LaguerreL3

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n,λ

For fixed n,z

For fixed λ,z

Values at infinities

General characteristics

Domain and analyticity

The function (Subscript[L, n]^λ)(z)is defined over ℕ⊗ℂ⊗ℂ. For fixed n,λ, the function (Subscript[L, n]^λ)(z) is a polynomial in z of degree n. For fixed n,z, the function (Subscript[L, n]^λ)(z) is a polynomial in λ of degree n.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For fixed λ the function (Subscript[L, n]^λ)(z) is polynomial and has pole of order n at z =(OverTilde[∞]).

With respect to λ

For fixed n, z, the function (Subscript[L, n]^λ)(z) has an infinite set of singular points:
a) λ==-n-k/;k∈(SuperPlus[ℕ]), are the simple poles with residues ((((-1))^(k-1))/(Γ(n+1) (k-1)!))(Subscript[, 1])(Subscript[(OverTilde[F]), 1])(-n;-k-n+1;z)/;k∈(SuperPlus[ℕ]);
b) λ== (OverTilde[∞]) is an essential singular point.

Branch points

With respect to z

For fixed n,λ, the function (Subscript[L, n]^λ)(z) does not have branch points.

With respect to λ

For fixed n,z, the function (Subscript[L, n]^λ)(z) does not have branch points.

Branch cuts

With respect to z

For fixed  n,λ, the function (Subscript[L, n]^λ)(z) does not have branch cuts.

With respect to λ

For fixed  n,z, the function (Subscript[L, n]^λ)(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at generic point λ==(Subscript[λ, 0])

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Expansions at z==∞

Expansions at λ==0

Expansions at λ==∞

Expansions at n==∞

Integral representations

On the real axis

Of the direct function

Integral representations of negative integer order

Rodrigues-type formula.

Limit representations

Generating functions

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Integral equations whose solutions contain the direct function

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Multiple arguments

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Additional relations between contiguous functions

Relations of special kind

Complex characteristics

Real part

Imaginary part

Argument

Conjugate value

Differentiation

Low-order differentiation

With respect to λ

With respect to z

Symbolic differentiation

With respect to λ

With respect to z

Fractional integro-differentiation

With respect to λ

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

Involving the direct function

Summation

Finite summation

Infinite summation

Operations

Limit operation

Orthogonality, completeness, and Fourier expansions

The set of functions (Subscript[L, n]^λ)(x), n=0,1,…, forms a complete, orthogonal (with weight ((n! )/(Γ(n+λ+1)))(x^λ)(e^(-x))) system on the interval (0,∞).

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric1F1Regularized

Involving Hypergeometric1F1

Through Meijer G

Classical cases involving Exp

Through other functions

Involving some hypergeometric-type functions

Theorems

Expansions in generalized Fourier series

f(x)==(∑_(k=0)^∞)(Subscript[c, k])(Subscript[ψ, k])(x)/;(Subscript[c, k])==(∫_0^∞)f(t)(Subscript[ψ, k])(t)dt,   (Subscript[ψ, k])(x)==((Sqrt[k!])/(Sqrt[Γ(λ+k+1)]))(x^(λ/2))(e^(-x/2))(Subscript[L, k]^λ)(x),  k∈ℕ.

Quantum mechanical eigenfunctions of the hydrogen atom

Quantum mechanical eigenfunctions (Subscript[ψ, n l m])(r,θ,ϕ) of the hydrogen atom are:

(Subscript[ψ, n l m])(r,θ,ϕ)==(((2τ))^3)(((n-l-1)!)/(2((n(n+l)!)^3)))(((e^(-τ r))(2 τ r))^l)(Subscript[L, n-l-1]^(2l+1))(2τ r) (Subscript[Y, l]^m)(θ,ϕ)/;τ>0, n∈(SuperPlus[ℕ]),l∈ℕ,l≤n-1,m∈ℤ,|m|≤l

The intensity of the Laguerre-Gauss modes

The intensity of the Laguerre-Gauss modes (Subscript[ℒ, n]^m)(x,y)=(((e^(-((x^2)+(y^2))))(x+i y))^m)(Subscript[L, n]^m)(2((x^2)+(y^2))) are shape invariant under a Fresnel transformation:

|(k/(2π i))(∫_(-∞)^∞)(∫_(-∞)^∞)(e^(i k((((x-ξ))^2)+(((y-η))^2))))(Subscript[ℒ, n]^m)(x,y)dxdy((|)^2)==(1/(|α((|)^2)))|(Subscript[ℒ, n]^m)((ξ/(|α|)),(η/(|α|)))((|)^2)/;   α==1+((2i)/k)

The eigenfunctions of the Schrödinger equation of the one-dimensional hydrogen atom

The eigenfunctions of the Schrödinger equation of the one-dimensional hydrogen atom (the simplest example for a system with dynamical superselection rules) -(Subscript[ψ, n]^″)(x)-(1/x)(Subscript[ψ, n]^″)(x)==-(1/(2(n^2)))(Subscript[ψ, n])(x) are given by

(Subscript[ψ, n])(x)==(Subscript[α, 1])θ(x)(Sqrt[(4/((n^5)((n!)^2)))])(((-1))^(n-1))x (Subscript[L, n-1]^1)(((2 x)/n)) (e^(-(x/n)))+(Subscript[α, 2])θ(-x)(Sqrt[(4/((n^5)((n!)^2)))])(((-1))^(n-1))x (Subscript[L, n-1]^1)(-((2 x)/n)) (e^(x/n)).

History

– E. N. Laguerre (1879)
– N. J. Sonin (1880)

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