Contents
What are Laguerre polynomials?
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 !
Are Hermite polynomials orthogonal?
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
How do you solve equations with Legendre?
The associated Legendre functions are given by two integer indices Pnm (x). For positive m these are related to the Legendre polynomials by the formula, (6.29) P n m ( x ) = ( − 1 ) m ( 1 − x 2 ) m / 2 d m d x m p n ( x ) .
How do you solve a Bessel equation?
The general solution of Bessel’s equation of order n is a linear combination of J and Y, y(x)=AJn(x)+BYn(x).
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.
Why do we use Hermite polynomials?
Ordinary Special Functions ( n − 2 k ) ! Hermite polynomials are relevant for the analysis of the quantum harmonic oscillator, and the lowering and raising operators there correspond to creation and annihilation.
Can zero be a polynomial?
Like any constant value, the value 0 can be considered as a (constant) polynomial, called the zero polynomial. It has no nonzero terms, and so, strictly speaking, it has no degree either.
What is an Indicial equation?
An indicial equation, also called a characteristic equation, is a recurrence equation obtained during application of the Frobenius method of solving a second-order ordinary differential equation.