UA MATH524 复变函数3 复变函数的极限与可微性

复数域上的集合
复变函数的极限
- 连续性
复变函数的可微性
- Cauchy-Riemann方程
- Laplace方程与调和函数

复数域上的集合

概念	定义
开圆盘(open disk)	B(z0,r)={z:∥z−z0∥<r}B(z_0,r)=\{z:\left\\| z-z_0\right\\|<r\}B(z0,r)={z:∥z−z0∥<r}
w0w_0w0为集合DDD的内点(interior point)	∃z0∈D,r>0,w0∈B(z0,r)\exists z_0 \in D,r >0,w_0 \in B(z_0,r)∃z0∈D,r>0,w0∈B(z0,r)
DDD为开集(open set)	DDD中的所有点都是它的内点
p0p_0p0为集合SSS的边界点(boundary point)	∀r>0,B(p0,r)∩S≠∅,B(p0,r)∩S∁≠∅\forall r>0,B(p_0,r) \cap S \ne \varnothing,B(p_0,r) \cap S^{\complement} \ne \varnothing∀r>0,B(p0,r)∩S=∅,B(p0,r)∩S∁=∅
集合SSS的边界(boundary)∂S\partial S∂S	SSS的所有边界点的集合
SSS为闭集(closed set)	∂S⊂S\partial S \subset S∂S⊂S
集合VVV的闭包(closure)V‾\overline VV	V‾=V∪∂V\overline V=V \cup \partial VV=V∪∂V
集合VVV的内部(interior)intV\text{int} VintV	intV=V‾∖∂V\text{int} V=\overline V \setminus \partial VintV=V∖∂V
折线(polygonal curve)	P1P2⋯PnP_1P_2\cdots P_nP1P2⋯Pn代表依次经过点P1,⋯,PnP_1,\cdots,P_nP1,⋯,Pn的折线
DDD为连通集(connected set)	∀p,q∈D\forall p,q \in D∀p,q∈D，存在折线pP1⋯PnqpP_1 \cdots P_n qpP1⋯Pnq为DDD的子集
DDD为凸集(convex set)	∀p,q∈D\forall p,q \in D∀p,q∈D，折线pqpqpq 为DDD的子集

∥z−z0∥\left\| z-z_0\right\|∥z−z0∥其实就是z−z0z-z_0z−z0的模，只是它正好是(Re,Im)(Re,Im)(Re,Im)坐标系中两个点的距离，所以写成这样，在这一篇的所有表格中，∥∥\left\| \right\|∥∥都是如此
开集的另一种判断方法：D∩∂D=∅D \cap \partial D=\varnothingD∩∂D=∅，则DDD为开集
闭集的另一种判断方法：S∁S^{\complement}S∁为开集，则SSS为闭集

复变函数的极限

f:C→Cf:\mathbb{C} \to \mathbb{C}f:C→C被称为复变函数（因为自变量维数为1，所以也叫单复变函数，这个课程只讨论单复变函数）。与实变函数类似，它是一种多对一（比如eze^zez）或者一对一（比如f(z)=zˉf(z)=\bar zf(z)=zˉ）的映射，因此z1n,log⁡(z)z^{\frac{1}{n}},\log(z)zn1,log(z)都不是函数，并且基于log⁡(z)\log(z)log(z)定义的反三角函数也不是函数。

