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单词 EulerRelation
释义

Euler relation


Euler’s relation (also known as Euler’s formula) is considered the first between the fields of algebra and geometry, as it relates the exponential functionDlmfDlmfMathworldPlanetmathPlanetmath to the trigonometric sine and cosine functions.

Euler’s relation states that

eix=cosx+isinx

Start by noting that

ik={1if k0(mod4)iif k1(mod4)-1if k2(mod4)-iif k3(mod4)

Using the Taylor seriesMathworldPlanetmath expansions of ex, sinx and cosx (see the entries on the complex exponential function and the complex sine and cosine), it follows that

eix=n=0inxnn!
=n=0(x4n(4n)!+ix4n+1(4n+1)!-x4n+2(4n+2)!-ix4n+3(4n+3)!)

Because the series expansion above is absolutely convergent for all x,we can rearrange the terms of the series as

eix=n=0(-1)nx2n(2n)!+in=0(-1)nx2n+1(2n+1)!
=cosx+isinx

As a special case, we get the beautiful and well-known identity, often called Euler’s identity:

eiπ=-1
TitleEuler relation
Canonical nameEulerRelation
Date of creation2013-03-22 11:57:05
Last modified on2013-03-22 11:57:05
Ownerrm50 (10146)
Last modified byrm50 (10146)
Numerical id17
Authorrm50 (10146)
Entry typeDefinition
Classificationmsc 30B10
SynonymEuler’s formula
Related topicTaylorSeries
Related topicDeMoivreIdentity
Related topicComplexSineAndCosine
DefinesEuler identity
DefinesEuler’s identity
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更新时间:2025/5/4 2:19:26