De Moivre's Theorem (DP IB Analysis & Approaches (AA): HL): Revision Note

De Moivre's theorem

What is de Moivre’s theorem?

• De Moivre’s theorem states that for$z=r\mathrm{cis}\theta =r\left(\cos \theta +\mathrm{isin}\theta \right)$ 

  • $z^n=\left[r \right(\cos \theta +\mathrm{isin}\theta ){]}^n =r^n(\cos n\theta +\mathrm{isin}n\theta )$

  • where

    • $n\in \mathrm{ℝ}$

    • $r=\left|z\right|$

    • $\theta =\arg z$

    • $z\ne 0$

• In exponential (Euler’s) form this is simply

  • ${{\left(r{\mathrm{e}}^{\mathrm{i}\theta }\right)}^n= r}^n{\mathrm{e}}^{\mathrm{i}n\theta }$

    • It confirms that you can use 'index laws' with complex exponentials

How do I find positive powers of a complex number?

• To raise a complex number $z$ to a power, $n$

  • write $z$ in modulus-argument (polar) form

  • apply de Moivre's theorem in the form $z^n=r^n\left(\cos n\theta +\mathrm{i}\sin n\theta \right)$

  • simplify the real and imaginary parts

• e.g. if $z=2\left(\cos \frac{\pi }{12}+\mathrm{i}\sin \frac{\pi }{12}\right)$, find $z^8$

  • $z^8=2^8\left(\cos \left(8\times \frac{\pi }{12}\right)+\mathrm{i}\sin \left(8\times \frac{\pi }{12}\right)\right)$

    • using $z^n=r^n\left(\cos n\theta +\mathrm{i}\sin n\theta \right)$

  • which simplifies to $256\left(\cos \left(\frac{2\pi }{3}\right)+\mathrm{i}\sin \left(\frac{2\pi }{3}\right)\right)=256\left(-\frac{1}{2}+\frac{\mathrm{i}\sqrt{3}}{2}\right)$

    • giving $z^8=-128+128\sqrt{3}\mathrm{i}$

How do I find negative powers of a complex number?

• $z^n=r^n\left(\cos n\theta +\mathrm{i}\sin n\theta \right)$ works for negative powers

  • e.g. $\frac{1}{z}=z^{-1}=r^{-1}\left(\cos \left(-\theta \right)+\mathrm{i}\sin \left(-\theta \right)\right)$

    • recall that $\cos \left(-\theta \right)=\cos \theta$ and $\sin \left(-\theta \right)=-\sin \theta$

    • giving $\frac{1}{z}=r^{-1}\left(\cos \theta -\mathrm{i}\sin \theta \right)$