考虑复数序列{zn}\{z_n\}{zn}与定义在SSS上的函数f(z)f(z)f(z)，下表列出了与它们相关的极限的定义。

符号	定义
lim⁡n→∞zn=A\lim_{n \to \infty}z_n=Alimn→∞zn=A	∀ϵ>0,∃N(ϵ)∈Z,∥zn−A∥<ϵ,∀n≥N(ϵ)\forall \epsilon>0,\exists N(\epsilon) \in \mathbb Z, \left\\| z_n-A\right\\|<\epsilon,\forall n \ge N(\epsilon)∀ϵ>0,∃N(ϵ)∈Z,∥zn−A∥<ϵ,∀n≥N(ϵ)
lim⁡z→z0f(z)=L\lim_{z \to z_0}f(z)=Llimz→z0f(z)=L	∀ϵ>0,∃δ(ϵ)>0,∥f(z)−L∥<ϵ,∀z∈B(z0,δ(ϵ))\forall \epsilon>0,\exists \delta(\epsilon)>0,\left\\| f(z)-L \right\\|<\epsilon,\forall z \in B(z_0,\delta(\epsilon))∀ϵ>0,∃δ(ϵ)>0,∥f(z)−L∥<ϵ,∀z∈B(z0,δ(ϵ))
lim⁡z→∞f(z)=L\lim_{z \to \infty}f(z)=Llimz→∞f(z)=L	∀ϵ>0,∃δ(ϵ)>0,∥f(z)−L∥<ϵ,∀zs.t.∥z∥≥δ(ϵ)\forall \epsilon>0,\exists \delta(\epsilon)>0,\left\\| f(z)-L \right\\|<\epsilon,\forall z\ s.t.\left\\| z \right\\| \ge \delta(\epsilon)∀ϵ>0,∃δ(ϵ)>0,∥f(z)−L∥<ϵ,∀z s.t.∥z∥≥δ(ϵ)

性质

如果zn=xn+iynz_n=x_n+iy_nzn=xn+iyn，则{zn}\{z_n\}{zn}收敛的充要条件是{xn},{yn}\{x_n\},\{y_n\}{xn},{yn}收敛
如果{zn}\{z_n\}{zn}收敛，则{∣zn∣}\{|z_n|\}{∣zn∣}收敛，反之不一定成立
如果zn→A,wn→Bz_n \to A,w_n \to Bzn→A,wn→B，则zn+λwn→A+λBz_n+\lambda w_n \to A+\lambda Bzn+λwn→A+λB, znwn→ABz_nw_n \to ABznwn→AB，如果B≠0B \ne 0B=0，则zn/wn→A/Bz_n/w_n \to A/Bzn/wn→A/B
如果lim⁡z→z0f=L,lim⁡z→z0g=M\lim_{z \to z_0}f=L,\lim_{z \to z_0}g=Mlimz→z0f=L,limz→z0g=M，则z→z0z \to z_0z→z0时，f+λg→L+λMf+\lambda g \to L+\lambda Mf+λg→L+λM, fg→LMfg \to LMfg→LM，如果M≠0M \ne 0M=0，则f/g→L/Mf/g \to L/Mf/g→L/M

评注这几个性质的证明与实数与实变函数序列极限四则运算的证明类似，以第三条为例，
∣znwn−AB∣=∣(zn−A)B+(wn−B)zn∣|z_nw_n-AB|=|(z_n-A)B+(w_n-B)z_n|∣znwn−AB∣=∣(zn−A)B+(wn−B)zn∣

根据极限的定义，∃Nz(ϵ′),Nw(ϵ′)∈Z,∀ϵ′∈(0,1)\exists N_z(\epsilon'),N_w(\epsilon') \in \mathbb Z,\forall \epsilon' \in (0,1)∃Nz(ϵ′),Nw(ϵ′)∈Z,∀ϵ′∈(0,1),
∣zn−A∣<ϵ′,∀n≥Nz(ϵ′)∣wn−B∣<ϵ′,∀n≥Nw(ϵ′)|z_n-A|<\epsilon',\forall n\ge N_z(\epsilon') \\ |w_n-B|<\epsilon',\forall n \ge N_w(\epsilon')∣zn−A∣<ϵ′,∀n≥Nz(ϵ′)∣wn−B∣<ϵ′,∀n≥Nw(ϵ′)

