SphericalHarmonicYGeneral

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed λ,μ,ϑ

For fixed λ,μ,φ

For fixed λ,ϑ,φ

For fixed μ,ϑ,φ

For fixed λ,φ

For fixed ϑ,φ

General characteristics

Domain and analyticity

(Subscript[Y, λ]^μ)(ϑ,φ) is an analytical function of λ,μ,ϑ,φ which is defined over (ℂ^4).

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

(Subscript[Y, λ]^μ)(ϑ,φ) is a periodic function with respect to ϑ and φ with periods 2 π and 2π/μ respectively.

Phase shifts

Poles and essential singularities

With respect to φ

For fixed λ,μ,ϑ, the function (Subscript[Y, λ]^μ)(ϑ,φ) has only one singular point at φ = (OverTilde[∞]).  It is an essential singular point.

With respect to ϑ

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

With respect to μ

For fixed λ,ϑ,φ, the function (Subscript[Y, λ]^μ)(ϑ,φ) has only one singular point at μ= (OverTilde[∞]).  It is an essential singular point. .

With respect to λ

For fixed μ,ϑ,φ, the function (Subscript[Y, λ]^μ)(ϑ,φ) has only one singular point at λ= (OverTilde[∞]).  It is an essential singular point. .

Branch points

With respect to φ

For fixed λ,μ,ϑ, the function (Subscript[Y, λ]^μ)(ϑ,φ) does not have branch points.

With respect to ϑ

For fixed generic λ,φ,μ/;(μ/2)∉ℤ, the function (Subscript[Y, λ]^μ)(ϑ,φ) has the set of  branch points where: cos(ϑ)==-1, cos(ϑ)==1and cos(ϑ)==(OverTilde[∞]). For fixed φ, noninteger λ and integer (μ/2), the function (Subscript[Y, λ]^μ)(ϑ,φ) has the set of  branch points where: cos(ϑ)==-1and cos(ϑ)==(OverTilde[∞]). For fixed φ, integers λ and integers (μ/2), the function (Subscript[Y, λ]^μ)(ϑ,φ) does not have branch points.

With respect to μ

With respect to λ

Branch cuts

With respect to φ

For fixed λ,μ,ϑ, the function (Subscript[Y, λ]^μ)(ϑ,φ) does not have branch cuts.

With respect to ϑ

For fixed generic λ,φ,μ/;(μ/2)∉ℤ, the function (Subscript[Y, λ]^μ)(ϑ,φ) is a single-valued function on the ϑ-plane cut along the intervals -∞<cos(ϑ)<-1 and 1<cos(ϑ)<∞.

For fixed φ, noninteger λ and integer (μ/2), the function (Subscript[Y, λ]^μ)(ϑ,φ) is a single-valued function on the ϑ-plane cut along the interval -∞<cos(ϑ)<-1.

For fixed φ, integers λ and integers (μ/2), the function (Subscript[Y, λ]^μ)(ϑ,φ) is a polynomial and does not have branch cuts.

Series representations

Generalized power series

Expansions at sin((ϑ/2))==0

Expansions at cos((ϑ/2))==0

Expansions at tan((ϑ/2))==0

Expansions at cot((ϑ/2))==0

Expansions at sin(ϑ)==0

Expansions at cos(ϑ)==0

Expansions at tan(ϑ)==0

Expansions at cot(ϑ)==0

Expansions at ϑ==0

Expansions at ϑ==π

Expansions at cos(ϑ)==∞

Integral representations

On the real axis

Of the direct function

Involving the direct function

Multiple integral representations

Integral representations of negative integer order

Generating functions

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Partial differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Products, sums, and powers of the direct function

Products involving the direct function

Identities

Recurrence identities

Consecutive neighbors

Functional identities

Relations between contiguous functions

Additional relations between contiguous functions

Below relations are correct only under some restrictions on the parameters

Relations of special kind

Complex characteristics

Conjugate value

Differentiation

Low-order differentiation

With respect to λ

With respect to μ

With respect to ϑ

With respect to φ

Symbolic differentiation

With respect to φ

Fractional integro-differentiation

With respect to φ

Integration

Indefinite integration

Involving only one direct function with respect to φ

Involving one direct function and elementary functions with respect to φ

Involving functions of the direct function and elementary functions with respect to ϑ

