...(one writes $\Gamma(z) = (z-1)!$). It was introduced in 1729 by L. Euler in a letter to Ch. Goldbach, using the infinite product
...ain the integral representation (Euler integral of the second kind, cf. [[Euler integrals]])
10 KB (1,513 words) - 09:50, 5 December 2012
2) Euler base-$b$ pseudo-primes, whose [[Jacobi symbol]] with $b$ satisfies
...rmat) witness for $n$ if $n$ is composite and not a base-$b$ pseudo-prime. Euler and strong witnesses are similarly defined. If $W(n)$, the smallest strong
3 KB (516 words) - 17:58, 8 November 2014
...cit method for stiff initial-value problems is the backward [[Euler method|Euler method]]
...thereof is used, again for reasons of numerical stability. However, as the Euler method has only order of consistency $ p = 1 $,
5 KB (764 words) - 08:12, 6 June 2020
...e non-resonant case the Fuchsian system is locally gauge equivalent to the Euler system
;Example (Euler equation).
7 KB (1,237 words) - 11:48, 23 November 2023
taken in an arbitrary direction, by means of Euler's formula:
4 KB (610 words) - 14:54, 7 June 2020
...the number of which is $\phi(p^n-1)$, where $\phi$ is the [[Euler function|Euler function]]. Each primitive root of unity of degree $p^n-1$ is a primitive e
4 KB (749 words) - 18:32, 2 November 2014
Equation (4) is called the Euler–Ostrogradski equation (sometimes the Ostrogradski equation). This equatio
...(2). The Euler–Ostrogradski equation is analogous to the [[Euler equation|Euler equation]] for one-dimensional problems. In expanded form, (4) is a second-
4 KB (627 words) - 08:01, 6 June 2020
...bers. It was posed in 1742 by Ch. Goldbach in a communication to L. Euler. Euler replied by pointing out that in order to solve this problem it is sufficien
3 KB (514 words) - 21:14, 9 January 2015
...polygonal extremal is a piecewise-smooth solution of the [[Euler equation|Euler equation]] satisfying certain additional necessary conditions at the vertic
...ts determining the polygonal extremal — it is a particular solution of the Euler equation satisfying the boundary conditions. In fact, the equations (9) giv
7 KB (1,027 words) - 19:35, 5 June 2020
One of the simplest numerical methods — the [[
Euler method|
Euler method]] — is based on an approximate computation of the integral term in
The truncation error of
Euler's method is proportional to <img align="absmiddle" border="0" src="https://
32 KB (4,604 words) - 16:55, 7 February 2011
...e functional $W$ are called Willmore surfaces and are characterized by the Euler equation $\Delta H + 2 H ( H ^ { 2 } - K ) = 0$ corresponding to the variat
...hi ( \Sigma )$, where $\chi ( \Sigma )$ denotes the [[Euler characteristic|Euler characteristic]] of the surface, so that $W$ and $\tilde { W }$ have the sa
8 KB (1,192 words) - 16:56, 1 July 2020
L. Euler made an exceptionally important contribution to number theory. He proved Fe
...led to the creation of [[Analytic number theory|analytic number theory]]. Euler's generating functions (cf. [[Generating function|Generating function]]) ap
10 KB (1,503 words) - 08:03, 6 June 2020
Euler observed that the Euclidean algorithm applied to a pair of natural numbers
2 KB (351 words) - 20:40, 16 November 2023
...ents satisfy certain compatibility conditions. Generally, almost all known Euler systems satisfy the condition ES) described below. Let $K$ be a number fiel
...pping from $K ( L l )$ down to $K ( L )$. Next to condition ES), any given Euler system may have additional properties, cf. [[#References|[a4]]], [[#Referen
19 KB (2,901 words) - 17:41, 25 November 2023
...unity. The number of such roots in $U_n$ is given by the [[Euler function|Euler function]] $\phi(n)$, i.e. the number of residues $\bmod\,n$ which are rela
4 KB (680 words) - 13:40, 30 December 2018
...$H_1(K;\mathbf Z)=\mathbf Z\oplus\mathbf Z/(2)$, $H_2(K;\mathbf Z)=0$. Its Euler characteristic is $\chi(K)=0$. Together with the torus it is the only smoot
2 KB (269 words) - 19:36, 17 April 2014
.... [[Du Bois-Reymond lemma|du Bois-Reymond lemma]]) is the [[Euler equation|Euler equation]]:
5 KB (687 words) - 08:27, 6 June 2020
...so [[Convolution transform|Convolution transform]]; [[Euler transformation|Euler transformation]]; [[Fourier transform|Fourier transform]]; [[Gauss transfor
7 KB (959 words) - 19:34, 17 January 2024
...he constant $e^{-\gamma}$ as $u\rightarrow\infty$, where $\gamma$ is the [[Euler constant]]. The difference $\omega(u)-e^{-\gamma}$ behaves asymptotically l
2 KB (243 words) - 15:57, 22 September 2017
coincides with the reduction modulo 2 of the [[Euler class|Euler class]].
5 KB (692 words) - 17:18, 6 January 2024