最简复变

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一开始看到的“乱码”是latex代码，稍等一下，网页渲染需要一些时间

\[ \begin{align} \sqrt[n]{|Z|e^{i\theta}}&=\sqrt[n]{|Z|}e^{i\frac {\theta+2k\pi}{n}}, k从0到n-1\\ \text{Ln}z&=\ln |z|+i\arg z+2k\pi i\\ z^\mu&=e^{\mu \text{Ln}z}\\ 柯西黎曼方程&\frac {\partial u}{\partial x}=\frac {\partial v}{\partial y}, \frac {\partial u}{\partial y}=-\frac {\partial v}{\partial x} \\ 柯西不等式|f^{n}(z_0)|&\le \frac {n!}{R^n}\max\limits_{|z-z_0|=R}|f(z)|\\ 刘维尔定理&:有界整函数一定是常数函数\\ \oint_C f(z)dz&=2\pi i\sum \{f(z)在C内的留数\}\\ f(z)在m级极点z_0的留数Res[f(z);z_0]&=\frac 1 {(m-1)!}\lim\limits_{z\to z_0}\frac {d^{m-1}} {dz^{m-1}}(z-z_0)^mf(z)\\ 2级极点z_0的留数Res[\frac {g(z)}{h(z)};z_0]&=2\frac {g'(z_0)}{h''(z_0)}-\frac 2 3 \frac {g(z_0)h'''(z_0)}{[h''(z_0)]^2}, 其中g(z_0)\ne0, h(z_0)=h'(z_0)=0\\ \int_0^{2\pi}R(\cos\theta, \sin\theta)d\theta&=2\pi i\sum \{f(z)在单位圆内的留数\}, f(z)=\frac 1 {iz}R(\frac {z+\frac 1 z}2,\frac {z-\frac 1 z} {2i})\\ \int_{-\infty}^{+\infty}f(x)dx&=2\pi i\sum\{f(z)在上半平面的留数\}, f(z)的分母比分子高两次以上\\ \int_{-\infty}^{+\infty}e^{i\alpha x}f(x)dx&=2\pi i \sum\{e^{i\alpha x}f(z)在上半平面的留数\}, f(z)的分母比分子高一次以上\\ \int_{-\infty}^{+\infty}\cos(\alpha x) f(x)dx&=Re(2\pi i \sum\{e^{i\alpha x}f(z)在上半平面的留数\})\\ \int_{-\infty}^{+\infty}\sin(\alpha x) f(x)dx&=Im(2\pi i \sum\{e^{i\alpha x}f(z)在上半平面的留数\})\\ 把z_1, z_2, z_3映射成\omega_1, \omega_2, \omega_3的&分式线性映射为\frac {\omega-\omega_1} {\omega-\omega_2}\cdot\frac {\omega_3-\omega_2} {\omega_3-\omega_1}=\frac {z-z_1} {z-z_2}\cdot\frac {z_3-z_2} {z_3-z_1}\\ 把上半平面映射成单位圆,&且z_0映射到0的分式线性映射为\omega=e^{i\theta}\frac {z-z_0}{z-\overline{z_0}}\\ 把单位圆映射到单位圆，&且z_0映射到0的分式线性映射为\omega=e^{i\theta}\frac {z-z_0}{1-\overline{z_0}z}\\ L[f(t)]&=\int_0^{\infty}f(t)e^{-st}dt\\ f_1(t)\ast f_2(t)&=\int_0^tf_1(\tau)f_2(t-\tau)d\tau\\ L[u(t)]&=\frac 1 s\\ L[e^{kt}u(t)]&=\frac 1 {s-k}\\ L[t^n](n\in N)&=\frac {n!} {s^{n+1}}\\ L[\sin t]&=\frac 1 {s^2+1}\\ L[\sin \omega t]&=\frac \omega {s^2+\omega^2}\\ L[\cos t]&=\frac s {s^2+1}\\ L[\cos \omega t]&=\frac s {s^2+\omega^2}\\ 对于L[f(t)]&=F(s)\\ 线性L[\alpha_1f_1(t)+\alpha_2f_2(t)]&=\alpha_1F_1(s)+\alpha_2F_2(s)\\ 时移L[f(t-t_0)]或L[u(t-t_0)f(t-t_0)]&=e^{-st_0}F(s)\\ 频移L[e^{s_0t}f(t)]&=F(s-s_0)\\ 象原函数微分L[f'(t)]&=sF(s)-f(0^+)\\ 象原函数微分L[f^{(n)}t]&=s^nF(s)-s^{n-1}f(0^+)-s^{n-2}f'(0^+)-...-s^{0}f^{(n-1)}(0^+)\\ 象函数微分L[(-t)^nf(t)]&=F^{(n)}(s)\\ 象原函数积分L[\int_0^tf(\tau)d\tau]&=\frac 1 s F(s)\\ 象函数积分L[\frac {f(t)}{t}]&=\int_s^\infty F(u)du\\ 初值f(0^+)&=\lim\limits_{s\to \infty}sF(s)\\ 终值f(+\infty)&=\lim\limits_{s\to 0}sF(s)(运用前先判断终值f(+\infty)是否存在)\\ L[f(t)\ast g(t)]&=F(s)G(s) \end{align} \]