De Moivre formula

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The formula expressing the rule for raising a complex number, expressed in trigonometric form \begin{equation} z = \rho(\cos\varphi + i\sin\varphi), \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:

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 and in powers of and :

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

which is also sometimes called de Moivre's formula.



[a1] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
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De Moivre formula. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article