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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...

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...

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...

function Riesz function Hypergeometric functions: Versatile family of power series. Confluent hypergeometric function Associated Legendre functions Meijer G-function...

latitude and may be expressed as an infinite sum of associated Legendre polynomials; the functions are orthogonal over the sphere in the continuous case. Thus...

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...

1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles...

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

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

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)}...

Welsh Whipple, arise from a general expression, concerning associated Legendre functions. These formulae have been presented previously in terms of a...

P_{n}^{m}\left(\cos \theta \right)} are the Schmidt quasi-normalized associated Legendre functions of degree n {\displaystyle n} and order m {\displaystyle m}...

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

harmonic function of degree ℓ and order m, P ℓ m : [ − 1 , 1 ] → R {\displaystyle P_{\ell }^{m}:[-1,1]\to \mathbb {R} } is an associated Legendre polynomial...

rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind). Besides the Legendre form given...

supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. Legendre functions have widespread applications...

z) for reference: also P0n are the Legendre polynomials and Pmn for 1 ≤ m ≤ n are the associated Legendre functions. The first spherical harmonics with...

e^{-i\mu \phi }} Where P and Q are associated Legendre functions of the first and second kind. These Legendre functions are often referred to as toroidal...