg}(f(z))$ has a univalent continuous branch in a domain $D$ if and only if for any closed curve $\gamma:[0,1]\to D$, it holds $\alpha(\gamma,\arg(f(\gamma(0))),1)=\arg(f(\gamma(0)))$, where $\alpha(\gamma,\theta_0,t)$ is the unique continuous branch of $\mathrm{Arg}(f(\gamma))$ with $\alpha(\gamma,\theta_0,0)=\theta_0$ along $\gamma$.
- $\mathrm{Arg}(z)$($\mathrm{Log}(z)$) has a univalent continuous(analytic) branch in any simply connected domain without $0$. $$\mathrm{Arg}(z)=\arg(z_0)+i\int_{z_0}^z\dfrac{\mathrm dw}{w},\quad \mathrm{Log}(z)=\log|z_0|+i\int_{z_0}^z\dfrac{\mathrm dw}{w}.$$ Referring to the equvalent definitions of simply connected domain in $\overline{\mathbb C}$ below, we show the above statement is an equivalent property of simply connected domain without a given point $z_0$(Here $z_0=0$).
Geometry and Topology of $\mathbb C,\overline{\mathbb C}$ and $\mathbb S^2$
With $\infty$ as the point at infinity, the extended complex plane $\overline{\mathbb C}=\mathbb C\cup\{\infty\}$ is homeomorphic to the Riemann sphere $\mathbb S^2$ via stereographic projection, so the topology of $\overline{\mathbb C}$ is the same as that of $\mathbb S^2$, with the north pole corresponding to $\infty$
- From $\overline {\mathbb C}$ to $\mathbb S^2$ it holds $z\mapsto \left(\dfrac {2\Re(z)}{|z|^2+1},\dfrac {2\Im(z)}{|z|^2+1},\dfrac {|z|^2-1}{|z|^2+1}\right)$ and $\infty\to (0,0,1)$.
- From $\mathbb S^2$ to $\overline {\mathbb C}$ it holds $(X,Y,Z)\mapsto \dfrac {X+iY}{1-Z}$ and $(0,0,1)\to \infty$.
- $|z_1-z_2|_{\mathbb S^2}=\dfrac{2|z_1-z_2|}{\sqrt{(1+|z_1|^2)(1+|z_2|^2)}}$ for $z_1,z_2\in\mathbb C$ and $|z-\infty|_{\mathbb S^2}=\dfrac{2}{\sqrt{1+|z|^2}}$ for $z\in\mathbb C$.
- The metric leads to $\mathrm d|z|_{\mathbb S^2}=\dfrac{2\,\mathrm d|z|}{1+|z|^2}$ and $\mathrm d\arg(z)|_{\mathbb S^2}=\mathrm d\arg(z)$ for $z\in\mathbb C$, where the argument of $z$ in the Riemann sphere is defined as the longitude of the point, equivalent to the argument of $z$ in the complex plane.
By adding the point at infinity, the topology of $\overline{\mathbb C}$ differs from that of $\mathbb C$. The topology of $\overline{\mathbb C}$ is generated by adding neighborhoods $B(\infty,r)$, which makes it compact. For example, the real axis $\mathbb R$ separates $\mathbb C$ into two components, the upper and lower half-planes. However, $\mathbb R$ does not separate $\overline{\mathbb C}$, because the two half-planes are joined through the point at infinity.
Polylines and Curves
The polylines in $\overline{\mathbb C}$ is of importance in constructing, like the paths of integration in complex analysis.
- A domain $D\subset\overline{\mathbb C}$ is a connected open set in $\overline{\mathbb C}$.
- A simply connected domain $D\subset\overline{\mathbb C}$ is a domain with $\overline {\mathbb C}\setminus D$ connected.
- Jordan curve theorem: Any simple closed curve $\gamma$ in $\overline{\mathbb C}$ divides $\overline{\mathbb C}$ into two domains, one bounded and one unbounded, with $\gamma$ as their common boundary.(More discussion in the note of Topology)
- There exists a polyline contained in any domain connecting any two points of the domain.
Equivalent Definitions of Simply Connected Domain
More discussion of the simply connected domain in $\overline{\mathbb C}$, the following are equivalent:
- $D$ is simply connected.
- Any closed curve in $D$ is homotopic to a point(null-homotopic) in $D$.
