Recurrence relations for integrals of Associated Legendre functions

Recurrence relations for the evaluation of the integrals of associated Legendre functions over an arbitrary interval within (0°, 90°) have been derived which yield sufficiently accurate results throughout the entire range of their possible applications. These recurrence relations have been used to compute integrals up to degree 100 and similar computations can be carried out without any difficulty up to a degree as high as the memory in a computer permits. The computed values have been tested with independent check formulae, also derived in this work; the corresponding relative errors never exceed 10⁻²³ in magnitude. Contribution from the Earth Physics Branch No. 719