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Hace 1 día · Learn about Legendre polynomials in just a minute with IFAS Physics. Discover how these mathematical functions play a crucial role in various physics concept...
Hace 3 días · Legendre polynomials are eigenfunctions corresponding to eigenvalues λ = n ( n +1) of the singular Sturm--Liouville problem, (1 − x2)y ″ − 2xy. ′. + λy = 0, x ∈ ( − 1, 1), y( ± 1) < ∞, where n ∈ ℕ = {0, 1, 2, …} is a nonnegative integer. This equation can be written in a self-adjoint form.
Hace 5 días · This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre. The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics.
Hace 20 horas · This study is devoted to the numerical investigation of linear and nonlinear hyperbolic telegraph equation. We have proposed a wavelet collocation method based on Legendre polynomials for approximating the solution. Both the spatial and temporal variables, along with their derivatives, are approximated using the Legendre wavelet and its integration. The present approach is simple, consistent ...
Hace 20 horas · Quadratic Reciprocity (Legendre's statement). If p or q are congruent to 1 modulo 4, then: is solvable if and only if is solvable. If p and q are congruent to 3 modulo 4, then: is solvable if and only if is not solvable. The last is immediately equivalent to the modern form stated in the introduction above.
Hace 4 días · In this paper we investigate determinants whose entries are linear combinations of Legendre symbols. After a review of known results, we present some new results and pose many conjectures for further research. For example, we conjecture that det [ x + ( j − k p) + ( j p) − ( k p)] 0 ≤ j, k ≤ ( p − 1) / 2 = 4 for any prime p ≡ 3 ...
Hace 5 días · This section is devoted to expansions of real-valued functions into series over Laguerre polynomials. and Sonin polynomials. f(x) = ∑ k ≥ 0f ( α) k L ( α) k (x), f ( α) k = k! Γ(k + α + 1)∫∞0L ( α) k (x)e − xxαf(x)dx, k = 0, 1, 2, …. It is known that these series converge in 𝔏² sense for functions f ∈ 𝔏² (ℝ ...