- The bounded component of any simple closed curve in $D$ is contained in $D$.
- The winding number w.r.t. the point $z_0\in\overline{\mathbb C}\setminus D$ of any smooth closed curve in $D$ is zero.
- Any $C^1$ closed form $\omega$ in $D$ is exact, i.e., there exists a $C^1$ function $f$ in $D$ such that $\mathrm df=\omega$.
- Every harmonic function on $D$ has a conjugate harmonic function on $D$.
Curves should be able to parameterized by continuous functions in $[0,1]$. The definition of the winding number w.r.t. a point $z_0\in\overline{\mathbb C}$ for a smooth closed curve $\gamma$ not passing through $z_0$ is
$$ \mathrm{Ind}(\gamma,z_0)=\dfrac{1}{2\pi i}\oint_\gamma\dfrac{\mathrm dz}{z-z_0}=\dfrac{1}{2\pi}\oint_\gamma\mathrm d\arg(z-z_0)=\dfrac 1{2\pi}\oint_\gamma \mathrm d\arctan \dfrac {\Im(z-z_0)}{\Re(z-z_0)}=\dfrac 1{2\pi}\oint_\gamma \dfrac{\Re(z-z_0)\mathrm d\Im(z-z_0)-\Im(z-z_0)\mathrm d\Re(z-z_0)}{|z-z_0|^2}. $$and is invariant under homotopy of $\gamma$ in $\overline{\mathbb C}\setminus\{z_0\}$(parts of the curve do not pass through $z_0$ during the homotopy).
Derivatives, Differentiability and Integrals
- A function $f:D\to\mathbb C$ is analytic or holomorphic w.r.t. $z$ is that $f$ is derivable at a neighborhood of $z$ in $D$.
- Real differentiability of $f$ at $z$ is equivalent to the existence of the linear approximation as follows $$\begin{cases}\Delta u=a\Delta x+b\Delta y+o(|\Delta z|),\\ \Delta v=c\Delta x+d\Delta y+o(|\Delta z|)\end{cases},\quad \text{or equivalently,} \quad \Delta f=f_z\Delta z+f_{\bar z}\Delta \bar z+o(|\Delta z|).$$
- Complex differentiability of $f$ at $z$ is equivalent to the existence of the linear approximation as follows $$\Delta f=A\Delta z+o(|\Delta z|),\quad \text{where}\quad A=f_z=\dfrac {f_x-if_y}{2}.$$
- Cauchy-Riemann equations: For $f=u+iv$ with $u,v:D\to\mathbb R$, it holds that $$u_x=v_y,\quad u_y=-v_x,\quad \text{or equivalently,}\quad f_{\bar z}=\dfrac {f_x+if_y}{2}=0.$$ The version of polar coordinates is $$u_rr=v_\theta,\quad u_\theta=-v_rr,\quad \text{or equivalently,}\quad f_{\bar z}=\dfrac {e^{i\theta}(f_r+if_\theta/r)}{2}=0.$$
- A function is real differentiable and satisfies the Cauchy-Riemann equations at $z$ if and only if it is complex differentiable at $z$, if and only if it is derivable at $z$ in the complex sense.
- The formal quotient of $f$ at $z$ is based on(even though $z,\bar z$ are not independent variables) $$f(x,y)=f\left(\dfrac{z+\bar z}{2},\dfrac{z-\bar z}{2i}\right),\quad \text{so that}\quad f_z=\dfrac {f_x-if_y}{2},\quad f_{\bar z}=\dfrac {f_x+if_y}{2}.$$
Harmonic functions $\mathcal H(D)$, holomorphic functions $H(D)$ and maximum principles, sometimes view $\mathbb C$ as $\mathbb R^2$:
- For $u\in C^2(D)$, it holds $\Delta u=u_{xx}+u_{yy}=4u_{z\bar z}=4u_{\bar zz}$, and $u\in \mathcal H(D)$ if and only if $\Delta u=0$ in $D$.
- If $f\in H(D)$, then $\Re(f),\Im(f)\in \mathcal H(D)$. The converse is not true in general.
- If $u,v\in\mathcal H(D)$ are conjugate harmonic functions(not symmetric), then $f=u+iv\in H(D)$.
