Certain Numerical Properties of the Legendre Function and the Associated Legendre Function

摘要

The Legendre function and the associated Legendre function are orthogonal sets of functions, i.e., the integral of the square of a normalized function is unity and the integral of any pair of functions of different degree is zero. This will be only approximately true when ideal integration is replaced by a simple numerical summation. In doing this, certain curious numerical properties have been observed which are discussed by a few illustrative examples. The practical implication of these numerical aspects of the Legendre function, when performing harmonic analysis in spherical coordinates, is discussed. (Author)