复数及复空间

基础知识

\(z=x+iy=re^{i\theta}=r(\cos{\theta}+i\sin{\theta})\)

取\(-\pi\lt\theta_0\le\pi\)作为\(z\)的辐角主值, 则\(\arg z=\begin{cases}\arctan{\frac{y}{x}}&z\text{在1,4象限}\\\arctan{\frac{y}{x}}\pm\pi&z\text{在2,3象限}\end{cases}\)

\(\mathrm{Arg}z=\arg{z}+2k\pi\)

\(z^n=r^n(\cos{n\theta}+i\sin{n\theta})\)

\(w=\sqrt[n]{z}=r^{\frac{1}{n}}\left(\cos{\frac{\theta+2k\pi}{n}}+i\sin{\frac{\theta+2k\pi}{n}}\right)\)

\(z\overline{z}=|z|^2\)

\(\begin{cases}|z_1z_2|=|z_1||z_2|\\ \mathrm{Arg}(z_1z_2)=\mathrm{Arg}z_1+\mathrm{Arg}z_2\end{cases}\)

\(\begin{cases}|z_1/z_2|=|z_1|/|z_2|\\ \mathrm{Arg}\frac{z_1}{z_2}=\mathrm{Arg}z_1-\mathrm{Arg}z_2\end{cases}\)

  定理1 \[(1)\ |z_1+z_2|\leq|z_1|+|z_2|\] \[(2)\ ||z_1|-|z_2||\le|z_1-z_2|\]

  定理2 设 \(a_k,b_k(k=1,2,\cdots,n)\) 为复数, 则 \[\left|\sum_{k=1}^na_kb_k\right|\le\left(\sum_{k=1}^n|a_k|^2\right)\left(\sum_{k=1}^n|b_k|^2\right)\]

  证明: 设\(t\)为任意复数, 则 \[0\le\left|a_k-t\overline{b}_k\right|^2=(a_k-t\overline{b}_k)(\overline{a}_k-\overline{t}b_k)\\=|a_k|^2-2\operatorname{Re}\overline{t}a_kb_k+|t|^2|b_k|^2\] 累加得 \[0\le\sum_{k=1}^n|a_k|^2-2\operatorname{Re}(\overline{t}\sum_{k=1}^na_kb_k)+|t|^2\sum_{k=1}^n|b_k|^2\] 令\(t=\sum\limits_{k=1}^na_kb_k/\sum\limits_{k=1}^n|b_k|^2\), 代入得 \[0\le\sum_{k=1}^n|a_k|^2-\frac{2\operatorname{Re}\left|\sum\limits_{k=1}^na_kb_k\right|^2}{\sum\limits_{k=1}^n|b_k|^2}+\frac{\left|\sum\limits_{k=1}^na_kb_k\right|^2}{\sum\limits_{k=1}^n|b_k|^2},\] 化简即证.

圆周和直线方程

  定理3 给定方程 \(Az\overline{z}+\overline{B}z+B\overline{z}+C=0\), 其中 \(A,C\in\mathbb{R}\), \(B\in\mathbb{C}\), 且 \(|B|^2-AC\gt 0\), 则方程是一圆周方程, 即 \(\left|z+\frac{B}{A}\right|=\frac{\sqrt{|B|^2-AC}}{|A|}\).

关于圆周的对称点

  定理4 给定两点 \(z_1,z_2\) 和圆周 \(Az\overline{z}+\overline{B}z+B\overline{z}+C=0\), 其系数满足 \(|B|^2-AC\gt0\), 则\(z_1,z_2\)关于圆周对称的充分必要条件为\(Az_2\overline{z_1}+\overline{B}z_2+B\overline{z_1}+C=0\)

复数的球面表示与扩充复平面

 考虑 \(\mathbb{R}^3\) 中的单位球面 \(S=\{(x,y,z)|x^2+y^2+z^2=1\}\)，点 \(N(0,0,1)\) 称为北极，复平面 \(\mathbb{C}=\{(x,y,0)|x,y\in \mathbb{R}\}\) 表示复数集。\(\forall z\in\mathbb{C}\), 直线 \(zN\) 与球面 \(S\) 相交于一点 \(Z\)。若 \(|z|<1\), \(Z\) 在下半球面; 若 \(|z|>1\), \(Z\) 在上半球面; 若\(|z|=1\), 则 \(Z=z\). 在 \(\mathbb{C}\) 中引入理想点, 称为无穷远点, 记作 \(z=\infty\). 扩展复平面 \(\overline{\mathbb{C}}=\mathbb{C}\bigcup\{\infty\}\), \(\overline{\mathbb{C}}\) 与 \(S\) 上的点建立起一一对应关系, \(S\) 称为黎曼球面, \(\overline{\mathbb{C}}\) 与 \(S\) 这种一一对应称为球极射影.