- Conjugate differential form $*\mathrm d$, is a closed form for harmonic functions. $$\mathrm du=u_x\mathrm dx+u_y\mathrm dy,\quad *\mathrm du=-u_y\mathrm dx+u_x\mathrm dy,\quad \text{and (for harmonic $u$)}\quad \mathrm d(*\mathrm du)=0.$$
- $\mathrm d\mathrm{Arg}(z)=*\mathrm d\log|z|$.
- By the Cauchy integral formula, $H\subset C^\infty$, and thus $\mathcal H\subset C^\infty\subset C^2$.
Conformal Mappings and Elementary Functions
Function $f$ is real differentiable and $\mathrm df\neq 0$ at $z$, then the following are equivalent:
- $f$ is complex differentiable at $z$.
- $f$ preserves angles between curves at $z$ including the orientation.
- The dilatation ratio of $f$ at $z$ is $1$.(When $-1$, the orientation is reversed, so $f$ is anti-conformal at $z$.) $$\text{dilatation ratio}=\dfrac{\text{maximal stretching}}{\text{minimal stretching}}=\dfrac{|f_z|+|f_{\bar z}|}{|f_z|-|f_{\bar z}|}\in\overline {\mathbb R}.$$
Conformal mapping in $D$ is an alias of injective holomorphic mapping in $D$, also known as univalent holomorphic mapping. Conformal mapping always has a non-zero derivative anywhere in $D$. Consider conformal mapping like: $e^z,\ \log z,\ z^\alpha$ and Mobius transformation $f(z)=\dfrac{az+b}{cz+d}$ with $ad-bc\neq 0$.
Mobius Transformation
- Mobius transformation $\mathcal M$ is a subgroup of bijective conformal mappings in $\overline{\mathbb C}$, with representation $$\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathcal M\to \begin{cases}\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}a'&b'\\c'&d'\end{pmatrix}=\begin{pmatrix}aa'+bc'&ab'+bd'\\ca'+dc'&cb'+dd'\end{pmatrix},&\text{composition};\\[10pt] \begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\dfrac 1{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix},&\text{inverse},\end{cases}\quad \text{where } ad-bc\neq 0.$$
- $\mathcal M$ preserves generalized circles, i.e., circles, point and lines in $\overline{\mathbb C}$. They have a uniform representation as $$Az\bar z-B\bar z-\bar B z+C=0,\quad \text{where }A,C\in\mathbb R,\ B\in\mathbb C,\ AC\leq |B|^2.$$
- $T\in \mathcal M$ preserves angles in $\overline{\mathbb C}$, thus conformal in $\overline{\mathbb C}$, since $T^*:=P^{-1}TP$ preserves metric and angles in $\mathbb S^2$.
- $T\in \mathcal M$ preserves cross ratio of four points in $\overline{\mathbb C}$, i.e., for $z_1,z_2,z_3,z_4\in\overline{\mathbb C}$, it holds $$[T(z_1),T(z_2),T(z_3),T(z_4)]=[z_1,z_2,z_3,z_4],\quad \text{where }[z_1,z_2,z_3,z_4]=\dfrac{z_1-z_3}{z_1-z_4}:\dfrac{z_2-z_3}{z_2-z_4}.$$ A cross ratio $[z:z_2:z_3:z_4]$ has no more than two fixed points, so there exists a unique $T\in \mathcal M$ satisfying for given $z_1,z_2,z_3\in\overline{\mathbb C}$ and $w_1,w_2,w_3\in\overline{\mathbb C}$, with $w_i\neq w_j,z_i\neq z_j$ for $i\neq j$, it holds that $T(z_i)=w_i.$
Examples of Conformal Mappings/Mobius Transformations
- Joukowski function $w(z)=\dfrac {z+z^{-1}}2$ is a Mobius transformation that biholomorphically maps $$z\mapsto w:\quad \begin{cases}\mathbb A(0,1)\cap \mathbb H^+\xrightarrow{1:1}\mathbb H^-,\\ \mathbb A(0,1)\cap \mathbb H^-\xrightarrow{1:1}\mathbb H^+,\end{cases}\quad \begin{cases}\mathbb A(1,\infty)\cap \mathbb H^+\xrightarrow{1:1}\mathbb H^+,\\ \mathbb A(1,\infty)\cap \mathbb H^-\xrightarrow{1:1}\mathbb H^-,\end{cases}\quad \begin{cases}\mathbb S^1\cap \mathbb H^+\xrightarrow{1:1}[-1,1],\\ \mathbb S^1\cap \mathbb H^-\xrightarrow{1:1}[-1,1].\end{cases}$$ It is useful for triangular functions and elliptic functions, and the inverse function is $z(w)=w\pm \sqrt{w^2-1}$, so there are two branches of the inverse function $$w\mapsto z: \quad \begin{cases}\mathbb H^+\xrightarrow{1:1}\mathbb A(1,\infty)\cap \mathbb H^+,\\ \mathbb H^-\xrightarrow{1:1}\mathbb A(1,\infty)\cap \mathbb H^-,\end{cases}\quad \text{for $z_+$},\quad \begin{cases}\mathbb H^+\xrightarrow{1:1}\mathbb A(0,1)\cap \mathbb H^-,\\ \mathbb H^-\xrightarrow{1:1}\mathbb A(0,1)\cap \mathbb H^+,\end{cases}\quad \text{for $z_-$}.$$ We choose the branch $z_+(w):=w+\sqrt{w^2-1}$ as principal branch.
