# 幅角

## 定义

$z = x+y\mathbf{i} = \sqrt{x^2 + y^2}(\cos\phi + \mathbf{i}\sin\phi)$

## 幅角主值

$\operatorname{Arg}z = \{ \operatorname{arg}z + 2k\pi ; k\in \mathbb{N}\}.$

## 幅角的计算

$\operatorname{Arg}(x+y \mathbf{i} ) = \begin{cases} \arccos \frac{x}{\sqrt{x^2+y^2}} & \qquad y > 0 \\ -\arccos \frac{x}{\sqrt{x^2+y^2}} & \qquad y < 0 \\ 0 & \qquad x>0, \, y = 0 \\ \pi & \qquad x<0, \, y = 0 \\ \end{cases} .$

$\operatorname{Arg}(x + y\mathbf{i}) = \begin{cases} 2 \arctan \left( \frac{y}{\sqrt{x^2+y^2}+x} \right) & \qquad y \ne 0 \\ 0 & \qquad x>0, \, y = 0 \\ \pi & \qquad x<0, \, y = 0 \\ \end{cases}.$

## 性质

$z = |z| e^{\mathbf{i}\phi }$

$\arg\left(z_1 z_2\right) = \arg(z_1) + \arg(z_2) \pmod {2\pi} ,$以及
$\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) \pmod {2\pi} .$

$\arg\left(z^n\right) = n \arg(z) \pmod {2\pi} .$

$\arg(\bar{z}) = - \arg(z) \pmod {2\pi} .$

## 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.