De Moivre formula

The formula expressing the rule for raising a complex number, expressed in trigonometric form \begin{equation} z = \rho(\cos\phi + i\sin\phi), \end{equation} to an $n$-th power. According to de Moivre's formula the modulus $\rho$ of the complex number is raised to that power and the argument $\varphi$ is multiplied by the exponent:

\[ z^n = [\rho(\cos \phi + i \sin \phi)]^n = \rho^n(\cos n\phi + i \sin n \phi). \]

The formula was found by A. de Moivre (1707), its modern notation was suggested by L. Euler (1748).

De Moivre's formula can be used to express $ \cos n \phi $ and $ \sin n \phi $ in powers of $ \cos \phi $ and $ \sin \phi $:

\[ \cos n\phi = \cos^n \phi - \binom{n}{2} \cos^{n-2} \phi \sin^2 \phi + \binom{n}{4}\cos^{n-4}\phi \sin^4\phi - \dots, \]

\[ \sin n\phi = \binom{n}{1}\cos^{n-1}\phi \sin \phi - \binom{n}{3} \cos^{n-3}\phi \sin^3\phi + \dots. \]

Inversion of de Moivre's formula leads to a formula for extracting roots of a complex number:

\[ [\rho (\cos \phi + i \sin \phi)]^{1/n} = \rho^{1/n}\left( \cos \frac{\phi + 2 \pi k}{n} + i \sin \frac{\phi + 2 \pi k}{n} \right), \quad k = 0, 1, \dots, \] which is also sometimes called de Moivre's formula.

References

[a1]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)