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Spherical harmonic

From Academic Kids

In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplace's equation represented in a system of spherical coordinates. The solutions are generally expressed in terms of trigonometric functions and Legendre polynomials. This form comes from separation of variables once the Laplacian is written in the spherical coordinate system. 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>\phi<math> the azimuth or longitude, ranging from <math>0\leq\varphi<2\pi<math>.

The spherical harmonic with parameters l, m can be written as:

<math> Y_{\ell,m}( \theta , \varphi ) = e^{i m \varphi } \cdot P_\ell^m ( \cos{\theta} ) <math>

where <math>P_\ell^m<math> are the associated Legendre polynomials.

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.

Y1 Missing image
Legendre_Y1_xy.png


Missing image
Legendre_Y1_polaire.png


Y2 Missing image
Legendre_Y2_xy.png


Missing image
Legendre_Y2_polaire.png


Y3 Missing image
Legendre_Y3_xy.png


Missing image
Legendre_Y3_polaire.png


In space

Missing image
Harmoniques_spheriques_positif_negatif.png
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 alternatively positive and negative between two circles.
Missing image
Traces_harmonique_spherique.png
the Y32 with four sections

See also

References

Template:Math-stubde:Kugelflächenfunktionen

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