LegendreQ2General

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[Q, ν]^μ)(z) is an analytical function of ν, μ and z which is defined over (ℂ^3).  For integer ν and noninteger μ, (Subscript[Q, ν]^μ)(z) degenerates to a sum of two polynomial in z multiplied on functions ((((1+z))^(μ/2))/(((1-z))^(μ/2)))and ((((1-z))^(μ/2))/(((1+z))^(μ/2)))correspondently. For integers ν,(μ/2)/;μ>-ν, (Subscript[Q, ν]^μ)(z) becomes the meromorthic function.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

With respect to μ

For fixed ν, z, the function (Subscript[Q, ν]^μ)(z) has an infinite set of singular points:
a) μ==-ν-k/;k∈(SuperPlus[ℕ]), are the simple poles with residues -((π csc(π ν))/(2 (k-1)! Γ(k+2 ν+1)))(Subscript[P, ν]^(k+ν))(z);
b) μ==(OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

With respect to ν

For fixed μ, z, the function (Subscript[Q, ν]^μ)(z) has an infinite set of singular points:
a) ν==-μ-k/;k∈(SuperPlus[ℕ]), are the simple poles with residues (((((-1))^k) π csc(π μ))/(2 (k-1)! Γ(1-k-2 μ)))(Subscript[P, -k-μ]^(-μ))(z);
b) ν==(OverTilde[∞]) is the point of accumulation of poles, which is an essential singular point.

Branch points

With respect to z

For fixed ν, μ (except ν∈ℤ∧(μ/2)∈(SuperPlus[ℕ])∧μ>-ν), the function (Subscript[Q, ν]^μ)(z) has three singular branch points: z==-1, z==1and z==(OverTilde[∞]). 
For integer ν and integer  (μ/2)/;μ>1∧μ>-ν, the function (Subscript[Q, ν]^μ)(z) does not have branch points.

With respect to μ

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

With respect to ν

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

Branch cuts

With respect to z

For fixed ν, μ (except ν∈ℤ∧(μ/2)∈(SuperPlus[ℕ])∧μ>-ν), the function  (Subscript[Q, ν]^μ)(z) is a single-valued function on the z-plane cut along the intervals (-∞,-1) and (1,∞). The function (Subscript[Q, ν]^μ)(z) is continuous from above on the interval (-∞,-1] and from below on the interval [1, ∞).

For integer ν and integer  (μ/2)/;μ>1∧μ>-ν, the function (Subscript[Q, ν]^μ)(z) is a meromorthic function and does not have branch cuts.

With respect to μ

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

With respect to ν

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

Series representations

Generalized power series

Expansions at z==0

Expansions at z==1

Expansions at z==-1

Expansions at z==∞

Integral representations

On the real axis

Of the direct function

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Pavlyk O. (2006)

Pavlyk O. (2006)

Relations of special kind

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

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

Theorems

The solution of Laplace equation

The solution of the Laplace equation using the method of separation of variables in toroidal coordinates

x=((c sinh(η) cos(ϑ))/(cosh(η)-cos(φ)))  y=((c sinh(η) sin(ϑ))/(cosh(η)-cos(φ)))  z=((c sin(φ))/(cosh(η)-cos(φ)))

is a linear combination of functions of the form

(Sqrt[2(cosh(η)-cos(φ))])((Subscript[c, 1])cos(φ)+(Subscript[c, 2])sin(φ))×((Subscript[c, 3])cos(ϑ)+(Subscript[c, 4])sin(ϑ))((Subscript[c, 5])(Subscript[P, n-(1/2)]^(|m|))(cosh(η))+(Subscript[c, 6])(Subscript[Q, n-(1/2)]^(|m|))(cosh(η)))/; m∈ℤ,n∈ℕ.

History

– E. Heine (1842)
– F. Neumann (1842)
– L. Schläfli (1881) used complex μ, ν
– R. Olbricht (1888)

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