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Powers of Complex Numbers

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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)

see also:

[Examples]
  automatically generated 10/ 4/2007