...elative to another one $0x'y'z'$ with the same origin and orientation. The Euler angles are regarded as the angles through which the former must be successi
These angles were introduced by L. Euler (1748).
2 KB (331 words) - 14:12, 13 November 2014
considered by L. Euler (1740). Its existence follows from the fact that the sequence
...mber $\gamma$ is also known as the ''Euler-Mascheroni'' constant, after L. Euler (1707–1783) and L. Mascheroni (1750–1800).
2 KB (328 words) - 11:50, 23 November 2023
$#C+1 = 61 : ~/encyclopedia/old_files/data/E036/E.0306620 Euler transformation
The Euler transformation of series. Given a series
6 KB (972 words) - 12:59, 6 January 2024
Euler's theorem hold for polyhedrons of genus $0$; for polyhedrons of genus $p$ t
holds. This theorem was proved by L. Euler (1758); the relation \eqref{*} was known to R. Descartes (1620).
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''Euler circuit, Euler cycle, Eulerian cycle''
155 bytes (17 words) - 17:04, 7 February 2011
$#C+1 = 37 : ~/encyclopedia/old_files/data/E036/E.0306400 Euler characteristic
It was given this name in honour of L. Euler, who proved in 1758 that the number $ V $
4 KB (560 words) - 08:59, 4 November 2023
$#C+1 = 89 : ~/encyclopedia/old_files/data/E036/E.0306440 Euler equation
This equation was studied in detail by L. Euler, starting from 1740.
13 KB (1,798 words) - 19:06, 26 January 2022
$#C+1 = 34 : ~/encyclopedia/old_files/data/E036/E.0306530 Euler method
by a polygonal line (Euler's polygonal line) whose segments are rectilinear on the intervals $ [ x _
6 KB (877 words) - 19:38, 5 June 2020
#REDIRECT [[Euler straight line]]
33 bytes (4 words) - 19:04, 6 November 2016
...act that the set of [[prime number]]s is infinite. The partial sums of the Euler series satisfy the asymptotic relation
659 bytes (104 words) - 15:21, 10 April 2023
This formula was established by L. Euler (1760).
803 bytes (112 words) - 17:23, 30 July 2014
called the Euler integral of the first kind, or the [[Beta-function|beta-function]], and
called the Euler integral of the second kind. (The latter converges for $s>0$ and is a repre
431 bytes (67 words) - 21:21, 29 April 2012
-
31 bytes (3 words) - 19:11, 6 November 2016
...ues of the complex variable $z$. In particular, for a real value $z=x$ the Euler formulas become
These formulas were published by L. Euler in [[#References|[1]]].
1 KB (171 words) - 12:50, 10 August 2014
...itrary real number and the product extends over all prime numbers $p$. The Euler identity also holds for all complex numbers $s = \sigma + it$ with $\sigma
The Euler identity can be generalized in the form
2 KB (279 words) - 19:13, 14 December 2015
The first Euler substitution: If $a>0$, then
The second Euler substitution: If the roots $x_1$ and $x_2$ of the quadratic polynomial $ax^
2 KB (366 words) - 06:13, 10 April 2023
''Euler's totient function''
...ot exceeding $n$ and relatively prime to $n$ (the "totatives" of $n$). The Euler function is a [[multiplicative arithmetic function]], that is $\phi(1)=1$ a
2 KB (318 words) - 09:27, 10 November 2023
''Euler–Lagrange operator''
...for variational problems must satisfy (cf. also [[Euler–Lagrange equation|Euler–Lagrange equation]]).
17 KB (2,575 words) - 17:45, 1 July 2020
See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
557 bytes (85 words) - 18:50, 18 October 2014
...oriented $n$-dimensional [[Manifold|manifold]] may be calculated from the Euler class of the [[Tangent bundle|tangent bundle]] [[#References|[a1]]], p. 348
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often for $q=1$, used in the [[Euler summation method|Euler summation method]].
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The recurrence formula for the Euler numbers ($E^n\equiv E_n$ in symbolic notation) has the form
...\dots$; $E_2=-1$, $E_4=5$, $E_6=-61$, $E_8=1385$, and $E_{10}=-50521$. The Euler numbers are connected with the [[Bernoulli numbers|Bernoulli numbers]] $B_n
2 KB (297 words) - 11:53, 23 November 2023
...be a [[quadratic residue]] or non-residue modulo $p$. It was proved by L. Euler in 1761 (see [[#References|[1]]]).
