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 function to the trigonometric sine and cosine functions.
Euler’s relation states that
Start by noting that
Using the Taylor series expansions of , and (see the entries on the complex exponential function and the complex sine and cosine), it follows that
Because the series expansion above is absolutely convergent for all , we can rearrange the terms of the series as
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 |