## Euler Formula

The Euler formula states

 (1)

where i is the Imaginary Number. Note that the Euler 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 Euler formula can be demonstrated using a series expansion

 (4)

It can also be proven using a Complex integral. Let
 (5)

 (6)

 (7)

 (8)

so
 (9)

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. Philosophical Transactions 29, 32, 1714.

Euler, L. Miscellanea Berolinensia 7, 179, 1743.

Euler, L. Introductio in Analysin Infinitorum, Vol. 1. Lausanne, p. 104, 1748.