根据三角不等式，∣zn∣≤ϵ′+∣A∣<1+∣A∣|z_n| \le \epsilon'+|A|<1+|A|∣zn∣≤ϵ′+∣A∣<1+∣A∣，考虑ϵ=ϵ′(1+∣A∣+∣B∣)\epsilon=\epsilon'(1+|A|+|B|)ϵ=ϵ′(1+∣A∣+∣B∣), ∀N(ϵ)=Nz(ϵ′)+Nw(ϵ′)\forall N(\epsilon)=N_z(\epsilon')+N_w(\epsilon')∀N(ϵ)=Nz(ϵ′)+Nw(ϵ′),
∣(zn−A)B+(wn−B)zn∣<ϵ′∣B∣+ϵ′(1+∣A∣)=ϵ\begin{aligned} & |(z_n-A)B+(w_n-B)z_n| < \epsilon'|B|+\epsilon'(1+|A|)=\epsilon\end{aligned}∣(zn−A)B+(wn−B)zn∣<ϵ′∣B∣+ϵ′(1+∣A∣)=ϵ

连续性

称定义在SSS上的函数在z0∈Sz_0 \in Sz0∈S处连续，如果
lim⁡z→z0f(z)=f(z0)\lim_{z \to z_0}f(z)=f(z_0)z→z0limf(z)=f(z0)

如果fff在SSS中的每一点处连续，则称fff在SSS上连续，记为f∈C(S)f \in C(S)f∈C(S)。

性质

如果f,gf,gf,g在z0z_0z0处连续，则f+λgf+\lambda gf+λg, fgfgfg在z0z_0z0处连续，如果g(z0)≠0g(z_0) \ne 0g(z0)=0，则f/gf/gf/g在z0z_0z0处也连续
如果∃r>0,h∈C(B(f(z0),r))\exists r>0,h \in C(B(f(z_0),r))∃r>0,h∈C(B(f(z0),r))，则h(f(z))h(f(z))h(f(z))在z0z_0z0处连续

复变函数的可微性

复变函数可微性的相关概念：

概念	定义
fff在其定义域DDD内的点z0z_0z0处可微	存在极限lim⁡h→0f(z0+h)−f(z0)h\lim_{h \to 0}\frac{f(z_0+h)-f(z_0)}{h}limh→0hf(z0+h)−f(z0)，记为f′(z0)f'(z_0)f′(z0)
fff为全纯函数(holomorphic function)	fff在其定义域DDD内的所有点处可微
fff为整函数(entire function)	fff在整个复平面C\mathbb CC上可微

性质

f(z)=zn,∀n∈Nf(z)=z^n,\forall n \in \mathbb Nf(z)=zn,∀n∈N为整函数，f′(z)=nzn−1f'(z)=nz^{n-1}f′(z)=nzn−1，
如果f,gf,gf,g均在z0z_0z0处可微，则(f+λg)′(z0)=f′(z0)+λg′(z0)(fg)′(z0)=f′(z0)g(z0)+f(z0)g′(z0)(fg)′(z0)=f′(z0)g(z0)−f(z0)g′(z0)[g(z0)]2(f+\lambda g)'(z_0)=f'(z_0)+\lambda g'(z_0) \\ (fg)'(z_0)=f'(z_0)g(z_0)+f(z_0)g'(z_0) \\ \left( \frac{f}{g}\right)'(z_0)=\frac{f'(z_0)g(z_0)-f(z_0)g'(z_0)}{[g(z_0)]^2}(f+λg)′(z0)=f′(z0)+λg′(z0)(fg)′(z0)=f′(z0)g(z0)+f(z0)g′(z0)(gf)′(z0)=[g(z0)]2f′(z0)g(z0)−f(z0)g′(z0)
如果ggg在z0z_0z0处可微，fff在g(z0)g(z_0)g(z0)处可微，则[f(g(z0))]′=f′(g(z0))g′(z0)[f(g(z_0))]'=f'(g(z_0))g'(z_0)[f(g(z0))]′=f′(g(z0))g′(z0)
根据1，2可知，多项式函数是整函数，有理函数在除了奇异点之外所有点可微

Cauchy-Riemann方程

Cauchy-Riemann方程 f=u(x,y)+iv(x,y)f=u(x,y)+iv(x,y)f=u(x,y)+iv(x,y)是其定义域DDD上的全纯函数，则
∂u∂x=∂v∂y,∂u∂y=−∂v∂x\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}∂x∂u=∂y∂v,∂y∂u=−∂x∂v

