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Spherical harmonics

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The spherical harmonics Y_{l}^{m}(\theta ,\phi ) are the angular portion of the solution to Laplace's equation in spherical coordinates. They are given by

Y_{l}^{m}(\theta ,\phi )=(-1)^{m}{\sqrt  {{\frac  {2n+1}{4\pi }}{\frac  {(n-m)!}{(n+m)!}}}}P_{n}^{m}(\cos \theta )e^{{im\phi }},

where P_{n}^{m} is the associated Legendre function.

The first few spherical harmonics are given by:

Y_{0}^{0}(\theta ,\phi )={\frac  {1}{2}}{\frac  {1}{{\sqrt  {\pi }}}}
Y_{1}^{{-1}}(\theta ,\phi )={\frac  {1}{2}}{\sqrt  {{\frac  {3}{2\pi }}}}\sin \theta e^{{-i\phi }}
Y_{1}^{0}(\theta ,\phi )={\frac  {1}{2}}{\sqrt  {{\frac  {3}{\pi }}}}\cos \theta
Y_{1}^{1}(\theta ,\phi )=-{\frac  {1}{2}}{\sqrt  {{\frac  {3}{2\pi }}}}\sin \theta e^{{i\phi }}

See also[edit]