Editing Spherical harmonics (section)

==Visualization of the spherical harmonics==
[[Image:Spherical harmonics positive negative.svg|thumb|right|Schematic representation of <math>Y_{\ell m}</math> on the unit sphere and its nodal lines. <math>\Re [Y_{\ell m}]</math> is equal to 0 along {{mvar|m}} [[great circle]]s passing through the poles, and along ''ℓ''−''m'' circles of equal latitude. The function changes sign each time it crosses one of these lines.]]
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[[Image:Traces harmonique spherique.png|300px|thumb|right|The real spherical harmonic function <math>Y_{32}</math> shown in four cross sections.]]-->
[[Image:Spherical harmonics.png|300px|thumb|right|3D color plot of the spherical harmonics of degree {{math|1=''n'' = 5}}. Note that {{math|1=''n'' = ''ℓ''}}.]]

The Laplace spherical harmonics <math>Y_\ell^m</math> can be visualized by considering their "[[nodal line]]s", that is, the set of points on the sphere where <math>\Re [Y_\ell^m] = 0</math>, or alternatively where <math>\Im [Y_\ell^m] = 0</math>. Nodal lines of <math>Y_\ell^m</math> are composed of ''ℓ'' circles: there are {{math|{{abs|''m''}}}} circles along longitudes and ''ℓ''−|''m''| circles along latitudes. One can determine the number of nodal lines of each type by counting the number of zeros of <math>Y_\ell^m</math> in the <math>\theta</math> and <math>\varphi</math> directions respectively. Considering <math>Y_\ell^m</math> as a function of <math>\theta</math>, the real and imaginary components of the associated Legendre polynomials each possess ''ℓ''−|''m''| zeros, each giving rise to a nodal 'line of latitude'. On the other hand, considering <math>Y_\ell^m</math> as a function of <math>\varphi</math>, the trigonometric sin and cos functions possess 2|''m''| zeros, each of which gives rise to a nodal 'line of longitude'.

{{anchor|Zonal|Tesseral|Sectoral}}When the spherical harmonic order ''m'' is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as '''''[[zonal spherical harmonics|zonal]]'''''. Such spherical harmonics are a special case of [[zonal spherical function]]s. When {{math|1=''ℓ'' = {{abs|''m''}}}} (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as '''''sectoral'''''. For the other cases, the functions [[English draughts|checker]] the sphere, and they are referred to as '''''tesseral'''''.

More general spherical harmonics of degree {{mvar|ℓ}} are not necessarily those of the Laplace basis <math>Y_\ell^m</math>, and their nodal sets can be of a fairly general kind.<ref>{{harvnb|Eremenko|Jakobson|Nadirashvili|2007}}</ref>