Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number $x$ , one has

e^{ix}=\cos x+i\sin 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. This complex exponential function is sometimes denoted

cis x

("cosine plus i sine"). The formula is still valid if

x

is a complex number, and is also called Euler's formula in this more general case.^[1]

Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".^[2]

When $x = π$ , Euler's formula may be rewritten as $e iπ + 1 = 0$ or $e iπ = -1$ , which is known as Euler's identity.

^ Moskowitz, Martin A. (2002). A Course in Complex Analysis in One Variable. World Scientific Publishing Co. p. 7. ISBN 981-02-4780-X.
^ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley. p. 22-10. ISBN 0-201-02010-6.

[1] Moskowitz, Martin A. (2002). A Course in Complex Analysis in One Variable. World Scientific Publishing Co. p. 7. ISBN 981-02-4780-X.

[2] Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley. p. 22-10. ISBN 0-201-02010-6.

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