# Spherical harmonics

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, and the approximation of the Earth's gravitational field and the geoid.

## Introduction

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Top to bottom: l=0 to 4
Left to right: m=0 to ±4 (non-imaginary harmonics)
The two non-imaginary spherical harmonics that are linear combinations of Yl,m and Yl,-m are equivalent to each other but rotated 90 degrees relative to the z axis relative to each other.

Laplace's equation in spherical coordinates is:

$\displaystyle {1 \over r^2}{\partial \over \partial r}\left(r^2 {\partial f \over \partial r}\right) + {1 \over r^2\sin\theta}{\partial \over \partial \theta}\left(\sin\theta {\partial f \over \partial \theta}\right) + {1 \over r^2\sin^2\theta}{\partial^2 f \over \partial \varphi^2} = 0$

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

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

$\displaystyle r^{-1-\ell} \cos (m \varphi) P_\ell^m (\cos{\theta} )$

and

$\displaystyle r^{-1-\ell} \sin (m \varphi) P_\ell^m (\cos{\theta} )$

where $\displaystyle P_\ell^m$ are the associated Legendre polynomials, and with integer parameters $\displaystyle \ell \ge 0$ and m from 0 to $\displaystyle \ell$ .

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

$\displaystyle U_{\ell,m}(r,\theta , \varphi ) = r^{-1-\ell} Y_\ell^m( \theta , \varphi )$

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

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

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

$\displaystyle \int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi}Y_\ell^mY_{\ell'}^{m'*}\,d\Omega=\delta_{\ell\ell'}\delta_{mm'}\quad\quad d\Omega=\sin\theta\,d\varphi\,d\theta$
 Y1 Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Y2 Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Y3 Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination

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

$\displaystyle Y_\ell^0\quad\quad for\ 0\le\ell\le\infin$

and

$\displaystyle {1\over\sqrt2}\left((-1)^mY_\ell^m+Y_\ell^{-m}\right)\quad\quad \mbox{ for } \ 0\le\ell\le\infin,\ 1\le m\le \ell$

and

$\displaystyle {1\over i\sqrt2}\left((-1)^mY_\ell^m-Y_\ell^{-m}\right)\quad\quad \mbox{ for } \ 0\le\ell\le\infin,\ 1\le m\le \ell$

The spherical harmonics in cartesian coordinates may be obtained by substituting

$\displaystyle \cos\theta={z\over r},\qquad e^{\pm ni\varphi}\cdot\sin^n\theta={(x\pm iy)^n\over r^n},\qquad r=\sqrt{x^2+y^2+z^2}$ .

## First few spherical harmonics

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Representation as ρ = ρ0 + ρ1·Ylm(θ,φ)
then the representative surface looks like a "battered" sphere;
Ylm is equal to 0 along circles (the representative surface intersects the ρ = ρ0 sphere at these circles). Ylm is alternately positive and negative between two circles.
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the Y32 with four sections

These are the first few spherical harmonics:

$\displaystyle Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over \pi}$

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

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

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

## 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.