Complex Math Simple Life

De Moivre’s Formula

Posted in complex, de moivre, theorem, trigonometry by beauangel on January 17, 2007

For any complex number (and in particular any real number) x, and any integer n,

\left( \cos x + i \sin x \right)^n = \cos (nx) + i \sin (nx)

This is the equation that Abraham de Moivre came up with to link two branches of mathematics together, that of trigonometry and complex numbers.

This formula can be used to find the nth root of a complex number say z.

Written in polar form,

z = r (\cos x + i \sin x)

then,

z^{\tfrac{1}{n}} = \left( r(\cos x + i \sin x) \right)^{\tfrac{1}{n}} = r^{\tfrac{1}{n}} \left\{ \cos \left( \frac{x+2k\pi}{n} \right) + i \sin \left( \frac{x+2k\pi}{n} \right)\right\}

where k varies from 0 to n — 1 to give the n roots of the complex number.

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