证明：取任意z0∈Dz_0 \in Dz0∈D，h∈Rh \in \mathbb Rh∈R，根据全纯函数的定义，
f′(z0)=lim⁡h→0f(z0+h)−f(z0)h\begin{aligned}f'(z_0) & =\lim_{h \to 0} \frac{f(z_0+h)-f(z_0)}{h}\end{aligned}f′(z0)=h→0limhf(z0+h)−f(z0)

假设h∈Rh \in \mathbb Rh∈R，则
f′(z0)=lim⁡h→0[u(x0+h,y0)−u(x0,y0)h+iv(x0+h,y0)−v(x0.y0)h]=∂u∂x(x0,y0)+i∂v∂x(x0,y0)\begin{aligned}f'(z_0)& = \lim_{h \to 0} \left[ \frac{u(x_0+h,y_0)-u(x_0,y_0)}{h} + i \frac{v(x_0+h,y_0)-v(x_0.y_0)}{h} \right] \\ & = \frac{\partial u}{\partial x}(x_0,y_0)+ i\frac{\partial v}{\partial x}(x_0,y_0)\end{aligned}f′(z0)=h→0lim[hu(x0+h,y0)−u(x0,y0)+ihv(x0+h,y0)−v(x0.y0)]=∂x∂u(x0,y0)+i∂x∂v(x0,y0)

假设h=ikh=ikh=ik为纯虚数，则
f′(z0)=lim⁡h→0[u(x0,y0+k)−u(x0,y0)ik+iv(x0,y0+k)−v(x0.y0)ik]=−i∂u∂y(x0,y0)+∂v∂y(x0,y0)\begin{aligned}f'(z_0)& = \lim_{h \to 0} \left[ \frac{u(x_0,y_0+k)-u(x_0,y_0)}{ik} + i \frac{v(x_0,y_0+k)-v(x_0.y_0)}{ik} \right] \\ & =-i \frac{\partial u}{\partial y}(x_0,y_0)+\frac{\partial v}{\partial y}(x_0,y_0)\end{aligned}f′(z0)=h→0lim[iku(x0,y0+k)−u(x0,y0)+iikv(x0,y0+k)−v(x0.y0)]=−i∂y∂u(x0,y0)+∂y∂v(x0,y0)

根据极限的唯一性，
∂u∂x(x0,y0)+i∂v∂x(x0,y0)=−i∂u∂y(x0,y0)+∂v∂y(x0,y0)\frac{\partial u}{\partial x}(x_0,y_0)+ i\frac{\partial v}{\partial x}(x_0,y_0)=-i \frac{\partial u}{\partial y}(x_0,y_0)+\frac{\partial v}{\partial y}(x_0,y_0)∂x∂u(x0,y0)+i∂x∂v(x0,y0)=−i∂y∂u(x0,y0)+∂y∂v(x0,y0)

实部等于实部，虚部等于虚部，所以Cauchy-Riemann方程成立。

定理根据Cauchy-Rienmann方程判断可微性
f=u(x,y)+iv(x,y)f=u(x,y)+iv(x,y)f=u(x,y)+iv(x,y)在z0z_0z0处可微，如果

u,v,ux=∂u∂x,uy=∂u∂y,vx=∂v∂x,vy=∂v∂yu,v,u_x=\frac{\partial u}{\partial x},u_y=\frac{\partial u}{\partial y},v_x=\frac{\partial v}{\partial x},v_y=\frac{\partial v}{\partial y}u,v,ux=∂x∂u,uy=∂y∂u,vx=∂x∂v,vy=∂y∂v在z0z_0z0的某个邻域内连续
u,vu,vu,v满足Cauchy-Riemann方程

