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复变函数与积分变换

2022年11月13日大约 5 分钟学习数学

复变函数与积分变换

基础计算

sinz=eiz−e−iz2i , cosz=eiz+e−iz2\displaystyle sinz=\frac{e^{iz}-e^{-iz}}{2i} \ , \ cosz=\frac{e^{iz}+e^{-iz}}{2}

lnz=ln∣z∣+iargz , Lnz=lnz+i2kπ\displaystyle lnz = ln|z| + iarg z \ , \ Lnz = lnz+i2k\pi

z1/n=r1/n[cos(1n(θ+2kπ))+isin(1n(θ+2kπ))]\displaystyle z^{1/n}=r^{1/n}[cos(\frac{1}{n}(\theta+2k\pi)) + isin(\frac{1}{n}(\theta+2k\pi))]   k=0,1..=n-1

za=eaLnz\displaystyle z^a=e^{aLnz}

解析与调和

(holomorphic and harmonic)

Cauchy–Riemann equations: f(z)=u(x,y)+v(x,y) is holomorphic⇔{∂u∂x=∂v∂y∂u∂y=−∂v∂x⇐\displaystyle f(z)=u(x,y)+v(x,y) \ is \ holomorphic\Leftrightarrow \begin{cases}\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\end{cases} \Leftarrow (u,v is differentiable)

Harmonic function: u(x,y) is harmonic⇔∂2u∂x2+∂2u∂y2=0⇐\displaystyle u(x,y) \ is \ harmonic \Leftrightarrow \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0 \Leftarrow (x, y is real, u is twice continuously differentiable function)

留数

奇点类型：lim⁡z→z0f(z)={有限值可去奇点∞极点不存在本性奇点\displaystyle \lim_{z\to z_0} f(z) = \begin{cases} \text{有限值} & \text{可去奇点} \\ \infty & \text{极点} \\ \text{不存在} & \text{本性奇点}\end{cases}

留数定理：∮Cf(z)dz=2πi∑Res[f(z),zk]\displaystyle \oint_{C}{f(z)dz} = 2\pi i\sum Res[f(z),z_k]；如果 f(z)在扩充复平面内只有有限个孤立奇点，则 f(z)在所有各奇点(包括 ∞ 点)的留数总和必等于零。

留数计算：Res[f(z),z0]={0可去奇点lim⁡z→z0(z−z0)f(z)一级极点P(z0)Q′(z0)该值存在且不为0;一级极点;f(z)=P(z)Q(z)1(m−1)!lim⁡z→z0dm−1dzm−1[(z−z0)mf(z)]m级极点洛朗展开本性奇点\displaystyle Res[f(z),z_0] = \begin{cases} 0 & \text{可去奇点} \\ \lim_{z\to z_0}(z-z_0)f(z) & \text{一级极点} \\ \frac{P(z_0)}{Q'(z_0)} & \text{该值存在且不为}0;\text{一级极点};f(z) = \frac{P(z)}{Q(z)} \\ \frac{1}{(m-1)!}\lim_{z\to z_0}\frac{d^{m-1}}{dz^{m-1}}[(z-z_0)^{m}f(z)] & m\text{级极点} \\ \text{洛朗展开} & \text{本性奇点} \end{cases}

无穷远点的留数：Res[f(z),∞]=−Res[f(1z)⋅1z2,0]\displaystyle Res[f(z),\infty] = -Res[f(\frac{1}{z})\cdot\frac{1}{z^2},0]

解定积分

∫02πR(cosθ,sinθ)dθ=∮∣z∣=1R(z2+12z,z2−12iz)1izdz\displaystyle \int_{0}^{2\pi}R(cos\theta,sin\theta)d\theta = \oint_{|z|=1}R(\frac{z^2+1}{2z},\frac{z^2-1}{2iz})\frac{1}{iz}dz

∫−∞+∞P(z)Q(z)dz=2πi∑Res[P(z)Q(z),zk]  ⇐  zk\displaystyle \int_{-\infty}^{+\infty}\frac{P(z)}{Q(z)}dz = 2\pi i\sum Res[\frac{P(z)}{Q(z)},z_k] \ \ \Leftarrow \ \ z_k 为上半平面奇点，Q(z) 比 P(z) 高至少两次

