Euler's formula

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This article is about Euler's formula in complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics

see Euler characteristic.

Part of a series of articles on The mathematical constant e Natural logarithm ·Exponential function Applications in: compound interest · Euler's identity & Euler's formula · half-lives & exponential growth / decay Defining e: proof that e is irrational · representations of e · Lindemann–Weierstrass theorem People John Napier · Leonhard Euler Schanuel's conjecture

Euler's formula, named after Leonhard Euler, is a mathematicalformula in complex analysis that establishes the deep relationship between the trigonometric functions and the complexexponential function. Euler's formula states that, for any real number x,

where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis (x). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula. ^[1]

Richard Feynman called Euler's formula " our jewel" and " one of the most remarkable, almost astounding, formulas in all of mathematics." ^[2]