Open AccessOn Legendre's functions P n ( θ ), when n is great and θ has any value
Publication year - 1916
Publication title -
proceedings of the royal society of london. series a, containing papers of a mathematical and physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9150
pISSN - 0950-1207
DOI - 10.1098/rspa.1916.0028
Subject(s) - laplace transform , bessel function , generality , legendre function , mathematics , legendre polynomials , series (stratigraphy) , expression (computer science) , function (biology) , term (time) , value (mathematics) , identity (music) , special functions , pure mathematics , calculus (dental) , combinatorics , mathematical analysis , computer science , statistics , physics , psychology , paleontology , dentistry , quantum mechanics , evolutionary biology , acoustics , psychotherapist , biology , programming language , medicine
As is well known, an approximate formula for Legendre's function Pn (θ), whenn , 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 ofn , not only for Pn 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 whennθ is at least moderately large. On the other hand, whenθ is small,pn tends to identity itself with the Bessel's function J0 (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 forn = 20 is exhibited.