“Euler Formula”的概念、定义、习题解题方法及翻译-数学辞典

单词

Euler Formula

释义

Euler Formula

The Euler formula states

(1)

where i is the Imaginary Number. Note that the Euler Polyhedral Formula is sometimes also calledthe 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 beautifulidentity

(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)

(9)

See also de Moivre's Identity, Euler Polyhedral Formula

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.

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