∫−∞+∞P(z)Q(z)eiαzdz=2πi∑Res[P(z)Q(z)eiαz,zk]   ⇐  zk\displaystyle \int_{-\infty}^{+\infty}\frac{P(z)}{Q(z)}e^{i\alpha z}dz = 2\pi i\sum Res[\frac{P(z)}{Q(z)}e^{i\alpha z},z_k] \ \ \ \Leftarrow \ \ z_k 为上半平面奇点，Q(z) 比 P(z) 高至少一次，P(x),Q(x) 为有理函数

儒歇定理

Fourier transform

Fourier transform: F(ω)=F(f(t))=∫−∞+∞f(t)e−iωtdt\displaystyle F(\omega)=\mathscr{F}(f(t))=\int_{-\infty}^{+\infty}f(t)e^{-i\omega t}dt

inverse Fourier transform: f(t)=F−1(F(ω))=12π∫−∞+∞F(ω)eiωtdω\displaystyle f(t)=\mathscr{F}^{-1}(F(\omega))=\frac{1}{2\pi}\int_{-\infty}^{+\infty}F(\omega)e^{i\omega t}d\omega

Dirichlet integral: ∫0+∞sinωωdω=π2\displaystyle \int_{0}^{+\infty}\frac{sin\omega}{\omega}d\omega=\frac{\pi}{2}

δ 函数筛选性质: ∫−∞+∞δ(t−t0)f(t)dt=f(t0)  ⇐ f(t)在t0连续\displaystyle \int_{-\infty}^{+\infty}\delta(t-t_0)f(t)dt=f(t_0) \ \ \Leftarrow \ f(t)\text{在}t_0\text{连续}

位移性质：{F(f(t±a))=e±iωaF(ω)F(e±iω0tf(t))=F(ω∓ω0)\displaystyle \begin{cases}\mathscr{F}(f(t\pm a))=e^{\pm i\omega a}F(\omega)\\ \mathscr{F}(e^{\pm i\omega_0 t}f(t))=F(\omega\mp\omega_0)\end{cases}

微分性质：F(f′(t))=iωF(ω) ⇐ lim∣t∣→+∞f(t)=0\displaystyle \mathscr{F}(f'(t))=i\omega F(\omega) \ \Leftarrow\ lim_{|t| \to +\infty}f(t)=0

像函数的微分性质：F(tnf(t))=inF(n)(ω)\displaystyle \mathscr{F}(t^nf(t))=i^nF^{(n)}(\omega)

积分性质：F(∫−∞tf(t)dt)=1iωF(ω) ⇐ ∫−∞+∞f(t)dt=0\displaystyle \mathscr{F}(\int_{-\infty}^t f(t)dt)=\frac{1}{i\omega}F(\omega) \ \Leftarrow\ \int_{-\infty}^{+\infty}f(t)dt=0

对称性质：F(F(t))=2πf(−ω)\displaystyle \mathscr{F}(F(t))=2\pi f(-\omega)

相似性质：F(f(at))=1∣a∣F(ω/a)\displaystyle \mathscr{F}(f(at))=\frac{1}{|a|}F(\omega/a)

翻转性质：F(f(−t))=F(−ω)\displaystyle \mathscr{F}(f(-t))=F(-\omega)

Laplace transform

Laplace transform: F(s)=L(f(t))=∫0+∞f(t)e−stdt\displaystyle F(s)=\mathscr{L}(f(t))=\int_{0}^{+\infty}f(t)e^{-st}dt

信号与系统中，积分下限可从 -∞ 开始

线性性质，收敛域至少为两个收敛域的交

时域平移性质：L(x(t−t0))=e−st0X(s)\displaystyle \mathscr{L}(x(t-t_0))=e^{-st_0}X(s)，收敛域不变

s 域平移性质：L(eatx(t))=X(s−a)\displaystyle \mathscr{L}(e^{at}x(t))=X(s-a)，收敛域 R+Re(a)

尺度变换：L(f(at))=1∣a∣F(s/a)\displaystyle \mathscr{L}(f(at))=\frac{1}{|a|}F(s/a)，收敛域 aR

时域微分性质（单边）：L(f′(t))=sF(s)−f(0−)\displaystyle \mathscr{L}(f'(t))=sF(s)-f(0_-)

