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Finding large order spherical harmonics

  1. Jul 30, 2013 #1
    is there an approximation for spherical harmonics for very large l and m in closed form?
  2. jcsd
  3. Aug 1, 2013 #2


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    Homework Helper

    sure see The Theory of Spherical and Ellipsoidal Harmonics by E. W. Hobson
    and learn such things as
    $$l^{-m}\mathrm{P}_l^m(\cos(\theta)=\sqrt{\frac{2}{l \pi \sin(\theta)}}\cos \left( \left( l+\frac{1}{2} \right)\theta-\frac{\pi}{4}+m\frac{\pi}{2} \right)+{O}(l^{-3/2}) \\
    l^{-m}\mathrm{Q}_l^m(\cos(\theta)=\sqrt{\frac{2}{l \pi \sin(\theta)}}\cos \left( \left( l+\frac{1}{2} \right)\theta+\frac{\pi}{4}+m\frac{\pi}{2} \right)+{O}(l^{-3/2}) \\
    \theta \in (\epsilon,\pi-\epsilon) \\
    of course there are endless variations if you need more accuracy or l or theta complex and so on.
  4. Aug 1, 2013 #3
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