Search results
Jul 9, 2024 · In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a real variable.
1 day ago · Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial P m ℓ (cos θ). Finally, the equation for R has solutions of the form R(r) = A r ℓ + B r −ℓ − 1; requiring the solution to be regular throughout R 3 forces B = 0.
Jul 13, 2024 · It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation ; it is indeed derived using the product rule.
2 days ago · The eigenfunction corresponding to eigenvalues λ n = n ( n +1), n = 0, 1, 2, …, are convensionally called associated Legendre functions. They usually are denoted by Pmn(x), m = 0, 1, 2, …, n; or by Pn ( m, x ). The function Pn ( m, x) is a polynomial of degree n if and only if m is even.
Jun 30, 2024 · Abstract. In this chapter we introduce classical field theory, where the covariant formulation requires the Lagrangian formalism. We shall discuss the Bogomolny bound, basic topology in field theory, as well as scalar kinks, and lumps in the non–linear sigma model.
Jul 6, 2024 · The ordinary generating function for Legendre polynomials is. \begin {equation} \label {EqLege1.1} G (x,t) = \frac {1} {\sqrt {1 -2xt + t^2}} = \sum_ {n\ge 0} P_n (x)\, t^n , \end {equation} where Pn ( x is the Legendre polynomial of degree n. Legendre's polynomials are orthogonal.
2 days ago · Legendre's equation comes up in many physical situations involving spherical symmetry. Legendre Polynomials. Legendre's polynomial can be defined explicitly: Pn(x) = 1 2n ⌊ n / 2 ⌋ ∑ k = 0 ( − 1)k(n k)(2n − 2k n)xn − 2k, where ⌊ · ⌋ is the floor function, and (n k) =n!k!(n is the binomial coefficient.
