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.

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".

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.

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mathematical constant $e$
Part of a series of articles on the

Properties
Natural logarithm Exponential function
Applications
compound interest Euler's identity Euler's formula half-lives exponential growth and decay
Defining $e$
proof that $e$ is irrational representations of $e$ Lindemann–Weierstrass theorem
People
John Napier Leonhard Euler
Related topics
Schanuel's conjecture