  设 \(z=x+iy\), 对应的 \(Z=(x_1,x_2,x_3)\), 过 \(z,N\) 的直线上的点为 \[tN+(1-t)z,\quad-\infty\lt t\lt\infty\] 即 \[((1-t)x,(1-t)y,t),\quad-\infty\lt t\lt\infty\] \(\exists\ t\in(-\infty,\infty)\), 使 \[x_1=(1-t)x,\quad x_1=(1-t)y,\quad x_3=t.\]

\(Z\) 在 \(S\) 上, 坐标满足

\[1=(1-t)^2x^2+(1-t)^2y^2+t^2=(1-t)^2|z|^2+t^2\]

解得 \(t=1\) 或 \(t=\displaystyle\frac{|z|^2-1}{|z|^2+1}\). 因\(Z\neq N\), 与 \(Z\) 对应的 \(t\) 只能为

\[t=\displaystyle\frac{|z|^2-1}{|z|^2+1},\quad 1-t=\displaystyle\frac{2}{|z|^2+1}\]

则 \(Z\) 的坐标

\[\begin{cases}x_1=(1-t)x=\displaystyle\frac{z+\overline{z}}{|z|^2+1},\\ x_2=(1-t)y=\displaystyle\frac{z-\overline{z}}{i(|z|^2+1)},\\ x_3=t=\displaystyle\frac{|z|^2-1}{|z|^2+1}\end{cases}\]

反之, 由 \(Z\) 的坐标 \((x_1,x_2,x_3)\)求对应点 \(z\) 的公式为: \[z=x+iy=\frac{x_1+ix_2}{1-t}=\frac{x_1+ix_2}{1-x_3}.\]

复平面的拓扑

开集与闭集

 open disc \(D_r(z_0)\) of radius \(r\) centered at \(z_0\) : \(D_r(z_0)=\{z\in\mathbb{C}:|z-z_0|\lt r\}\), closed disc \(\overline{D_r}(z_0)\) of radius r centered at \(z_0\) is defined by \(\overline{D_r}(z_0)=\{z\in\mathbb{C}:|z-z_0|\le r\}\), boundary \(C_r(z_0)=\{z\in\mathbb{C}:|z-z_0|\le r\}\), unit disc \(D=\{z\in\mathbb{C}:|z|\lt1\}\).

 The boundary of a set \(\Omega\) is equal to its closure minus its interior, and is often denoted by \(\partial\Omega\).

完备性

  序列\(\{z_n\}\)收敛于\(w\in\mathbb{C}\), 有\(\lim\limits_{n\to\infty}|z_n-w|=0\), 记为\(w=\lim\limits_{n\to\infty}z_n\), 容易证明其充分必要条件为 \[\lim_{n \to\infty}\operatorname{Re}z_n=\operatorname{Re}w,\quad\lim_{n\to\infty}\operatorname{Im}z_n=\operatorname{Im}w\]

  \(\mathbb{C}\)中序列\(\{z_n\}\)称为\(Cauchy\)序列, 如果\(\forall\varepsilon\gt0,\exists\)正整数\(N\), 使得\(n,m\ge N\)时, 有\(|z_n-z_m|<\varepsilon\).

  定理1 设\(\{z_n\}\)为\(\mathbb{C}\)中\(Cauchy\)序列, 则序列\(\{z_n\}\)收敛到\(w\), 或序列极限存在.

 If \(\Omega\subset\mathbb{C}\) is bounded, we define its diameter by \[\operatorname{diam}(\Omega)=\sup_{z,w\in\Omega}|z-w|\]

  Cantor定理 若\(\Omega\subset\mathbb{C}(n=1,2,\cdots)\)为闭集, 且\(\Omega_1\supset\Omega_2\supset\cdots\supset\Omega_n\supset\cdots, \operatorname{diam}(\Omega_n)\to0\quad as\ n\to \infty\), 则\(\bigcap\limits_{n=1}^{\infty}\Omega_n\)由一点组成.

紧性

  \(\mathbb{C}\)或\(\overline{\mathbb{C}}\)中集合\(E\)称为紧集, 如果任一开集族\(\mathscr{G}\)覆盖\(E\), 即\(E\)中的每一点至少属于\(\mathscr{G}\)中某一开集, 则必能从\(\mathscr{G}\)中选出有穷个开集\(G_1,G_2,\cdots,G_n\)覆盖\(E\), 即\(E\subset\bigcup\limits_{j=1}^nG_j\).

  Heine-Borel定理 若\(E\subset\mathbb{C}\)是有界闭集, 则\(E\)为\(\mathbb{C}\)中的闭集.

