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Mathematik-Online lexicon:

Rational Power of a Complex Number

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In order to calculate

$\displaystyle z =(-1+\mathrm{i})^{2/3} $

we transform $ z$ into polar coordinates. This yields

$\displaystyle \left(\sqrt{2} \exp\left(\frac{3 \pi \mathrm{i}}{4}\right)\right)... ...sqrt[3]{2} \exp\left(\frac{\mathrm{i}\pi}{2}\right) w_3^{2k}, \quad k=0,1,2\,, $

with $ w_3 = \exp(2\pi\mathrm{i}/3)$. Thus, possible values are

$\displaystyle z_0 =$ $\displaystyle \sqrt[3]{2}\; \mathrm{i} \,,$    
$\displaystyle z_1 =$ $\displaystyle \sqrt[3]{2} \; \mathrm{i} \exp\left( \frac{4 \pi \mathrm{i}}{3} \right) = \sqrt[3]{2}\left( \frac{\sqrt{3}}{2}-\frac{\mathrm{i}}{2} \right)\,,$    
$\displaystyle z_2 =$ $\displaystyle \sqrt[3]{2}\; \mathrm{i} \exp\left( \frac{8 \pi \mathrm{i}}{3} \right) = \sqrt[3]{2}\left( -\frac{\sqrt{3}}{2} -\frac{\mathrm{i}}{2} \right)\,.$    

The results can be easily checked. For example, we have

$\displaystyle z_1^3 = \left(\sqrt[3]{2} \;\mathrm{i} \exp\left( \frac{4 \pi \mathrm{i}}{3} \right) \right)^3 = -2 \,\mathrm{i} \,, $

which equals $ (-1+\mathrm{i})^2$ .

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  automatisch erstellt am 15. 11. 2013