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

This article is about Euler's formula in complex analysis. For other uses, see List of things named after Leonhard Euler § Formulae.

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.

$x = Re z$ is the real part,

$y = Im z$ is the imaginary part,

$r = | z | = \sqrt x 2 + y 2$ is the of $z$ and

magnitude

$φ = arg z = (y, x)$ .

atan2

Complex number

Euler's identity

Integration using Euler's formula

History of Lorentz transformations

List of things named after Leonhard Euler

Nahin, Paul J. (2006). . Princeton University Press. ISBN 978-0-691-11822-2.

Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills

Wilson, Robin (2018). Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics. Oxford: Oxford University Press. 978-0-19-879492-9. MR 3791469.