Euler also obtained a more general result: A number $a \not\equiv 0 \pmod p$ is a
1 KB (217 words) - 07:30, 19 December 2014
$#C+1 = 23 : ~/encyclopedia/old_files/data/E036/E.0306550 Euler polynomials
are the [[Euler numbers]]. The Euler polynomials can be computed successively by means of the formula
3 KB (477 words) - 08:36, 6 January 2024
-
54 bytes (8 words) - 18:01, 2 May 2012
is summable by means of the Euler summation method ($(E,q)$-summable) to the sum $S$ if
...hen its terms $a_n$ satisfy the condition $a_n=o((2q+1)^n)$, $q\geq0$. The Euler summation method can also be applied for analytic continuation beyond the d
2 KB (358 words) - 17:36, 14 February 2020
$#C+1 = 34 : ~/encyclopedia/old_files/data/E036/E.0306520 Euler\ANDMacLaurin formula
then the Euler–MacLaurin formula becomes
5 KB (745 words) - 09:18, 6 January 2024
#REDIRECT [[Euler function]]
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...s or right-angled, or both right-angled and isosceles. The segments of the Euler line satisfy the relation
This line was first considered by L. Euler (1765).
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A generalization of the Goldbach–Euler problem (cf. [[Goldbach problem|Goldbach problem]]) according to which any
...of the distribution of prime numbers allows one to solve both the Goldbach–Euler problem and the more general problem of solvability of the linear Diophanti
1 KB (201 words) - 20:18, 14 October 2014
...(x,y)$ is zero. Particular integrals for it were obtained by G. Monge. The Euler–Lagrange equation was systematically investigated by S.N. Bernshtein, who
The Euler–Lagrange equation can be generalized with respect to the dimension: The e
3 KB (501 words) - 18:46, 13 November 2014
The Euler–Frobenius polynomials $p _ { m } ( x )$ of degree $m - 1 \geq 0$ are char
...invariance of this kind is to look for an appropriate space with which the Euler–Frobenius polynomials $( p _ { m } ( x ) ) _ { m \geq 1 }$ are attached i
5 KB (742 words) - 07:27, 25 January 2024
...3]]], the colliding of gravitational waves [[#References|[a6]]], etc.. The Euler–Poisson–Darboux equation has rather interesting properties, e.g. in rel
A formal solution to the Euler–Poisson–Darboux equation has the form [[#References|[a8]]]
9 KB (1,341 words) - 20:49, 23 January 2024
-
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...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
...ars the name of Euler–Knopp summation method, see [[Euler summation method|Euler summation method]].
246 bytes (39 words) - 15:19, 1 May 2014
...numbers, where $\phi(m)$ is Euler's $\phi$-function (cf. [[Euler function|Euler function]]). One usually takes the numbers mutually prime with $m$ in the c
521 bytes (82 words) - 12:45, 23 November 2014
...oriented $n$-dimensional [[Manifold|manifold]] may be calculated from the Euler class of the [[Tangent bundle|tangent bundle]] [[#References|[a1]]], p. 348
794 bytes (108 words) - 18:57, 17 April 2014
Euler's theorem hold for polyhedrons of genus $0$; for polyhedrons of genus $p$ t
holds. This theorem was proved by L. Euler (1758); the relation \eqref{*} was known to R. Descartes (1620).
418 bytes (68 words) - 17:34, 14 February 2020
called the Euler integral of the first kind, or the [[Beta-function|beta-function]], and
called the Euler integral of the second kind. (The latter converges for $s>0$ and is a repre
431 bytes (67 words) - 21:21, 29 April 2012
''Euler circuit, Euler cycle, Eulerian cycle''
155 bytes (17 words) - 17:04, 7 February 2011
#REDIRECT [[Euler constant]]
28 bytes (3 words) - 19:25, 29 December 2014
#REDIRECT [[Euler function]]
28 bytes (3 words) - 21:24, 23 December 2015
...s or right-angled, or both right-angled and isosceles. The segments of the Euler line satisfy the relation
This line was first considered by L. Euler (1765).
730 bytes (113 words) - 20:16, 16 January 2016
#REDIRECT [[Euler straight line]]
33 bytes (4 words) - 19:04, 6 November 2016
often for $q=1$, used in the [[Euler summation method|Euler summation method]].
