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单词 DeMoivreIdentity
释义

de Moivre identity


From the Euler relation

eiθ=cosθ+isinθ

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,

cos2θ+isin2θ=(cosθ+isinθ)2=cos2θ+2isinθcosθ-sin2θ.

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

cos2θ=cos2θ-sin2θ

and

sin2θ=2sinθcosθ.
Titlede Moivre identityMathworldPlanetmath
Canonical nameDeMoivreIdentity
Date of creation2013-03-22 12:20:45
Last modified on2013-03-22 12:20:45
OwnerDaume (40)
Last modified byDaume (40)
Numerical id11
AuthorDaume (40)
Entry typeTheorem
Classificationmsc 12E10
Synonymde Moivre’s theorem
Synonymde Moivre’s formula
Related topicEulerRelation
Related topicDoubleAngleIdentity
Related topicArgumentOfProductAndSum
Related topicArgumentOfProductAndQuotient
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更新时间:2025/5/4 6:07:27