Euler's formula

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  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 mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential 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 called 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"[2] and "one of the most remarkable, almost astounding, formulas in all of mathematics."[3]


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