This formula was established by L. Euler (1760).
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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
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-
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...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]]].
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...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
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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^
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''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
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''Euler–Lagrange operator''
...for variational problems must satisfy (cf. also [[Euler–Lagrange equation|Euler–Lagrange equation]]).
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See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
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...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
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...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
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$#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
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...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).
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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).
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$#C+1 = 61 : ~/encyclopedia/old_files/data/E036/E.0306620 Euler transformation
The Euler transformation of series. Given a series
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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''
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$#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 $
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...ars the name of Euler–Knopp summation method, see [[Euler summation method|Euler summation method]].
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...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
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...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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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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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''
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#REDIRECT [[Euler constant]]
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#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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#REDIRECT [[Euler straight line]]
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often for $q=1$, used in the [[Euler summation method|Euler summation method]].
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...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]]].
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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
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''Euler totient function, Euler totient''
Another frequently used named for the [[Euler function]] $\phi(n)$, which counts a [[reduced system of residues]] modulo
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''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
$#C+1 = 23 : ~/encyclopedia/old_files/data/E036/E.0306550 Euler polynomials
are the [[Euler numbers]]. The Euler polynomials can be computed successively by means of the formula
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...act that the set of [[prime number]]s is infinite. The partial sums of the Euler series satisfy the asymptotic relation
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$#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 $
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