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In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplace's equation represented in a system of spherical coordinates. Spherical harmonics are important in many theoretical and practical applications, particularly in the computation of atomic electron configurations, the representation of the gravitational field, geoid, and magnetic field of planetary bodies, as well as characterization of the cosmic microwave background radiation.

    Spherical harmonics
        Introduction
        Normalizations
        Spherical harmonics expansion
        Spectrum Analysis
        Addition theorem
        First few spherical harmonics
        Generalizations
        See also
        Software

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Introduction

Laplace's equation in spherical coordinates is:

abla^2 f = left(r^2 ight) + left(sin heta ight) + = 0

(see also nabla in cylindrical and spherical coordinates). For

f(r,	heta,phi)=R(r)Theta(	heta)Phi(phi)

, the angular portion of Laplace's equation satisfies

$left(sin heta$

ight) + + l(l+1)Theta( heta)Phi(varphi) = 0.

Using the technique of separation of variables, the angular solutions can be shown to be a products of trigonometric functions and associated Legendre functions:

$Y_ell^m ( heta, varphi ) = N , e^ , P_ell^m (cos )$ ,

where

Y_ell^m

is a called a spherical harmonic function of degree

ell

and order

m

P_ell^m

is an associated Legendre function,

N

is a normalization constant, and

heta

and

varphi

represent colatitude and longitude, respectively. The spherical coordinates used in this article are consistent those used by physicists, but differ from those employed by mathematicians (see spherical coordinates). In particular, the colatitude

heta

, or polar angle, ranges from

0leq	hetaleqpi

and the longitude

varphi

, or azimuth, ranges from

0leqvarphi<2pi

.

When Laplace's equation is solved on the surface of the sphere, the periodic boundary conditions in

varphi

, as well as regularity conditions at both the north and south poles, ensure that the degree

ell

and order

m

are integers that satisfy

ell ge 0

and

|m| le ell

. In contrast, if the function

f

were only to have been defined for

heta le 	heta_0

, then the resulting spherical cap harmonics would have been defined for integer order, but non-integer degree. The general solution to Laplace's equation is a linear combination of the spherical harmonic functions multplied by the solutions of

R(r)

$f(r, heta, varphi) = sum_^infty sum_^ell r^ , f_ell^m , Y_ell^m ( heta, varphi ) +$

sum_^infty sum_^ell r^ell , f_ell^ , Y_ell^m ( heta, varphi ) ,

where

f_ell^m

and

f_ell^

are constants. The terms in the first summation approach zero as

r

goes to infinity, whereas the terms in the second summation approach zero at the origin.

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Normalizations
Several different normalizations are in common use for the spherical harmonic functions. In physics and seismology, these functions are generally defined as

$Y_ell^m( heta , varphi ) = sqrt , P_ell^m ( cos ) , e^$

which are orthonormalized

$int_^piint_^Y_ell^m , Y_^ , dOmega=delta_, delta_$ ,

where δ_aa = 1, δ_ab = 0 if a ≠ b, and

dOmega=sin	heta,dvarphi,d	heta

. The disciplines of geodesy and spectral analysis use

$Y_ell^m( heta , varphi ) = sqrt , P_ell^m ( cos ), e^$

which possess unit power

$int_^piint_^Y_ell^m , Y_^ dOmega=delta_, delta_$ .

The magnetics community, in contrast, uses Schmidt semi-normalized harmonics

$Y_ell^m( heta , varphi ) = sqrt , P_ell^m ( cos ) , e^$

which have the normalization

$int_^piint_^Y_ell^m , Y_^dOmega=delta_, delta_$ .

Using the identity (see associated Legendre functions)

$P_ell ^ = (-1)^m rac P_ell ^$

it can be shown that all of the above normalized spherical harmonic functions satisfy

$Y_ell^ ( heta, varphi) = (-1)^m Y_ell^ ( heta, varphi)$ ,

where the superscript

denotes complex conjugation.

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Spherical harmonics expansion
The spherical harmonics form a complete set of orthonormal functions and thus form a vector space analogue to unit basis vectors. On the unit sphere, any square-integrable function can thus be expanded as a linear combination of these:

$f( heta,varphi)=sum_^ sum_^ell f_ell^m , Y_ell^m( heta,varphi)$ .

