On the computation of derivatives of Legendre functions

Abstract

Analysis and evaluation of spherical harmonics are important for Earth sciences and potential theory. Depending on the functional of the harmonic series, Legendre functions, their derivatives or their integrals must be computed numerically which in general is based on recurrence relations. Numerical stability and optimization of such recurrence relations become more and more important with increasing degree and order. In this paper, a simple relation is recovered to obtain first and higher order derivatives of Legendre functions. The relation is shown to be numerical stable, it does not cause a singularity at the poles, and can be applied recursively to obtain second and higher order derivatives. Moreover, it can be applied to compute integrals over derivatives of Legendre functions, quantities required if, for example, mean values of deflections of the vertical are to be analyzed or evaluated. A sample FORTRAN code is given and a few additional formulas used to verify the code and to investigate round-off errors for degree and order up to 360.