Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
Mathematics-Online lexicon:

Powers of Complex Numbers

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

For $ z = r e^{\mathrm{i}\varphi}$ and a rational exponent $ p/q$ ,

$\displaystyle z^{p/q} = r^{p/q} \exp(p(\varphi+2\pi j)\mathrm{i}/q),\quad j=0,\ldots,q-1 \,. $

The fractional power is ambiguous since $ \varphi=\operatorname{arg}(z)$ is determined only up to multiples of $ 2\pi$ . Possible values differ by powers of the $ q$ -th roots of unity:

$\displaystyle w_q^0,w_q^p,\ldots,w_q^{(q-1)p},\quad w_q = \exp(2\pi\mathrm{i}/q) \,. $

(Authors: Höllig/Hörner/Kopf/Abele)

Examples:

[Links]
  automatically generated 10/ 4/2007