404.
ON LEGENDRE'S FUNCTION Pn(0), WHEN n IS GREAT AND 0 HAS ANY VALUE*.
[Proceedings of the Royal Society, A, Vol. xcn. pp. 433—437, 1916.]
As is well known, an approximate formula for Legendre's function Pn(0), when n is very large, was given by Laplace. The subject has been treated with great generality by Hobsonf, who has developed the complete series proceeding by descending powers of n, 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 when 116 is at least moderately large. On the other hand, when 6 is small, Pn tends to identify itself with the Bessel's function J"0(w0), as 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 = 20J is exhibited.
The differential equation satisfied by Legendre's function Pn is
~j/fa + °Ot 0 -JTJ +
If we assume u = v (sin 6) ~ , and write m for n + |, we have
v
•(1)
+ mzv
4 sin2 6'
-(2)
* [1917. It would be more correct to say Pn (oos 0), where cos0 lies between i 1.]
+ " On a Type of Spherical Harmonics of Unrestricted Degree, Order, and Argument," Phil,
Trans. A, Vol. OLXXXVII. (1896).
J " On the Acoustic Shadow of a Sphere," Phil. Trans. A, Vol. com. (1904); Scientific Papers,
Vol. v. p. 163. by On, so that only Jp appears, and