This expansion is exact as long as

ell

goes to infinity. Truncation errors will arise when limiting the sum over

ell

to a finite bandwidth

L

. The expansion coefficients can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle

Omega!,

, and utilizing the above orthogonality relationships. For the case of orthonormalized harmonics, this gives

$f_ell^m=int_ f( heta,varphi), Y_ell^( heta,varphi)dOmega = int_0^dvarphiint_0^d hetasin heta f( heta,varphi)Y_ell^ ( heta,varphi)$ .

An alternative set of spherical harmonics for real functions may be obtained by taking the set

$Y_ = egin$

Y_ell^0 qquadqquadqquadqquadqquadqquadqquadqquadqquadqquadquadquadmbox m=0\ left(Y_ell^m+(-1)^m , Y_ell^ ight) = sqrt N P_ell^m( heta) cos mvarphi qquadquadquad mbox m>0 \ left(Y_ell^-(-1)^, Y_ell^ ight) = sqrt N P_ell^( heta) sin |m|varphi quadmbox m<0. end

These functions have the same normalization properties as the complex ones above. In this notation, a real square-integrable function can be expressed as an infinite sum of real spherical harmonics as

$f( heta, varphi) = sum_^infty sum_^ell f_ , Y_( heta, varphi)$ .

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Spectrum Analysis
The total power of a function

f

is defined in the signal processing literature as the integral of the function squared, divided by the area it spans. Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem:

$rac int_Omega f(Omega)^2, dOmega = sum_^infty S_(l)$ ,

where

$S_(l) = sum_^l f_^2$

is defined as the angular power spectrum. In a similiar manner, one can define the cross-power of two functions as

$rac int_Omega f(Omega) , g(Omega) , dOmega = sum_^infty S_(l)$ ,

where

$S_(l) = sum_^l f_ g_$

is defined as the cross-power spectrum. If the functions

f

and

g

have a zero mean (i.e., the spectral coefficients

f_

and

g_

are zero), then

S_(l)

and

S_(l)

represent the contributions to the function's variance and covariance for degree

ell

, respectively. It is common that the (cross-)power spectrum is well approximated by a power law of the form

$S_(l) = C , ell^$ .

When

eta=0

, the spectrum is "white" as each degree possesses equal power. When

eta<0

, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Finally, when

eta>0

, the spectrum is termed "blue".

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Addition theorem
A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. Two vectors r and r', with spherical coordinates

(r,	heta,varphi)

and

(r ',	heta ',varphi ')

,respectively, have an angle

gamma

between them given by

$cosgamma=cos hetacos heta'+sin hetasin heta'cos(varphi-varphi')$ .

The addition theorem expresses a Legendre polynomial of order

l

in the angle

gamma

in terms of products of two spherical harmonics with angular coordinates

(	heta,varphi)

and

(	heta',varphi')

P_l( cos gamma ) = racsum_^l Y_^

( heta',phi') , Y_( heta,varphi) .

This expression is valid for both real and complex harmonics. However, it should be emphasized that the quoted form above is valid only for the orthonormalized spherical harmonics. For unit power harmonics it is only necessary to remove the factor of

4 pi

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First few spherical harmonics

Analytic expressions for the first few orthonormalized spherical harmonics:

$Y_^( heta,varphi)=sqrt$

$Y_^(x)=sqrt , sin heta , e^ quad=sqrtcdot$

$Y_^(x)=sqrt, cos hetaquad=sqrtcdot$

$Y_^(x)=sqrt, sin heta, e^quad=sqrtcdot$

$Y_^( heta,varphi)=sqrt , sin^ heta , e^$

$Y_^( heta,varphi)=sqrt, sin heta, cos heta, e^$

$Y_^( heta,varphi)=sqrt, (3cos^ heta-1)$

$Y_^( heta,varphi)=sqrt, sin heta,cos heta, e^$

$Y_^( heta,varphi)=sqrt, sin^ heta , e^$

$Y_^( heta,varphi)=sqrt, (5cos^ heta-3cos heta)$

More spherical harmonics up to Y₁₀

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Generalizations
The spherical harmonics map can be seen as representations of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). As such they capture the symmetry of the two-dimensional sphere (or two-sphere). Each set of spherical harmonics with a given value for the l-parameter map onto a different irreducible representation of SO(3).

In addition, the two-sphere is equivalent to the Riemann sphere. The complete set of symmetries of the Riemann sphere are described by the Mobius transformation group SL(2,C), of which the Lorentz group is a representation. The analog of the spherical harmonics for the Lorentz group are given by the hypergeometric series; indeed, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as SO(3) is a subgroup of SL(2,C).

More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in particular, hypergeometric series can be developed for any Lie group.

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