Difference between revisions of "Euler's Identity - Adam Frey" - Rhea

Revision as of 11:10, 20 July 2009

Adam Frey's Euler Identity Summary

The identity is a special case of Euler's formula from complex analysis, which states that where j = i = $ \sqrt{-1} $

$ e^{jx} = \cos x + j \sin x \,\! $

for any real number x. (Note that sine and cosine should be in radians)

In particular,

$ e^{j \pi} = \cos \pi + j \sin \pi.\,\! $

We know from trig identities that:

$ \cos \pi = -1 \, \! $

and

$ \sin \pi = 0,\,\! $

which results in

$ e^{j \pi} = -1,\,\! $

which gives the identity

$ e^{j \pi} +1 = 0.\,\! $

Also useful is the relationship in splitting sine and cosine is where

$ \cos wt = \frac{e^{j w}}{2} + \frac{e^{-j w}}{2} \,\! $

and

$ \sin wt = \frac{e^{j w}}{2j} - \frac{e^{-j w}}{2j} \,\! $

--Freya 15:04, 20 July 2009 (UTC)

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