LegendreP2General

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed ν,μ

For fixed ν,z

For fixed μ,z

For fixed z

General characteristics

Domain and analyticity

(Subscript[P, ν]^μ)(z) is an analytical function of ν, μ and z which is defined over (ℂ^3). For integer ν, (Subscript[P, ν]^μ)(z) degenerates to a polynomial in z multiplied on function ((((1+z))^(μ/2))/(((1-z))^(μ/2))).

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For integer ν and integer  (μ/2), the function (Subscript[P, ν]^μ)(z) is polynomial and has pole of order ν at z =(OverTilde[∞]) (for ν∈(SuperPlus[ℕ])) or order -ν-1 at z =(OverTilde[∞]) (for -ν∈(SuperPlus[ℕ])).

With respect to μ

For fixed ν, z, the function (Subscript[P, ν]^μ)(z) has only one singular point at μ= (OverTilde[∞]).  It is an essential singular point. .

With respect to ν

For fixed μ, z, the function (Subscript[P, ν]^μ)(z) has only one singular point at ν= (OverTilde[∞]).  It is an essential singular point. .

Branch points

With respect to z

For fixed generic ν,μ/;(μ/2)∉ℤ, the function (Subscript[P, ν]^μ)(z) has three branch points: z==-1, z==1and z==(OverTilde[∞]). For fixed noninteger ν and integer (μ/2), the function (Subscript[P, ν]^μ)(z) has two branch points: z==-1and z==(OverTilde[∞]). For fixed integers ν and integers (μ/2), the function (Subscript[P, ν]^μ)(z)does not have branch points.

With respect to μ

For fixed ν, z, the function (Subscript[P, ν]^μ)(z) does not have branch points.

With respect to ν

For fixed μ, z, the function (Subscript[P, ν]^μ)(z) does not have branch points.

Branch cuts

With respect to z

For fixed generic ν,μ/;(μ/2)∉ℤ, the function (Subscript[P, ν]^μ)(z) is a single-valued function on the z-plane cut along the intervals (-∞,-1) and (1,∞). The function (Subscript[P, ν]^μ)(z) is continuous from above on the interval (-∞,-1] and from below on the interval [1, ∞).

For fixed noninteger ν and integer (μ/2), the function (Subscript[P, ν]^μ)(z) is a single-valued function on the z-plane cut along the interval (-∞,-1), where it is continuous from above.

For fixed integers ν and (μ/2), the function (Subscript[P, ν]^μ)(z) is a polynomial and does not have branch cuts.

With respect to μ

For fixed ν, z, the function (Subscript[P, ν]^μ)(z) does not have branch cuts.

With respect to ν

For fixed μ, z, the function (Subscript[P, ν]^μ)(z) does not have branch cuts.

Series representations

Generalized power series

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

Expansions on branch cuts

Expansions at z==0

Expansions at z==1

Expansions at z==-1

Expansions at z==∞

Asymptotic series expansions

Expansions at ν==-(1/2)+i ∞

Integral representations

On the real axis

Of the direct function

Integral representations of negative integer order

Rodrigues-type formula.

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Pavlyk O. (2006)

Pavlyk O. (2006)

Additional relations between contiguous functions

Differentiation

Low-order differentiation

With respect to ν

With respect to μ

With respect to z

Symbolic differentiation

With respect to ν

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Definite integration

Involving the direct function

Summation

Finite summation

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1Regularized

Involving Hypergeometric2F1

Through Meijer G

Classical cases for the direct function itself

Classical cases involving algebraic functions

Classical cases involving unit step UnitStep

Classical cases involving  Sign

Generalized cases involving algebraic functions

Generalized cases involving unit step UnitStep

Generalized cases involving  Sign

Through other functions

Involving some hypergeometric-type functions

Involving spheroidal functions

Representations through equivalent functions

With related functions

Involving Gegenbauer functions

Involving Legendre functions

Involving spherical harmonic functions

Theorems

One infinite sum

(∑_(n=0)^∞)(2n+1) (i^n)(∑_(m=0)^n)(2-(Subscript[δ, m]))(((n-m)!)/((n+m)!))cos(φ-α) (Subscript[P, n]^m)(cos(β))(Subscript[P, n]^m)(cos(ϑ))(Subscript[J, n+(1/2)])(|k||r|)==(e^(i k.r))/;r==|r|{cos(φ)sin(ϑ),sin(φ)sin(ϑ),cos(ϑ)}∧k==|k|{cos(α)sin(β),sin(α)sin(β),cos(β)}

Eigenfunctions of the Schrödinger equation

Legendre functions (Subscript[P, n]^m) are eigenfunctions of the Schrödinger equation with reflectionless, shape invariant, supersymmetric potential:-(((∂^2)ψ(x))/(∂(x^2)))-(n(n+1)(sech^2)(x))ψ(x)==-(m^2)ψ(x).

The solution of Dirichlet problem for Laplace equation in spherical coordinates

The solution of the Dirichlet problem ψ(1,ϕ,ϑ)==(Subscript[ψ, 0])(ϕ,ϑ) for the Δψ(ϕ,ϑ)==0 on unit sphere in spherical coordinates is given by:

ψ(r,ϕ,ϑ)==(∑_(n=0)^∞)(∑_(m=0)^n)((2n+1)/(2π(1+(Subscript[δ, m]))))(((n-m)!)/((n+m)!))(r^n)(Subscript[P, n]^m)(cos(ϑ) ) (∫_0^π)((∫_(-π)^π)(Subscript[ψ, 0])((Overscript[ϕ, ∼]),(Overscript[ϑ, ∼]))cos(m(ϕ-(Overscript[ϕ, ∼])))d(Overscript[ϕ, ∼]))(Subscript[P, n]^m)(cos((Overscript[ϑ, ∼])))sin((Overscript[ϑ, ∼]))d(Overscript[ϑ, ∼])

History

– D. Bernoulli (1748)
– A. M. Legendre (1782)
– E. Heine (1842)
– F. Neumann (1848)
– L. Schläfli (1881) used complex μ, ν
– E. Hobson (1896)

Copyright