证明：考虑一个模非常小的复数h=a+ibh=a+ibh=a+ib，在(x0,y0)(x_0,y_0)(x0,y0)附近对u,vu,vu,v做Taylor展开，
{u(x0+a,y0+b)=u(x0,y0)+aux(x0,y0)+buy(x0,y0)+o(∣h∣)v(x0+a,y0+b)=v(x0,y0)+avx(x0,y0)+bvy(x0,y0)+o(∣h∣)\begin{cases} u(x_0+a,y_0+b)=u(x_0,y_0)+a u_x(x_0,y_0)+bu_y(x_0,y_0)+o(|h|) \\ v(x_0+a,y_0+b)=v(x_0,y_0)+a v_x(x_0,y_0)+bv_y(x_0,y_0)+o(|h|) \end{cases}{u(x0+a,y0+b)=u(x0,y0)+aux(x0,y0)+buy(x0,y0)+o(∣h∣)v(x0+a,y0+b)=v(x0,y0)+avx(x0,y0)+bvy(x0,y0)+o(∣h∣)

考虑z0=x0+iy0z_0=x_0+iy_0z0=x0+iy0，用Cauchy-Riemann方程计算，
f(z0+h)−f(z0)=[u(x0+a,y0+b)−u(x0,y0)]+i[v(x0+a,y0+b)−v(x0,y0)]=aux(x0,y0)+buy(x0,y0)+i[avx(x0,y0)+bvy(x0,y0)]+o(∣h∣2)=aux(x0,y0)−bvx(x0,y0)+i[−auy(x0,y0)+bux(x0,y0)]+o(∣h∣2)=[ux(x0,y0)+ivx(x0,y0)]h+o(∣h∣)\begin{aligned}f(z_0+h)-f(z_0) & =[u(x_0+a,y_0+b)-u(x_0,y_0)]+i[v(x_0+a,y_0+b)-v(x_0,y_0)] \\ & = a u_x(x_0,y_0)+bu_y(x_0,y_0)+i[a v_x(x_0,y_0)+bv_y(x_0,y_0)]+o(|h|^2) \\ & = au_x(x_0,y_0)-bv_x(x_0,y_0)+i[-au_y(x_0,y_0)+bu_x(x_0,y_0)]+o(|h|^2) \\ & = [u_x(x_0,y_0)+iv_x(x_0,y_0)]h+o(|h|)\end{aligned}f(z0+h)−f(z0)=[u(x0+a,y0+b)−u(x0,y0)]+i[v(x0+a,y0+b)−v(x0,y0)]=aux(x0,y0)+buy(x0,y0)+i[avx(x0,y0)+bvy(x0,y0)]+o(∣h∣2)=aux(x0,y0)−bvx(x0,y0)+i[−auy(x0,y0)+bux(x0,y0)]+o(∣h∣2)=[ux(x0,y0)+ivx(x0,y0)]h+o(∣h∣)

根据导数的定义，
f′(z0)=lim⁡h→0f(z0+h)−f(z0)h=ux(x0,y0)+ivx(x0,y0)f'(z_0) = \lim_{h \to 0} \frac{f(z_0+h)-f(z_0)}{h}=u_x(x_0,y_0)+iv_x(x_0,y_0)f′(z0)=h→0limhf(z0+h)−f(z0)=ux(x0,y0)+ivx(x0,y0)

Laplace方程与调和函数

Laplace方程 假设f=u+ivf=u+ivf=u+iv在z0=x0+iy0z_0=x_0+iy_0z0=x0+iy0处可微，则uuu在(x0,y0)(x_0,y_0)(x0,y0)处满足Laplace方程：
Δu=∂2u∂x2+∂2u∂y2=0\Delta u = \frac{\partial^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}=0Δu=∂x2∂2u+∂y2∂2u=0

证明：用Cauchy-Riemann方程直接计算验证，
∂2u∂x2+∂2u∂y2=∂ux∂x+∂uy∂y=∂vy∂x−∂vx∂y=0\begin{aligned} \frac{\partial^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2} & = \frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y} = \frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}=0\end{aligned}∂x2∂2u+∂y2∂2u=∂x∂ux+∂y∂uy=∂x∂vy−∂y∂vx=0

二阶导可交换次序，所以最后一个等号成立。

称二阶连续可微且满足Laplace方程的函数为Harmonic Function，上述推导说明holomorphic function的实部是harmonic function。