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


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!

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)!

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

Title Euler relation
Canonical name EulerRelation
Date of creation 2013-03-22 11:57:05
Last modified on 2013-03-22 11:57:05
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 17
Author rm50 (10146)
Entry type Definition
Classification msc 30B10
Synonym Euler’s formula
Related topic TaylorSeries
Related topic DeMoivreIdentity
Related topic ComplexSineAndCosine
Defines Euler identity
Defines Euler’s identity