science and mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions Pμ λ, Qμ λ, and Legendre functions of the second kind, Qn, are...

related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials...

variables, then the Legendre transform with respect to this variable is applicable to the function. In physical problems, the Legendre transform is used...

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 chi function is a special function whose Taylor series is also a Dirichlet series, given by χ ν ( z ) = ∑ k = 0 ∞ z 2 k +...

Bessel–Clifford function Kelvin functions Legendre function: From the theory of spherical harmonics. Scorer's function Sinc function Hermite polynomials...

the Legendre rational functions are a sequence of orthogonal functions on [0, ∞). They are obtained by composing the Cayley transform with Legendre polynomials...

In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p:...

Adrien-Marie Legendre (/ləˈʒɑːndər, -ˈʒɑːnd/; French: [adʁiɛ̃ maʁi ləʒɑ̃dʁ]; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous...

between the vectors x and x1. The functions P i : [ − 1 , 1 ] → R {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } are the Legendre polynomials, and they can be...

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

three-variable Laplace equation, is given in terms of the generating function for Legendre polynomials, 1 | x − x ′ | = ∑ l = 0 ∞ r < l r > l + 1 P l ( cos...

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

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

as kernels of the transform. Legendre transform is a special case of Jacobi transform. The Legendre transform of a function f ( x ) {\displaystyle f(x)}...

theorem of algebra. The name gamma function and the symbol Γ were introduced by Adrien-Marie Legendre around 1811; Legendre also rewrote Euler's integral definition...

in families of rational orthogonal functions called Legendre rational functions and Chebyshev rational functions. Solutions of linear differential equations...

(z_{1}),\operatorname {Re} (z_{2})>0} . The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its...

functions. These include most of the commonly used functions of mathematical physics. Legendre functions are solutions of a second order differential equation...

} Here Yℓm is called a spherical harmonic function of degree ℓ and order m, Pℓm is an associated Legendre polynomial, N is a normalization constant,...

integrable functions on the unit sphere S2 is a Hilbert space. The basis functions in this case are the spherical harmonics. The Legendre polynomials...

the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl Friedrich Gauss introduced the square bracket...

In the theory of special functions, Whipple's transformation for Legendre functions, named after Francis John Welsh Whipple, arise from a general expression...

Gauss–Legendre algorithm Gauss–Legendre method Gauss–Legendre quadrature Legendre (crater) Legendre chi function Legendre duplication formula Legendre–Papoulis...

)}}\right)} Legendre function Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "Ferrers Function", NIST Handbook...

The Gauss–Legendre algorithm is an algorithm to compute the digits of π. It is notable for being rapidly convergent, with only 25 iterations producing...

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

rational functions and the three Legendre canonical forms, also known as the elliptic integrals of the first, second and third kind. Besides the Legendre form...

arcsine and the lemniscate arccosine can also be expressed by the Legendre-Form: These functions can be displayed directly by using the incomplete elliptic integral...

on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Legendre functions (Handbuch der...