egendre.html">plegendre</a>. In this expression, the associated Legendre polynomial itself does not include the Condon-Shortley phase.
</p>

<p>
The real-valued spherical harmonics are computed as follows. For M&gt;0
</p>

<div class="eqn">
<img src="eqn/spherharm5.png" alt="[eqn]"><!--MATH
\begin{equation*}
Y_{L,M}(\theta,\phi) =
\sqrt{2}
\sqrt{\frac{(2L+1)}{4\pi}\frac{(L-M)!}{(L+M)!}}
P_L^M(\cos\theta)
\cos(M\phi)
\end{equation*}
-->
</div>

   
<p>
(without the first factor for M=0), and for M&lt;0
</p>
