How are the "real" spherical harmonics derived? How were the real spherical harmonics derived?
The complex spherical harmonics:
$$
Y_l^m( \theta, \phi ) = K_l^m P_l^m( \cos{ \theta } ) e^{im\phi}
$$
But the "real" spherical harmonics are given on this wiki page as
$$
Y_{lm} =
\begin{cases}
\frac{1}{\sqrt{2}} ( Y_l^m + (-1)^mY_l^{-m} ) & \text{if } m > 0 \\
Y_l^m        & \text{if } m = 0 \\
\frac{1}{i \sqrt{2}}( Y_l^{-m} - (-1)^mY_l^m)       & \text{if } m < 0
\end{cases}
$$


*

*Note: $Y_{lm} $ is the real spherical harmonic function and $Y_l^m$ is the complex-valued version (defined above)


What's going on here?  Why are the real spherical harmonics defined this way and not simply as $ \Re{( Y_l^m )} $ ?

As far as I remember, real Spherical Harmonics are real functions that still have an eigenvalue respect to L operator (angular moment). They are no longer eigenfunctions of L_z operator (M value, as they mix m and -m). On another hand, real Spherical Harmonics are also orthonormal.
Real part of Spherical Harmonics are not eigenfunctions of L operator, nor orthonormal.
