Euler's formula

related topics
{math, number, function}
{math, energy, light}
{school, student, university}

  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]

Contents

Full article ▸

related documents
Primitive recursive function
Dual space
Continuous function
Convolution
Fundamental theorem of algebra
Basis (linear algebra)
BCH code
Ackermann function
Hyperreal number
Bessel function
Fundamental group
Computable number
Multivariate normal distribution
Halting problem
Fermat number
Dynamic programming
Lp space
Probability theory
Prime number theorem
Group action
Abelian group
Subset sum problem
Monte Carlo method
Permutation
Uniform space
Truth table
Factorial
Taylor series
Frame problem
Support vector machine