de Moivre identity

From the Euler relation


it follows that

eiθn =(eiθ)n
cosnθ+isinnθ =(cosθ+isinθ)n

where n. This is called de Moivre’s formula, and besides being generally useful, it’s a convenient way to remember double- (and higher-multiple-) angle formulas. For example,


Since the imaginary partsMathworldPlanetmath and real parts on each side must be equal, we must have



Title de Moivre identityMathworldPlanetmath
Canonical name DeMoivreIdentity
Date of creation 2013-03-22 12:20:45
Last modified on 2013-03-22 12:20:45
Owner Daume (40)
Last modified by Daume (40)
Numerical id 11
Author Daume (40)
Entry type Theorem
Classification msc 12E10
Synonym de Moivre’s theorem
Synonym de Moivre’s formula
Related topic EulerRelation
Related topic DoubleAngleIdentity
Related topic ArgumentOfProductAndSum
Related topic ArgumentOfProductAndQuotient