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 \[Y_l^m (\theta,\phi) = (-1)^m \sqrt{\frac{2n+1}{4\pi}\frac{(n-m)!}{(n+m)!}} P^m_n(\cos\theta) e^{i m \phi},\] where \( P^m_n \) is the associated Legendre function.

The first few spherical harmonics are given by:

\[Y_0^0 (\theta,\phi) = \frac{1}{2} \frac{1}{\sqrt{\pi}}\]

\[Y_1^{-1} (\theta,\phi) = \frac{1}{2} \sqrt{\frac{3}{2\pi}} \sin \theta e^{-i\phi} \]

\[Y_1^0 (\theta,\phi) = \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos \theta \]

\[Y_1^1 (\theta,\phi) = -\frac{1}{2} \sqrt{\frac{3}{2\pi}} \sin \theta e^{i\phi} \]

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