Statements (62)
| Predicate | Object |
|---|---|
| gptkbp:instance_of |
gptkb:Marxism
|
| gptkbp:application |
gptkb:Quantum_Mechanics
numerical analysis electrostatics |
| gptkbp:defined_on |
interval [-1, 1]
|
| gptkbp:first_few_polynomials |
P_0(x) = 1
P_1(x) = x P_2(x) = (3x^2 -1)/2 P_3(x) = (5x^3 -3x)/2 P_4(x) = (35x^4 -30x^2 + 3)/8 |
| gptkbp:has_produced |
orthogonal functions
Legendre series |
| gptkbp:has_property |
P_n(-1) = (-1)^n
P_n(0) = (-1)^{n/2} if n is even P_n(0) = 0 if n is odd P_n(1) = 1 P_n(x) is a polynomial of degree n |
| https://www.w3.org/2000/01/rdf-schema#label |
Legendre polynomials
|
| gptkbp:named_after |
Adrien-Marie Legendre
|
| gptkbp:offers_degree |
n
|
| gptkbp:orthogonal_with_respect_to |
weight function 1 on [-1, 1]
|
| gptkbp:orthogonality_condition |
∫_{-1}^{1} P_n(x) P_m(x) dx = 0 for n ≠ m
|
| gptkbp:recurrence_relation |
P_{n+1}(x) = (2n + 1)x P_n(x) -n P_{n-1}(x) / (n + 1)
|
| gptkbp:related_to |
gptkb:Chebyshev_polynomials
Jacobi polynomials |
| gptkbp:roots |
real and distinct
located in the interval (-1, 1) |
| gptkbp:satisfy |
Legendre's differential equation
|
| gptkbp:used_in |
gptkb:Graphics_Processing_Unit
finite element analysis signal processing approximation theory solving differential equations geophysics solving partial differential equations solving boundary value problems solving engineering problems solving optimization problems spectral methods solving integral equations solving systems of equations expansion of functions solving Laplace's equation solving astrophysics problems solving chaos theory problems solving control theory problems solving cosmology problems solving electromagnetism problems solving fluid dynamics problems solving heat equation solving initial value problems solving mathematical physics problems solving nonlinear dynamics problems solving problems in physics solving quantum field theory problems solving relativity problems solving statistical mechanics problems solving string theory problems solving thermodynamics problems solving wave equation |
| gptkbp:bfsParent |
gptkb:Bessel_functions_of_the_first_kind
|
| gptkbp:bfsLayer |
5
|