Spherical harmonics

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In mathematics, the spherical harmonics are 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 the computation of atomic electron configurations, the approximation of the Earth's gravitational field and the geoid, and the parameter fitting for the anisotropy map of the cosmic microwave background.

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Introduction

Image:Rotating spherical harmonics.gif

Laplace's equation in spherical coordinates is:

<math>{1 \over r^2}{\partial \over \partial r}\left(r^2 {\partial f \over \partial r}\right) 
 + {1 \over r^2\sin\theta}{\partial \over \partial \theta}\left(\sin\theta {\partial f \over \partial \theta}\right) 
 + {1 \over r^2\sin^2\theta}{\partial^2 f \over \partial \varphi^2} = 0</math>

(see nabla in cylindrical and spherical coordinates).

Separation of variables leads to solutions expressed in terms of trigonometric functions and Legendre polynomials. Note that the spherical coordinates <math>\theta\!\,</math> and <math>\varphi\!\,</math> in this article are used in the physicist's way, as opposed to the mathematician's definition of spherical coordinates. That is, <math>\theta\!\,</math> is the colatitude or polar angle, ranging from <math>0\leq\theta\leq\pi</math> and <math>\varphi\!\,</math> the azimuth or longitude, ranging from <math>0\leq\varphi<2\pi</math>.

The general solution which remains finite towards infinity is a linear combination of functions of the form

<math> r^{-1-\ell} \cos (m \varphi) P_\ell^m (\cos{\theta} ) </math>

and

<math> r^{-1-\ell} \sin (m \varphi) P_\ell^m (\cos{\theta} ) </math>

where <math>P_\ell^m</math> are the associated Legendre polynomials, and with integer parameters <math>\ell \ge 0</math> and m from 0 to <math>\ell</math>.

Put in another way, the solutions with integer parameters <math>\ell \ge 0</math> and m from <math>- \ell</math> to <math>\ell</math>, can be written as linear combinations of:

<math> U_{\ell,m}(r,\theta , \varphi ) = r^{-1-\ell} Y_\ell^m( \theta , \varphi )</math>

where the functions Y are the spherical harmonics with parameters l, m, which can be written as:

<math> Y_\ell^m( \theta , \varphi ) = \sqrt{{(2\ell+1)\over 4\pi}{(\ell-m)!\over (\ell+m)!}} \cdot e^{i m \varphi } \cdot P_\ell^m ( \cos{\theta} ) </math>

The spherical harmonics obey the normalisation condition (δaa = 1 and δab = 0 if a ≠ b)

<math>\int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi}Y_\ell^mY_{\ell'}^{m'*}\,d\Omega=\delta_{\ell\ell'}\delta_{mm'}\quad\quad d\Omega=\sin\theta\,d\varphi\,d\theta</math>
Y1 Image:Legendre Y1 xy.png Image:Legendre Y1 polaire.png
Y2 Image:Legendre Y2 xy.png Image:Legendre Y2 polaire.png
Y3 Image:Legendre Y3 xy.png Image:Legendre Y3 polaire.png

An alternative set of spherical harmonics with no imaginary component may be obtained by taking the set

<math>Y_\ell^0\quad\quad for\ 0\le\ell\le\infin</math>

and

<math>{1\over\sqrt2}\left((-1)^mY_\ell^m+Y_\ell^{-m}\right)\quad\quad

\mbox{ for } \ 0\le\ell\le\infin,\ 1\le m\le \ell</math>

and

<math>{1\over i\sqrt2}\left((-1)^mY_\ell^m-Y_\ell^{-m}\right)\quad\quad

\mbox{ for } \ 0\le\ell\le\infin,\ 1\le m\le \ell</math>

The spherical harmonics in cartesian coordinates may be obtained by substituting

<math>\cos\theta={z\over r},\qquad e^{\pm ni\varphi}\cdot\sin^n\theta={(x\pm iy)^n\over r^n},\qquad r=\sqrt{x^2+y^2+z^2}</math>.
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First few spherical harmonics

Image:Harmoniques spheriques positif negatif.png

Image:Traces harmonique spherique.png

These are the first few spherical harmonics:

<math>Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over \pi}</math>


<math>Y_{1}^{-1}(x)={1\over 2}\sqrt{3\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\quad={1\over 2}\sqrt{3\over 2\pi}\cdot{(x-iy)\over r}</math>
<math>Y_{1}^{0}(x)={1\over 2}\sqrt{3\over \pi}\cdot\cos\theta\quad={1\over 2}\sqrt{3\over \pi}\cdot{z\over r}</math>
<math>Y_{1}^{1}(x)={-1\over 2}\sqrt{3\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\quad={-1\over 2}\sqrt{3\over 2\pi}\cdot{(x+iy)\over r}</math>


<math>Y_{2}^{-2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta</math>
<math>Y_{2}^{-1}(\theta,\varphi)={1\over 2}\sqrt{15\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot\cos\theta</math>
<math>Y_{2}^{0}(\theta,\varphi)={1\over 4}\sqrt{5\over \pi}\cdot(3\cos^{2}\theta-1)</math>
<math>Y_{2}^{1}(\theta,\varphi)={-1\over 2}\sqrt{15\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot\cos\theta</math>
<math>Y_{2}^{2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta</math>


<math>Y_{3}^{0}(\theta,\varphi)={1\over 4}\sqrt{7\over \pi}\cdot(5\cos^{3}\theta-3\cos\theta)</math>
More spherical harmonics up to Y10
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Generalizations

The spherical harmonics in a certain sense capture the symmetry properties of the two-sphere. The symmetry properties of the two-sphere are given by the Lie groups SO(3) and its double-cover SU(2). The spherical harmonic transform under the integer-spin representations of these groups; they are a part of the representation theory of these groups. However, the two-sphere can also be understood to be the Riemann sphere. The complete set of symmetries of the Riemann sphere can be understood to be described by the Mobius transformation group SL(2,C), of which the Lorentz group is but a representation. The analog of the spherical harmonics for the Lorentz group are given by the hypergeometric series; indeed, the spherical harmonics are easily 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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See also

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References

fr:Harmonique sphérique it:Armoniche sferiche ja:球面調和関数

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