This posting is a simple addendum to the much longer article, Notes on Calculating the Spherical Harmonics. Because it had already grown so long, I left out making any plots of individual eigenmodes, so here I simply present plots of the first 50 degrees of the pure spherical harmonic eigenmodes, which can be useful to help visualize some of the symmetry properties that were discussed in the longer article.
In this article I review the critical properties of the Spherical Harmonics. In particular, I concentrate on filling in a couple of details regarding numerical computation of the spherical harmonic transformations that I have found to be unstated or less clear than ideal in the literature.
In the previous part we computed Legendre polynomials via a few recurrence relations, but the polynomial values grow rapidly with degree and quickly overflow the range of standard finite floating point numbers. In this posting we explore baking in a normalization factor into the recurrence relation — such as used when the Legendre polynomials are used to calculate the spherical harmonics — that eliminates the overflow.