In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the...

In mathematics, the Euler function is given by ϕ ( q ) = ∏ k = 1 ∞ ( 1 − q k ) , | q | < 1. {\displaystyle \phi (q)=\prod _{k=1}^{\infty }(1-q^{k}),\quad...

absolutely, and is known as the Euler integral of the second kind. (Euler's integral of the first kind is the beta function.) Using integration by parts...

Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation...

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined...

notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal...

fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one...

exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal...

{\displaystyle \cosh(t)} is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely: E n = 2 n E n ( 1...

\mathrm {d} x.\end{aligned}}} Here, ⌊·⌋ represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: 0...

In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose...

In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary...

series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in particular...

exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers...

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...

the exponential function x ↦ e x {\displaystyle x\mapsto e^{x}} are not homogeneous. Roughly speaking, Euler's homogeneous function theorem asserts that...

{\displaystyle x=2\pi i\tau } in Euler Pentagonal number theorem with the definition of eta function. Another way to see the Eta function is through the following...

the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial...

ϕ {\displaystyle \phi } is the Euler function, η {\displaystyle \eta } is the Dedekind eta function, and the function Δ ( z ) {\displaystyle \Delta (z)}...

Euler Mathematical Toolbox (or EuMathT; formerly Euler) is a free and open-source numerical software package. It contains a matrix language, a graphical...

type of superspiral that has the property of a monotonic curvature function. The Euler spiral has applications to diffraction computations. They are also...

proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function. In general, if...

dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular...

In mathematics, Euler's pentagonal number theorem relates the product and series representations of the Euler function. It states that ∏ n = 1 ∞ ( 1 −...

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...

denotes Euler's totient function; that is a φ ( n ) ≡ 1 ( mod n ) . {\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}.} In 1736, Leonhard Euler published...

algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant...

Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations...

Ramanujan's lost notebook and a relevant reference at Euler function. The Ramanujan results quoted at Euler function plus a few elementary operations give the results...

numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the...