Definite integration

Involving the direct function

Multiple integration

Orthonormality relations:

Summation

Finite summation

Involving the direct function

Involving Clebsch-Gordan functions

The inverse Clebsch-Gordan series:

Infinite summation

Multiple infinite summation

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1Regularized

Involving Hypergeometric2F1

Through Meijer G

Classical cases

Through other functions

Involving Legendre functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Involving Wigner-D functions

Zeros

When (Subscript[Y, n]^m)(ϑ,φ) is not identically zero, it possesses a finite number of zeros in the interval  0<ϑ<π, all of which are nondegenerate.

For integers m and n with n≥|m|, the function (Subscript[Y, n]^m)(ϑ,φ) has n-|m| zeros in the interval  0<ϑ<π. If m≠0, there are also two more zeros at  ϑ=0,π. All of these zeros are symmetric about  ϑ=π/2.

Theorems

Eigenfunction to the angular part of the Laplace operator in spherical coordinates

The function (Subscript[Y, n]^m)(ϑ,φ) is an eigenfunction to the angular part (L^2)of the Laplace operator in spherical coordinates -(L^2)==(1/((sin(ϑ))^2))((∂^2)/(∂(φ^2)))+(1/(sin(ϑ)))(∂/(∂ϑ))(sin(ϑ)(∂/(∂ϑ))) with eigenvalue n (n+1).

Eigenfunction of the z component of the quantum mechanical angular momentum operator

The function (Subscript[Y, n]^m)(ϑ,φ) is an eigenfunction to the z-component of the quantum mechanical angular momentum operator (Subscript[(OverHat[L]), z])=-i ∂/∂φ with eigenvalue m.

Eigenfunctions of the curl operator in spherical coordinates

The function u(r,ϑ,φ)==λ r ×∇ψ(r,ϑ,φ)+∇×(r×∇ψ(r,ϑ,φ)) with ψ(r,ϑ,φ)==g(r)(Subscript[Y, n]^m)(ϑ,φ) and g(r)==(((Sqrt[2])i)/(λ(Sqrt[n (n+1)])))((Subscript[c, 1])(((Subscript[J, n+1/2])(λ r))/(Sqrt[λ r]))+(((Subscript[Y, n+1/2])(λ r))/(Sqrt[λ r])))  are eigenfunctions of the curl operator in spherical coordinates ∇×u(r,ϑ,φ)==λ u(r,ϑ,φ).

Multiple expansion theorem

Any function f(ϑ,φ) that is square integrable over  0≤ϑ≤π,0≤φ≤2π can be expanded in a series of spherical harmonics (Subscript[Y, n]^m)(ϑ,φ), with series coefficients  (Subscript[a, n,m]) called the multipole moments:

f(ϑ,φ)==(∑_(n=0)^∞)(∑_(m=-n)^n)(Subscript[a, n,m]) (Subscript[Y, n]^m)(ϑ,φ)/;(Subscript[a, n,m])==(∫_0^(2 π))(∫_0^π)sin(ϑ) (OverBar[(Subscript[Y, n]^m)(ϑ,φ)]) f(ϑ,φ)dϑdφ.

History

A.M. Legendre (1785); P.S. Laplace (1785) gave the name “spherical harmonic”;  K.F. Gauss (1828).

References

L.C. Biedenharn and J.D. louck, Angular Momentum in Quantum Physics, Addison-Wesley, Reading, 1981.

L.C. Biedenharn and J.D. louck, The Racah-Wigner Algebra in Quantum Theory, Addison-Wesley, Reading, 1981.

E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, Cambridge, 1955.

M.E. Rose, Elementary Theory of Angular Momentum, Dover, New York, 1995.

D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore, 1988.

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