初等代數/複數

• ${\displaystyle \left.(a,b)+(c,d)=(a+c,b+d)\right.}$
• ${\displaystyle \left.(a,b)(c,d)=(ac-bd,bc+ad)\right.}$

複數的性質

• 加法交換律${\displaystyle \left.u+v=v+u\right.}$
• 加法結合律${\displaystyle \left.(u+v)+w=u+(v+w)\right.}$
• 乘法交換律${\displaystyle \left.uv=vu\right.}$
• 乘法結合律：${\displaystyle \left.u(vw)=(uv)w\right.}$
• 分配律：${\displaystyle \left.(u+v)w=uw+vw\right.}$${\displaystyle \left.u(v+w)=uv+uw\right.}$

棣美弗定理

${\displaystyle z^{n}=\left|z\right|^{n}(\cos x+i\sin x)^{n}=\left|z\right|^{n}(\cos nx+i\sin nx)}$

${\displaystyle \left|z\right|(\cos x+i\sin x)=e^{ix+\ln \left|z\right|}}$

歐拉公式

${\displaystyle e^{iz}=\sum _{n=0}^{+\infty }{\frac {(iz)^{n}}{n!}}=\sum _{n=0}^{+\infty }(-1)^{n}{\frac {z^{2n}}{(2n)!}}+i\sum _{n=1}^{+\infty }(-1)^{n-1}{\frac {z^{2n-1}}{(2n-1)!}}=\cos(z)+i\sin(z)}$

${\displaystyle \Rightarrow \left|\alpha \right|e^{i\beta }=\left|\alpha \right|\cos(\beta )+i\sin(\beta )}$

複數的乘冪與開方

${\displaystyle \left.z=x+iy\right.}$，設${\displaystyle \left|z\right|=r}$${\displaystyle \theta =\arctan {\frac {y}{x}}}$則：

${\displaystyle z^{n}=r^{n}e^{i\cdot n\theta }}$
${\displaystyle {\sqrt[{n}]{z}}={\sqrt[{n}]{r}}e^{i{\frac {2k\pi +\theta }{n}}}}$（其中${\displaystyle \left.k=0,1,\cdots n-1\right.}$