On Legendre's functions P n ( θ ), when n is great and θ has any value

Abstract

As is well known, an approximate formula for Legendre's function P n (θ), when n , is very large, was given Laplace. The subject has been treated with great generality by Hobson, who has developed the complete series proceeding by descending powers of n , not only for P n but also for the "associated functions." The generality aimed at by Hobson requires the Use of advanced mathematical methods. I have thought that a simpler derivation, sufficient for practical purposes and more within the reach of physicists with a smaller mathematical equipment, may be useful. It had, indeed, been worked out independently. The series, of which Laplace's expression constitutes the first term, is arithmetically useful only when nθ is at least moderately large. On the other hand, when θ is small, p n tends to identity itself with the Bessel's function J 0 ( nθ ), is was first remarked by Mehler. A further development of this approximation is here proposed. Finally, a comparison of the results of the two methods of approximation with the numbers calculated by A. Lodge for n = 20 is exhibited.