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The spherical harmonics, $Y^m_n(\theta, \varphi)$, are only defined when $m$ $\in$ $[n,n-1,...,-n+1,-n]$. However, the derivative relation with respect to $\theta$ requires $Y^{m+1}_n(\theta, \varphi)$ (see http://functions.wolfram.com/Polynomials/SphericalHarmonicY/20/ShowAll.html).

Does this mean that $\frac{\partial Y^m_n(\theta, \varphi)}{\partial \theta}$ is not defined for $n = m$?

I may have some misunderstanding here.

D. Price
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    $\frac{\partial Y^n_n(\theta, \varphi)}{\partial \theta}=n\cot\theta ;Y^n_n(\theta,\varphi)$; the term with $Y_n^{m+1}$ has a prefactor $n-m$, so it does not contribute when $m=n$. – Carlo Beenakker Jun 05 '19 at 15:33
  • @CarloBeenakker Thanks! I really appreciate this –  Jun 05 '19 at 15:42

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