Associated Legendre polynomialsCanonical solutions of the general Legendre equationIn mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre differential equation(1−x2)d2dx2Pℓm(x)−2xddxPℓm(x)+[ℓ(ℓ+1)−m21−x2]Pℓm(x)=0,{\displaystyle \left(1-x^{2} ight){\frac {d^{2}}{dx^{2}}}P_{\ell }^{m}(x)-2x{\frac {d}{dx}}P_{\ell }^{m}(x)+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}} ight]P_{\ell }^{m}(x)=0,}or equivalently ddx[(1−x2)ddxPℓm(x)]+[ℓ(ℓ+1)−m21−x2]Pℓm(x)=0,{\displaystyle {\frac {d}{dx}}\left[\left(1-x^{2} ight){\frac {d}{dx}}P_{\ell }^{m}(x) ight]+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}} ight]P_{\ell }^{m}(x)=0,}where the indices ℓ and m (which are integers) are referred to as the order and degree of the associated Legendre polynomial respectively.
Spherical harmonicsIn mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics contains a list of common spherical harmonics.
Legendre functionIn physical science and mathematics, the Legendre functionsPλ, Qλ and associated Legendre functionsPλ, Qλ, and Legendre functions of the second kind, Qn, are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications.
Laplace's equationIn mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786.
Generating functionFormal power seriesIn mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series.
Multipole expansionA multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, R3{\displaystyle \mathbb {R} ^{3}}. Multipole expansions are useful because, similar to Taylor series, often times only the first few terms are needed to provide a good approximation of the original function.
Rodrigues' formulaFormula for the Legendre polynomialsIn mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Olinde Rodrigues (1816), Sir James Ivory (1824) and Carl Gustav Jacobi (1827). The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out in 1865 that Rodrigues was the first to discover it.
Orthogonal polynomialsIn mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials.
Big q-Legendre polynomialsIn mathematics, the big q-Legendre polynomials are an orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as P n ( x ; c ; q ) = 3 ϕ 2 ( q − n , q n + 1 , x ; q , c q ; q , q ) {\displaystyle \displaystyle P_{n}(x;c;q)={}_{3}\phi _{2}(q^{-n},q^{n+1},x;q,cq;q,q)} .
Adrien-Marie LegendreAdrien-Marie Legendre (; French: [adʁiɛ̃ maʁi ləʒɑ̃dʁ]; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. He is also known for his contributions to the method of least squares, and was the first to officially publish on it, though Carl Friedrich Gauss had discovered it before him.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
Point particleA point particle, ideal particle or point-like particle (or pointlike particle) is an idealization used in physics. Its defining feature is negligible spatial extension or a body whose own rotation is irrelevant. A point particle is an appropriate representation of any object whenever its size, shape, and structure are irrelevant in a given context. For example, from far enough away, any finite-size object will look and behave as a point-like object.*As an Amazon Associate I earn from qualifying purchases.