Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric...

Euler's identity (also known as Euler's equation) is the equality e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} where e {\displaystyle e} is Euler's number...

been given simple yet ambiguous names such as Euler's function, Euler's equation, and Euler's formula. Euler's work touched upon so many fields that he is...

In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate...

has Euler characteristic 2. This viewpoint is implicit in Cauchy's proof of Euler's formula given below. There are many proofs of Euler's formula. One...

integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric...

way) as the basis for a proof of Euler's formula. Alternative proofs of Pick's theorem that do not use Euler's formula include the following. One can recursively...

developed by John Butler Johnson in 1893 as an alternative to Euler's critical load formula under low slenderness ratio (the ratio of radius of gyration...

mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different...

the complex exponential function. One can derive de Moivre's formula using Euler's formula and the exponential law for integer powers ( e i x ) n = e i...

in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in...

Eruditorum, 1744 The title page of Euler's Methodus inveniendi lineas curvas Euler's 1760 world map Euler's 1753 map of Africa Euler is listed by an academic genealogy...

Euler's critical load or Euler's buckling load is the compressive load at which a slender column will suddenly bend or buckle. It is given by the formula:...

infinite series. Of course, Euler's original reasoning requires justification (100 years later, Karl Weierstrass proved that Euler's representation of the sine...

limit as n goes to infinity, from which Euler's formula follows. Viète's formula may be obtained from this formula by the substitution x = π/2. Morrie's...

easily the rectangular form was deduced by an application of Euler's formula. Euler's formula introduces the possibility of negative ξ . {\displaystyle \xi...

characterize the planar graphs via a system of equations modulo 2. Euler's formula states that if a finite, connected, planar graph is drawn in the plane...

specific examples of compound crystals (aka double crystals) for which Euler's formula for convex polyhedra failed. In this case, the sum of the valence (degree)...

also referred to as Euler's totient function, the Euler totient, or Euler's totient. Jordan's totient is a generalization of Euler's. The cototient of n...

^{2}{A}} and sin 2 ⁡ A {\displaystyle \sin ^{2}{A}} , respectively. Euler's formula, which states that e i x = cos ⁡ x + i sin ⁡ x {\displaystyle e^{ix}=\cos...

connection with Euclid's demonstration of the Pythagorean theorem. Using Euler's formula e i θ = cos ⁡ θ + i sin ⁡ θ {\displaystyle e^{i\theta }=\cos \theta...

prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem. Euler's great interest in number theory...

be done readily on the power series representation; for instance, Euler's formula follows from Taylor series expansions for trigonometric and exponential...

recommended calling it "Euler's finite rotation formula." This proposal has received notable support, but some others have viewed the formula as just one of many...

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a...

a slightly modified linear Taylor approximation to the monthly payment formula that could be computed easily in their heads. In modern times, Albert Einstein's...

table Bhaskara I's sine approximation formula Discrete sine transform Dixon elliptic functions Euler's formula Generalized trigonometry Hyperbolic function...

sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant,...

The above definitions are related to the exponential definitions via Euler's formula (See § Hyperbolic functions for complex numbers below). It can be shown...

geometric argument that can be interpreted as a logarithmic version of Euler's formula. He was the first Plumian Professor at Cambridge University from 1707...