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 )} $ ?