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Okay I know that Real Spherical Harmonics are given by

  • If $m \lt 0$ $~$ then $\sqrt{2}$ $~$ $Im(\text{SphericalHarmonicY}[l,|m|])$

  • If $m=0$ $~$ then $~$ $\text{SphericalHarmonicY}[l,0]$

  • If $m \gt 0$ $~$ then $\sqrt{2}$ $~$ $Re(\text{SphericalHarmonicY}[l,|m|])$

I also know that the triple product integral of any $3$ Complex Spherical Harmonics is two Wigner-$3j$ symbols multiplied together with a normalizing constant

$\large \frac{\sqrt{2\times l_1+1}\times\sqrt{2 \times l_2+1}\times\sqrt{2 \times l_3+1}}{\sqrt{4 \pi}}$

So what is the triple product integral of any $3$ Real Spherical Harmonics?

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Since $Y_l^{m*} = (-1)^m Y_l^{-m}$, you can rewrite the real spherical harmonics as linear combination of $Y_l^m$ and $Y_l^{-m}$. –  achille hui Feb 10 '13 at 16:06
@Devsh: Welcome to MSE! Please make sure I updated your question properly using MathJax. Regards –  Amzoti Feb 10 '13 at 16:38

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