In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because[1] the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity (the ellipse being defined parametrically by , ).
In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric forms. A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals.
^Gratton-Guinness, Ivor (1997). The Fontana History of the Mathematical Sciences. Fontana Press. p. 308. ISBN 0-00-686179-2.
mathematics, the Legendreforms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the...
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number...
They are a modern alternative to the Legendreforms. The Legendreforms may be expressed in terms of the Carlson forms and vice versa. The Carlson elliptic...
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...
In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface...
integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals...
many applications. Alternate notations include: Carlson symmetric formLegendreform Nome Quarter period Elliptic functions: The inverses of elliptic integrals;...
calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form x 2 ≡ a mod p...
supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. Legendre functions have widespread applications...
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...
Newtonian potential for a point mass. Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of...
and foreshadowed the eventual development of infrastructure. In 1798, Legendre published Essai sur la théorie des nombres, which summarized the work of...
Sophus Lie, with the dual aims of studying differential equations (e.g. the Legendre transformation or canonical transformation) and describing the 'change...
In mathematics, Legendre moments are a type of image moment and are achieved by using the Legendre polynomial. Legendre moments are used in areas of image...
}}\;\Gamma (2z).} It is also called the Legendre duplication formula or Legendre relation, in honor of Adrien-Marie Legendre. The multiplication theorem is Γ...
process, then one obtains the Legendre polynomials. Another collection of orthogonal polynomials are the associated Legendre polynomials. The study of orthogonal...
In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior...
lemniscate arcsine and the lemniscate arccosine can also be expressed by the Legendre-Form: These functions can be displayed directly by using the incomplete elliptic...
of the prime field of K, a finite field of order p2. Suppose E is in Legendreform, defined by the equation y 2 = x ( x − 1 ) ( x − λ ) {\displaystyle...
^{2}(1-\lambda )^{2}}}\ .} which is the j-invariant of the elliptic curve of Legendreform y 2 = x ( x − 1 ) ( x − λ ) {\displaystyle y^{2}=x(x-1)(x-\lambda )}...
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