The orthogonality of the spherical harmonics states that the integral of the product of two distinct spherical harmonics over a specified interval is zero.
It is defined mathematically as:
Substituting eq400 in eq412 gives
In general, the integral over is
Since the associated Legendre polynomials are orthogonal to one another, we have
Conditions | |
zero | |
zero | |
zero | |
non-zero |
Therefore,
which is the expression showing the orthonormal property of the spherical harmonics.