188 bytes (34 words) - 14:11, 23 July 2014
...ues of the complex variable $z$. In particular, for a real value $z=x$ the Euler formulas become
These formulas were published by L. Euler in [[#References|[1]]].
1 KB (171 words) - 12:50, 10 August 2014
The recurrence formula for the Euler numbers ($E^n\equiv E_n$ in symbolic notation) has the form
...\dots$; $E_2=-1$, $E_4=5$, $E_6=-61$, $E_8=1385$, and $E_{10}=-50521$. The Euler numbers are connected with the [[Bernoulli numbers|Bernoulli numbers]] $B_n
2 KB (297 words) - 11:53, 23 November 2023
''Euler totient function, Euler totient''
Another frequently used named for the [[Euler function]] $\phi(n)$, which counts a [[reduced system of residues]] modulo
3 KB (519 words) - 10:04, 14 December 2014
''Euler's totient function''
...ot exceeding $n$ and relatively prime to $n$ (the "totatives" of $n$). The Euler function is a [[multiplicative arithmetic function]], that is $\phi(1)=1$ a
2 KB (318 words) - 09:27, 10 November 2023
...be a [[quadratic residue]] or non-residue modulo $p$. It was proved by L. Euler in 1761 (see [[#References|[1]]]).
Euler also obtained a more general result: A number $a \not\equiv 0 \pmod p$ is a
1 KB (217 words) - 07:30, 19 December 2014
The first Euler substitution: If $a>0$, then
The second Euler substitution: If the roots $x_1$ and $x_2$ of the quadratic polynomial $ax^
2 KB (366 words) - 06:13, 10 April 2023
$#C+1 = 23 : ~/encyclopedia/old_files/data/E036/E.0306550 Euler polynomials
are the [[Euler numbers]]. The Euler polynomials can be computed successively by means of the formula
3 KB (477 words) - 08:36, 6 January 2024
...act that the set of [[prime number]]s is infinite. The partial sums of the Euler series satisfy the asymptotic relation
659 bytes (104 words) - 15:21, 10 April 2023
$#C+1 = 37 : ~/encyclopedia/old_files/data/E036/E.0306400 Euler characteristic
It was given this name in honour of L. Euler, who proved in 1758 that the number $ V $
4 KB (560 words) - 08:59, 4 November 2023
See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
557 bytes (85 words) - 18:50, 18 October 2014
...that starts at the vertex. The centroid lies on the [[Euler straight line|Euler line]].
809 bytes (127 words) - 14:54, 2 May 2023
...itrary real number and the product extends over all prime numbers $p$. The Euler identity also holds for all complex numbers $s = \sigma + it$ with $\sigma
The Euler identity can be generalized in the form
2 KB (279 words) - 19:13, 14 December 2015
considered by L. Euler (1740). Its existence follows from the fact that the sequence
...mber $\gamma$ is also known as the ''Euler-Mascheroni'' constant, after L. Euler (1707–1783) and L. Mascheroni (1750–1800).
2 KB (328 words) - 11:50, 23 November 2023
A smooth solution of the [[Euler equation|Euler equation]], which is a necessary extremum condition in the problem of varia
Euler's equation has the form
7 KB (1,043 words) - 19:38, 5 June 2020
$#C+1 = 34 : ~/encyclopedia/old_files/data/E036/E.0306530 Euler method
by a polygonal line (Euler's polygonal line) whose segments are rectilinear on the intervals $ [ x _
6 KB (877 words) - 19:38, 5 June 2020
...ides the order of the group. Fermat's little theorem was generalized by L. Euler to the case modulo an arbitrary $m$. Namely, he proved that for every numbe
where $\phi(m)$ is the [[Euler function|Euler function]]. Another generalization of Fermat's little theorem is the equati
2 KB (257 words) - 18:02, 8 November 2014
..., 1740). There exists a constant $\gamma>0$, known as the [[Euler constant|Euler constant]], such that $S_n = \ln n + \gamma + \varepsilon_n$, where $\lim\l
1 KB (234 words) - 10:24, 10 December 2012
...$10^n-1$. Thus, the period length divides $\phi(q)$, the [[Euler function|Euler function]].