推论: L(f(n)(t))=snF(s)−s(n−1)f(0)−s(n−2)f′(0)−...−f(n−1)(0)\displaystyle \mathscr{L}(f^{(n)}(t))=s^nF(s)-s^{(n-1)}f(0)-s^{(n-2)}f'(0)-...-f^{(n-1)}(0)
时域微分性质（双边）：L(f′(t))=sF(s)\displaystyle \mathscr{L}(f'(t))=sF(s)

s 域微分性质：L(tnf(t))=(−1)nF(n)(s)\displaystyle \mathscr{L}(t^nf(t))=(-1)^nF^{(n)}(s)

积分性质：L(∫0tf(t)dt)=1sF(s)\displaystyle \mathscr{L}(\int_0^t f(t)dt)=\frac{1}{s}F(s)

象函数的积分性质：∫s∞F(s)ds=L(f(t)t)\displaystyle \int_s^{\infty}F(s)ds=\mathscr{L}(\frac{f(t)}{t})

初值定理：f(t)在[0,+∞]可微，则f(0)=lims→∞sF(s)\displaystyle f(0)=lim_{s\to\infty}sF(s) （若存在）

终值定理：若 sF(s)在 Re(s)≥0 的区域解析，f(∞)=lims→0sF(s)\displaystyle f(\infty)=lim_{s\to 0}sF(s)

inverse Laplace transform：f(t)=12πi∫β−i∞β+i∞F(s)estds=ΣRes[F(s)est,sk], sk\displaystyle f(t)=\frac{1}{2\pi i}\int_{\beta-i\infty}^{\beta+i\infty}F(s)e^{st}ds=\Sigma Res[F(s)e^{st},s_k], \ s_k is to the left of Re(s)=β

卷积性质：收敛域至少为两个收敛域的交

卷积

傅氏卷积：f1(t)∗f2(t)=∫−∞∞f1(τ)f2(t−τ)dτ\displaystyle f_1(t) * f_2(t) = \int_{-\infty}^{\infty}f_1(\tau)f_2(t-\tau)d\tau   （收敛）

拉氏卷积：f1(t)∗f2(t)=∫0tf1(τ)f2(t−τ)dτ ⇐ when  t<0,f1(t)=f2(t)=0\displaystyle f_1(t)*f_2(t) = \int_{0}^{t}f_1(\tau)f_2(t-\tau)d\tau\ \Leftarrow\ when\ \ t<0,f_1(t)=f_2(t)=0

卷积满足：交换律，结合律，分配率

（时域）卷积定理：F[f1(t)∗f2(t)]=F1(w)⋅F2(w)\displaystyle F[f_1(t)*f_2(t)]=F_1(w)\cdot F_2(w)

共形映射

旋转角=arg(f′(z0)),伸缩率=∣f′(z0)∣\displaystyle \text{旋转角}=arg(f'(z_0)),\text{伸缩率}=|f'(z_0)|

共形映射定义：w=f(z) 在区域内保角，且为一一映射

推论：f(z)解析,f′(z0)≠0⇒f(z)在z0处保角\displaystyle f(z)\text{解析},f'(z_0)\neq 0 \Rightarrow f(z)\text{在}z_0\text{处保角}

对应点公式：w−w1w−w2:w3−w1w3−w2=z−z1z−z2:z3−z1z3−z2\displaystyle \frac{w-w_1}{w-w_2}:\frac{w_3-w_1}{w_3-w_2}=\frac{z-z_1}{z-z_2}:\frac{z_3-z_1}{z_3-z_2}    其中 ∞ 替换为 1

上半平面 → 上半平面：w=az+bcz+d,ad−bc>0\displaystyle w=\frac{az+b}{cz+d},ad-bc>0

上半平面 → 单位圆：w=eiθ(z−λz−λˉ)\displaystyle w=e^{i\theta}(\frac{z-\lambda}{z-\bar{\lambda}})

特别的，w=z−iz+i\displaystyle w=\frac{z-i}{z+i}

单位圆 → 单位圆：w=eiφ(z−α1−αˉz),∣α∣<1\displaystyle w=e^{i\varphi}(\frac{z-\alpha}{1-\bar{\alpha}z}),|\alpha|<1

带形区域 → 角形区域：w=ez\displaystyle w=e^z , 0<Imz<a(0<a<2pi) → 0<arg w<a