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accuracy appropriate approximate associated Legendre functions asymptotic expansion asymptotic expression becomes Cambridge Carrus and Charlotte Charlotte G clearly coefficients complete Computation condition connection Consider constructed converges corresponding cot2 definite difficulties discrete equal equation 13 evaluated expand extend find it convenient foregoing FRACTIONAL ORDERS FUNCTIONS OF FRACTIONAL given Harmonics higher orders hypergeometric series implied INCOMPLETE INTEGRALS increases infinite interpolation investigations January Laboratory larger left-hand side Massachusetts Math Mathematical means method Modern Analysis neglecting noted obtain Pal's asymptotic formula Pal1 paper performed Physics Pierre Pn(x power series preparation present problems proved rapidly reason recursion formula Remembering respective root right-hand side roots of equation roots vk satisfy side of equation slow spectrum Spherical substitution sufficient suggestion TABLE terms of higher terms of order Treuenfels Watson written XXIX yn(l