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Euler Formula


The Euler formula, sometimes also called the Euler identity (e.g., Trott 2004, p. 174), states

 e^(ix)=cosx+isinx,
(1)

where i is the imaginary unit. Note that Euler's polyhedral formula is sometimes also called the Euler formula, as is the Euler curvature formula. The equivalent expression

 ix=ln(cosx+isinx)
(2)

had previously been published by Cotes (1714).

The special case of the formula with x=pi gives the beautiful identity

 e^(ipi)+1=0,
(3)

an equation connecting the fundamental numbers i, pi, e, 1, and 0 (zero), the fundamental operations +, ×, and exponentiation, the most important relation =, and nothing else. Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician (Derbyshire 2004, p. 202).

The Euler formula can be demonstrated using a series expansion

e^(ix)=sum_(n=0)^(infty)((ix)^n)/(n!)
(4)
=sum_(n=0)^(infty)((-1)^nx^(2n))/((2n)!)+isum_(n=1)^(infty)((-1)^(n-1)x^(2n-1))/((2n-1)!)
(5)
=cosx+isinx.
(6)

It can also be demonstrated using a complex integral. Let

z=costheta+isintheta
(7)
dz=(-sintheta+icostheta)dtheta
(8)
=i(costheta+isintheta)dtheta
(9)
=izdtheta
(10)
int(dz)/z=intidtheta
(11)
lnz=itheta,
(12)

so

z=e^(itheta)
(13)
=costheta+isintheta.
(14)

A mathematical joke asks, "How many mathematicians does it take to change a light bulb?" and answers "-e^(ipi)" (which, of course, equals 1).


See also

de Moivre's Identity, Euler Identity, Polyhedral Formula

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References

Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988.Conway, J. H. and Guy, R. K. "Euler's Wonderful Relation." The Book of Numbers. New York: Springer-Verlag, pp. 254-256, 1996.Cotes, R. "Logometria." Philos. Trans. Roy. Soc. London 29, 5-45, 1714.Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.Euler, L. "De summis serierum reciprocarum ex potestatibus numerorum naturalium ortarum dissertatio altera." Miscellanea Berolinensia 7, 172-192, 1743.Euler, L. Introductio in Analysin Infinitorum, Vol. 1. Bosquet, Lucerne, Switzerland: p. 104, 1748.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, p. 212, 1998.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.

Cite this as:

Weisstein, Eric W. "Euler Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerFormula.html

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