823 bytes (124 words) - 10:22, 27 September 2014
is summable by means of the Euler summation method ($(E,q)$-summable) to the sum $S$ if
...hen its terms $a_n$ satisfy the condition $a_n=o((2q+1)^n)$, $q\geq0$. The Euler summation method can also be applied for analytic continuation beyond the d
2 KB (358 words) - 17:36, 14 February 2020
''$B$-function, Euler $B$-function, Euler integral of the first kind''
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is equal to its [[Euler characteristic|Euler characteristic]]. Betti numbers were introduced by E. Betti [[#References|[
1 KB (172 words) - 13:05, 14 February 2020
The simplest method of computing a denumerant is by Euler's recurrence relation:
1 KB (185 words) - 16:46, 23 November 2023
...a is employed in the calculus of variations to derive the [[Euler equation|Euler equation]] in its integral form. In this proof it is not necessary to assum
1 KB (183 words) - 15:14, 27 August 2014
The spiral of Cornu is sometimes called the spiral of Euler after L. Euler, who mentioned it first (1744). Beginning with the works of A. Cornu (1874)
1 KB (205 words) - 10:58, 26 March 2023
where $c=0.5772\ldots$ is the [[Euler constant|Euler constant]] and $\operatorname{Ci}(x)$ is the [[Integral cosine|integral cos
1 KB (161 words) - 14:30, 14 February 2020
$#C+1 = 61 : ~/encyclopedia/old_files/data/E036/E.0306620 Euler transformation
The Euler transformation of series. Given a series
6 KB (972 words) - 12:59, 6 January 2024
...the existence of Euler cycles (Euler's theorem): A connected graph has an Euler cycle if and only if each of its vertices (except two) has even degree.
A graph is said to be Hamiltonian (Eulerian) if it has a Hamilton (Euler) cycle. A graph is said to be Hamilton-connected if any two of its vertices
4 KB (632 words) - 20:10, 15 March 2023
...ence is unsolvable, then $a$ is called a quadratic non-residue modulo $m$. Euler's criterion: Let $p>2$ be prime. Then an integer $a$ coprime with $p$ is a
1 KB (184 words) - 13:35, 14 September 2014
...n the $(x,y)$-plane for which the general integral of the [[Euler equation|Euler equation]] can be represented in the form
1 KB (227 words) - 19:28, 26 March 2023
Of course, in \eqref{*} $\phi$ denotes the [[Euler function|Euler function]].
1 KB (206 words) - 11:54, 2 January 2021
...|coprime]] $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's [[totient function]], which is $J_1$.
1 KB (172 words) - 13:07, 19 March 2023
of the Euler $ \phi $-function (cf. [[Euler function|Euler function]]).
4 KB (652 words) - 05:18, 7 March 2022
L. Euler [[#References|[1]]] was the first to study Bernoulli polynomials for arbitr
and are closely connected with the [[Euler polynomials|Euler polynomials]]
6 KB (828 words) - 10:58, 29 May 2020
A collection of solutions of the [[Euler equation|Euler equation]], depending on $n$ arbitrary constants and filling without mutual
depends. Euler's equation is understood in the vector sense, that is, it is a system of $n
2 KB (411 words) - 16:38, 24 November 2018
...of Venn diagrams goes back to L. Euler and they are sometimes also called Euler diagrams.
3 KB (416 words) - 12:22, 14 February 2020
...Irregularity|irregularity]] $q=5$, with topological [[Euler characteristic|Euler characteristic]] (in case $k=\mathbf C$) equal to 27. From the Fano surface
1 KB (212 words) - 15:54, 17 July 2014
...mal [[Euler identity]] beween the Dirichlet series $L(a,s)$ and a formal [[Euler product]] over primes
and is [[Totally multiplicative function|totally multiplicative]] if the Euler product is of the form
2 KB (358 words) - 17:25, 11 November 2023
...ence $x^3\equiv a$ ($\bmod\,p$) may be checked for solvability using the [[Euler criterion]]: The congruence $x^3\equiv a$ ($\bmod\,p$), $(a,p)=1$, is solva
...the form $p=x^2+27y^2$ with integers $x$ and $y$ (a result conjectured by Euler and first proved by Gauss).
1 KB (215 words) - 20:43, 5 December 2023
...[[Polyhedron, abstract|Polyhedron, abstract]]) with [[Euler characteristic|Euler characteristic]] equal to 2 can be realized as a convex polyhedron. Here, a
2 KB (277 words) - 15:40, 15 April 2014