Editing Spherical harmonics (section)

==Special cases and values==
# When <math>m = 0</math>, the spherical harmonics <math>Y_{\ell}^m: S^2 \to \Complex</math> reduce to the ordinary [[Legendre polynomials]]: <math display="block">Y_{\ell}^0(\theta, \varphi) = \sqrt{\frac{2\ell+1}{4\pi}} P_{\ell}(\cos\theta).</math>
# When <math>m = \pm\ell</math>, <math display="block">Y_{\ell}^{\pm\ell}(\theta,\varphi)
= \frac{(\mp 1)^{\ell}}{2^{\ell}\ell!} \sqrt{\frac{(2\ell+1)!}{4\pi}} \sin^{\ell}\theta\, e^{\pm i\ell\varphi},</math> or more simply in Cartesian coordinates, <math display="block">r^{\ell} Y_{\ell}^{\pm\ell}({\mathbf r})
= \frac{(\mp 1)^{\ell}}{2^{\ell}\ell!} \sqrt{\frac{(2\ell+1)!}{4\pi}} (x \pm i y)^{\ell}.</math>
# At the north pole, where <math> \theta = 0</math>, and <math>\varphi</math> is undefined, all spherical harmonics except those with <math>m = 0</math> vanish: <math display="block"> Y_{\ell}^m(0,\varphi) = Y_{\ell}^m({\mathbf z}) = \sqrt{\frac{2\ell+1}{4\pi}} \delta_{m0}.</math>