What is associated Laguerre polynomial?
The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space.
What is Laguerre equation?
The associated Laguerre differential equation has a regular singular point at 0 and an irregular singularity at . It can be solved using a series expansion, (12) (13)
What is Laguerre and legendre function?
The Legendre, Laguerre, and Hermite equations are all homogeneous second order Sturm-Liouville equations. In solving these equations explicit solutions cannot be found. That is solutions in in terms of elementary functions cannot be found. In many cases it is easier to find a numerical or series solution.
Are associated Laguerre polynomials orthogonal?
Orthogonality. The generalized Laguerre polynomials are orthogonal over [0, ∞) with respect to the measure with weighting function xα e−x: ∫ 0 ∞ x α e − x L n ( α ) ( x ) L m ( α ) ( x ) d x = Γ ( n + α + 1 ) n !
How do you know if a polynomial is orthogonal?
(c) A polynomial p \= 0 is an orthogonal polynomial if and only if (p,q) = 0 for any polynomial q with deg q < deg p. p(x)q(x)dx. Note that (xn,xm) = 0 if m + n is odd.
Are Legendre polynomials orthonormal?
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications.
What is the formula for the Laguerre polynomials?
Laguerre Polynomials. It is, in fact, a Rodrigues’ formula, like those used to define other kinds of orthogonal polynomials. From this formula, L 0 = 1, L 1 = 1 – x. The polynomials satisfy the recurrence relation L n+1 = (2n + 1 – x)L n – n 2 L n-1, and the differential equation xL” n + (1 – x)L’ n +nL n = 0.
Which is the formula for the Rodrigues polynomials?
It is, in fact, a Rodrigues’ formula, like those used to define other kinds of orthogonal polynomials. From this formula, L 0 = 1, L 1 = 1 – x. The polynomials satisfy the recurrence relation L n+1 = (2n + 1 – x)L n – n 2 L n-1, and the differential equation xL” n + (1 – x)L’ n +nL n = 0.
What are the solutions of Laguerre’s equation named after?
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre’s equation: which is a second-order linear differential equation.
Is the rodriguesformula a formula for a negative value of L?
The Rodriguesformula requires that l always be a positive value; the formula does not make sense for negative values of l. Combining the solutions to the Azimuthal and Colatitudeequations, produces a solution to the non-radial portion of the Schrodingerequation for the hydrogen atom: