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I was going through the answer to this problem, when I noticed that a factor of $\pi$ in the denominator disappeared and a factor of 4 appeared in the numerator when the author started evaluating the integral in equation (6). Is this something to do with integrating in spherical coordinates or is it some funky behaviour of spherical harmonics?

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  • $\begingroup$ "something to do with integrating in spherical coordinates": work it out in your question, if you insist on asking . $\endgroup$ – Cosmas Zachos Oct 14 '19 at 0:27
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$d\tau$ is the element of volume $dxdydz$, which in spherical coordinates is $r^2dr\sin{\theta}d\theta d\phi$. When integrating a function that doesn’t depend on the two angles, the integration over those angles gives $4\pi$.

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