# 辐角

## 定义

${\displaystyle z=x+yi={\sqrt {x^{2}+y^{2}}}(\cos \varphi +i\sin \varphi )}$

## 辐角主值

${\displaystyle \arg(z)=\{\operatorname {Arg} (z)+2k\pi \,|\,k\in \mathbb {Z} \}}$

## 辐角的计算

${\displaystyle \operatorname {Arg} (x+yi)={\begin{cases}\arccos {\dfrac {x}{\sqrt {x^{2}+y^{2}}}}&y>0\\-\arccos {\dfrac {x}{\sqrt {x^{2}+y^{2}}}}&y<0\\0&x>0\land y=0\\\pi &x<0\land y=0\\\end{cases}}}$

${\displaystyle \operatorname {Arg} (x+yi)={\begin{cases}2\arctan {\dfrac {y}{{\sqrt {x^{2}+y^{2}}}+x}}&y\neq 0\\0&x>0\land y=0\\\pi &x<0\land y=0\\\end{cases}}}$

## 性质

${\displaystyle z=|z|e^{i\varphi }}$

${\displaystyle \operatorname {Arg} (z_{1}z_{2})=\operatorname {Arg} (z_{1})+\operatorname {Arg} (z_{2}){\pmod {(-\pi ,\pi ]}}}$
${\displaystyle \operatorname {Arg} \left({\frac {z_{1}}{z_{2}}}\right)=\operatorname {Arg} (z_{1})-\operatorname {Arg} (z_{2}){\pmod {(-\pi ,\pi ]}}}$

${\displaystyle \operatorname {Arg} (z^{n})=n\operatorname {Arg} (z){\pmod {(-\pi ,\pi ]}}}$

${\displaystyle \operatorname {Arg} ({\overline {z}})=-\operatorname {Arg} (z){\pmod {(-\pi ,\pi ]}}}$

## 参考来源

• Ahlfors, Lars. Complex analysis: an introduction to the theory of analytic functions of one complex variable 3rd. New York, London: McGraw-Hill. 1979. ISBN 0-07-000657-1.
• Beardon, Alan. Complex analysis: the argument principle in analysis and topology. Chichester: Wiley. 1979. ISBN 0-471-99671-8.
• Borowski, Ephraim; Borwein, Jonathan. Mathematics. Collins Dictionary 2nd. Glasgow: HarperCollins. 2002 [1st ed. 1989 as Dictionary of Mathematics]. ISBN 0-00-710295-X.