Euler Formula

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

(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

(2)

had previously been published by Cotes (1714).

The special case of the formula with  gives the beautiful identity

(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

			(4)
			(5)
			(6)

It can also be demonstrated using a complex integral. Let

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)

so

			(13)
			(14)

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

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/.