  Bolzano-Weierstrass定理 任一无穷集至少有一极限点(或任一序列至少有一收敛子列, 子列可以收敛到\(\infty\))

曲线

  连续曲线 定义为区间\([a,b]\)上的连续复值函数\(z(t)=x(t)+iy(t)\quad(a\le t\le b)\)

  可求长曲线 给定曲线\(\gamma(t):a\le t\le b.\)对区间\([a,b]\)作分割 \[\Delta:a=t_0\le t_1\le\cdots\le t_n=b.\] 以\(z_j=z(t_j)(0\le j\le n)\)为顶点作折线\(P\), \(P\)的长度为 \[\sum_{j=1}^n|z(t_j)-z(t_{j-1})|.\] 并对\([a,b]\)的任意分割\(\Delta\), 上式有界, 则称曲线\(z(t)\)为可求长曲线, 并称上确界 \[L=\sup_{\{\Delta\}}\sum_{j=1}^n|z(t_j)-z(t_{j-1})|\] 为曲线\(z(t)\)的长度.

 We say that the parametrized curve is smooth if \(z'(t)\) exists and is continuous on \([a,b]\), and \(z'(t)\neq0\) for \(t\in[a,b]\).

 We can define a curve \(\gamma^-\) obtained from the curve \(\gamma\) by reversing the orientation. As a particular parametrization for \(\gamma^-\) we can take \(z^-:[a,b]\to\mathbb{R}^2\) defined by \[z^-(t)=z(b+a-t).\]

 A smooth or piecewise-smooth curve is closed if \(z(a)=z(b)\) for any of its parametrizations. Finally, a smooth or piecewise-smooth curve is simple(Jordan curve) if it is not self-intersecting, that is, \(z(t)\neq z(s)\ unless\ s=t\).

  引理 设\(f(t)\)为\([a,b]\)上的复值连续函数, 则 \[\left|\int_a^bf(t)\mathrm{d}t\right|\leq\int_a^b|f(t)|\mathrm{d}t.\]

  定理2 设\(z(t)(a\le t\le b)\)为光滑曲线, 则必为可求长曲线, 且长度为 \[L=\int_{a}^b|z'(t)|\mathrm{d}t.\]

连通性

  定义 设\(E\)为\(\mathbb{C}\)(或\(\overline{\mathbb{C}}\))中集合, 称\(E\)为连通集, 如果不存在\(\mathbb{C}\)(或\(\overline{\mathbb{C}}\))中满足下列条件的开集\(G_1,G_2\):

  (1) \(G_1\cap G_2=\varnothing;\)

  (2) \(E\cap G_1\neq\varnothing, E\cap G_2\neq\varnothing;\)

  (3) \(E\subset(G_1\cup G_2).\)

即不能用两个不相交非空集将其一分为二, 则称\(E\)为连通集.

  定义 称连通开集为区域, 称区域的闭包为闭区域.

  定理3 若\(D\)是开集, 则\(D\)的连通性与道路连通是等价的.

  定义 \(D\)为区域, 若\(\overline{\mathbb{C}}\setminus D\)是连通集, 则称\(D\)为单连通区域.

  Jordan定理 设\(\gamma\subset D\)为Jordan曲线, 它把\(\overline{\mathbb{C}}\)分成两个单连通区域, 其中一个是有界的, 称为\(\gamma\)的内部, 另一个是无界的, 称为\(\gamma\)的外部, \(\gamma\)是这两个单连通区域的共同边界.

  定理4 设\(D\subset\mathbb{C}\)为区域, 则\(D\)为单连通区域的充分必要条件是: 对任一Jordan曲线\(\gamma\subset D\), \(\gamma\)的内部属于\(D\).

  定义 集合\(E\)的最大连通子集称为\(E\)的一个分支.

  定义 设\(D\)为区域, 若\(\overline{\mathbb{C}}\setminus D\)由\(n\)个连通分支组成, 则称\(D\)为n连通区域.

连续函数

  如果\(f(z_1)=f(z_2)\)蕴含着\(z_1=z_2\), 即\(E\)中不同点的像也是\(F\)中的不同点, 则称映射\(f\)是一一的, 或单叶或双方单值的. 在这种情况下, \(w=f(z)\)有一个定义在\(f(E)\)上的反函数或逆映射, 记作\(z=f^{-1}(w)\).

解析函数

  定义: 当\(z\in D\)趋于\(z_0\)时, 若极限 \[\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}\] 存在, 则称\(f(z)\)在点\(z=z_0\)可导, 极限值称为\(f(z)\)在\(z_0\)点的导数, 记作\(f'(z_0)\).

