zn {\displaystyle \operatorname {zn} } . The Legendre's relation or Legendre Identity shows the relation of the integrals K and E of an elliptic modulus...

mathematics, Legendre's relation can be expressed in either of two forms: as a relation between complete elliptic integrals, or as a relation between periods...

]}_{y=0}^{y=1}=\arctan(1)={\frac {\pi }{4}}} For the Lemniscatic special case of Legendre's relation, this result emerges: K ( 1 2 2 ) [ 2 E ( 1 2 2 ) − K ( 1 2 2 ) ]...

the polynomials were first defined by Legendre in 1782. A third definition is in terms of solutions to Legendre's differential equation: This differential...

functional relation for elliptic integrals Legendre's conjecture Legendre sieve Legendre symbol Legendre's theorem on spherical triangles Saccheri–Legendre theorem...

kind, Qn, are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of...

had been discovered earlier and was known to Legendre, these two definitions are equivalent. Thus Legendre's contribution lay in introducing a convenient...

function q ( x ) {\displaystyle q(x)} is strictly left-curved. The Legendre's relation is defined that way: K E ′ + E K ′ − K K ′ = 1 2 π {\displaystyle...

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation ( 1 − x 2 ) d 2 d x 2 P ℓ m ( x ) − 2...

condition Legendre–Fenchel transformation Legendre's conjecture Legendre's constant Legendre's differential equation Legendre's equation Legendre's formula...

In numerical analysis, Gauss–Legendre quadrature is a form of Gaussian quadrature for approximating the definite integral of a function. For integrating...

depending on the application. 1 is the value of Legendre's constant, introduced in 1808 by Adrien-Marie Legendre to express the asymptotic behavior of the prime-counting...

In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface...

}}\;\Gamma (2z).} It is also called the Legendre duplication formula or Legendre relation, in honor of Adrien-Marie Legendre. The multiplication theorem is Γ...

function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel...

of the factorial function to the gamma function. Adrien-Marie Legendre included Legendre's formula, describing the exponents in the factorization of factorials...

logarithm of infinity!); Legendre's argument is heuristic; and Chebyshev's proof, although perfectly sound, makes use of the Legendre-Gauss conjecture, which...

"Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I". Constructive Approximation. 54 (1): 49–116. arXiv:1909.09107. doi:10...

\wp } -function The relation to elliptic integrals has mainly a historical background. Elliptic integrals had been studied by Legendre, whose work was taken...

meaning it is not formally defined, but rather informally said to be "a relation each thing bears to itself and nothing else". This characterization is...

polynomials of degree 2n − 1 or less. This exact rule is known as the Gauss–Legendre quadrature rule. The quadrature rule will only be an accurate approximation...

involution, on a set with n = 0, 1, 2, ... elements is given by a recurrence relation found by Heinrich August Rothe in 1800: a 0 = a 1 = 1 {\displaystyle a_{0}=a_{1}=1}...

integer, so it remains the same when transposed to the left, and the inverse relation follows upon making the substitution m3 → −m3: ⟨ j 1 m 1 j 2 m 2 | j 3...

gradient. Since 1997, the term has also been used to refer to Dufrêne & Legendre's indicator value, which is a quantitative index measuring the statistical...

instead use z!). Legendre's normalization does simplify some formulae, but complicates others. From a modern point of view, the Legendre normalization of...

{\mathbf {2} }} . The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis The expansion coefficients...

integer N and arbitrary s. See also Faulhaber's formula for a similar relation on finite sums of powers of integers. The Laurent series expansion can...

Gauss–Legendre quadrature. The Gauss–Legendre method based on s points has order 2s. All Gauss–Legendre methods are A-stable. The Gauss–Legendre method...

than the Heisenberg uncertainty relation. For a particle of spin- j , {\displaystyle j,} the following uncertainty relation holds ( Δ J x ) 2 + ( Δ J y )...

{1-t^{4}}}}=2.62205\ldots } The lemniscate functions satisfy the basic relation cl ⁡ z = sl ( 1 2 ϖ − z ) , {\displaystyle \operatorname {cl} z={\operatorname...