停留在 CA6
Taylor 展开与 Laurent 展开
对解析函数采用 Taylor 展开和 Laurent 展开，可以分析零点、极点的性质。考虑一般的幂级数
$$S(z)=\sum_{n=0}^{\infty}a_nz^n,\quad a_n\in\mathbb C.$$对于解析函数 $f(z)$，在 $z_0$ 的邻域内有唯一的展开式
$$f(z)=\sum_{n=0}^{\infty}a_n(z-z_0)^n,\quad a_n=\dfrac{f^{(n)}(z_0)}{n!}.$$对于 Laurent 展开式
$$f(z)=\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n,\quad a_n=\dfrac{1}{2\pi i}\oint_\gamma\dfrac{f(\zeta)\mathrm d\zeta}{(\zeta-z_0)^{n+1}},$$复数项级数，收敛，部分和，Cauchy收敛准则，绝对收敛，条件收敛，函数项级数，和函数，一致收敛，Cauhcy一致收敛准则，Weierstrass优级数判别法，一致收敛保证连续、内闭一致收敛
一致收敛保证连续和逐项积分：设 $\sum^\infty_{n=1}f_n(z)$ 在可求长曲线 $\gamma$ 上一致收敛到 $f(z)$，如果 $f_n(z)\in C(\gamma)$。那么 $f(z)\in C(\gamma)$，且
$$\int_\gamma f(z)\mathrm dz=\sum^\infty_{n=1}\int_\gamma f_n(z)\mathrm dz.$$内闭一致收敛的等价条件：$\sum^\infty_{n=1}f_n(z)$ 在 $D$ 内闭一致收敛到 $f(z)$，当且仅当 $\sum^\infty_{n=1}f_n(z)$ 在 $\overline {B(z,\delta)}\subset D$ 内一致收敛到 $f(z)$ 对任意 $z\in D$ 和 $\delta>0$ 成立。
Weierstrass 定理：如果 $f_n\in H(D)$ 且 $\sum^\infty_{n=1}f_n(z)$ 在 $D$ 内闭一致收敛到 $f(z)$，那么 $f\in H(D)$，且
$$f^{(k)}(z)=\sum^\infty_{n=1}f_n^{(k)}(z),\quad k\in \mathbb N.$$Weierstrass 定理用到了一致收敛保证连续和逐项积分的性质，从而由 Cauchy 积分公式得到逐项求导，这是实函数项级数所没有的性质，后者需要对导函数的函数项级数进行一致收敛的判别。
Abel 定理：设幂级数 $S(z)$：
- 如果在 $z_0$ 收敛，且 $z_0\neq 0$，则在 $B(0,|z_0|)$ 内绝对收敛、内闭一致绝对收敛；
- 如果在 $z_0$ 发散，则 $S(z)$ 在 $\overline {B(0,|z_0|)}$ 外发散。
Cauchy-Hadamard 定理：幂级数 $S(z)$ 的收敛半径为 $R$，满足
$$\dfrac 1R=\limsup_{n\to\infty}\sqrt[n]{|a_n|}\in [0,\infty].$$- $B(z_0,R)$ 内绝对收敛且内闭一致绝对收敛；
- $\overline {B(z_0,R)}$ 外发散；
- $\partial B(z_0,R)$ 上的收敛性不确定。
收敛半径上一定存在奇点。