  定义: 如果函数在\(z_0\)点的改变量可写成\(\Delta f=A(z_0)\Delta z+O(\Delta z)\), 则称\(f(z)\)在\(z=z_0\)点可微, 微分\(\mathbb{d}f(z_0)=A(z_0)\Delta z\).

  若函数\(f(z)\)在域\(D\)内的每一点可导, 则称函数\(f(z)\)在域\(D\)内是解析的(analytic)或全纯的(holomorphic). 函数\(f(z)\)在\(z_0\)邻域内解析, 则称\(f(z)\)在\(z_0\)点解析.

 we fix \(y_0\) and think of \(f\) as a complex-valued fuction of the single real variable \(x\).

\[f'(z_0)=\lim_{h_1\to0}\frac{f(x_0+h_1,y_0)-f(x_0,y_0)}{h_1}=\frac{\partial f}{\partial x}(z_0)\]

 Now taking h purely imaginary, say \(h=ih_2\), a similar argument yields

\[f'(z_0)=\lim_{h_2\to0}\frac{f(x_0,y_0+h_2)-f(x_0,y_0)}{ih_2}=\frac{1}{i}\frac{\partial f}{\partial y}(z_0)\]

 Therefore, if \(f\) is holomorphic we have shown that

\[\frac{\partial f}{\partial x}=\frac{1}{i}\frac{\partial f}{\partial y}\]

 Writing \(f=u+iv\)

\[f'(z)=\frac{\partial u}{\partial x}+\frac{1}{i}\frac{\partial v}{\partial x}=\frac{1}{i}\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\]

  定理1 设函数\(f(z)=u(z)+iv(z)\)定义在区域\(D\)内, 则\(f(z)\)在\(z_0\in D\)点可微的充要条件为: \(u(z),v(z)\)在\(z_0\)点可微, 且在该点偏导数满足Cauchy-Riemann方程(简称\(C-R\)方程)： \[\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\quad and\quad\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\]

 we can clarify the situation further by defining two differential operators \[\frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}+\frac{1}{i}\frac{\partial}{\partial y}\right)\quad and\quad\frac{\partial}{\partial{\overline{z}}}=\frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{1}{i}\frac{\partial}{\partial y}\right)\]

指数函数

  设\(z=x+iy\),指数函数\(e^z\)定义为: \[e^z=e^x(\cos{y}+i\sin{y})\] 易证如下性质

  (1) \(e^z\gt0,|e^z|=e^x\gt0\)

  (2) \(e^{z_1+z_2}=e^{z_1}\cdot e^{z_2}\)

  (3) \(e^z\)是以\(2\pi i\)为周期的周期函数

  (4) \(e^z\)在\(\mathbb{C}\)上解析, 且\((e^z)'=e^z\)

儒可夫斯基函数

  称函数 \[w=f(z)=\frac{1}{2}\left(z+\frac{1}{z}\right)\] 为儒可夫斯基函数

分式线性变换

  称函数 \[w=f(z)=\frac{az+b}{cz+d}\quad(ad-bc\neq0)\] 为分式线性变换, 也称为Mobius变换

三角函数

定义正弦函数和余弦函数 \[\sin{z}=\frac{e^{iz}-e^{-iz}}{2i},\quad\cos{z}=\frac{e^{iz}+e^{-iz}}{2}.\] 则 \[\mathrm{ch}{z}=\cos{iz}=\frac{e^z+e^{-z}}{2},\quad\mathrm{sh}{z}=\frac{\sin{iz}}{i}=\frac{e^z-e^{-z}}{2}\]

\[\begin{cases}\cos{(x+iy)}=\cos{x}\mathrm{ch}{y}-i\sin{x}\mathrm{sh}{y}\\ \sin{(x+iy)}=\sin{x}\mathrm{ch}{y}+i\cos{x}\mathrm{sh}{y}\end{cases}\]

\[\begin{cases}\mathrm{ch}{(x+iy)}=\mathrm{ch}{x}\cos{y}+i\mathrm{sh}{x}\sin{y}\\ \mathrm{sh}{(x+iy)}=\mathrm{sh}{x}\cos{y}+i\mathrm{ch}{x}\sin{y}\end{cases}\]

具有如下性质

  (1) \(\sin{z},\cos{z}\)在\(\mathbb{C}\)上解析, 且\((\sin{z})'=\cos{z},\quad(\cos{z})'=-\sin{z}\)

  (2) \(\sin{z},\cos{z}\)以\(2\pi\)为周期, \(\sin{z}\)为奇函数, \(\cos{z}\)为偶函数

  (3) 和角公式基本关系成立

  (4) \(|\sin{z}|\)和\(|\cos{z}|\)在\(\mathbb{C}\)上无界