A treatise on the theory of Bessel functions, by G. N. Watson.

MISCELLANEOUS THEOREMS

157

Another formula, slightly different from those just discussed, is
+ h = Z02
(5) lim P ( =2 ) (2z);
n — ~oo it 2 _Z /
this is due to Laurent*, and it may be proved by using the second of Murphy's
formulae, namely
Pn (cos 0) = cosn 0. 2F1 (- n, - n; 1; - tan2 1 ).
[NOTE. The existence of the formulae of this section must be emphasized because it
used to be generally believed that there was no connexion between Legendre functions and
Bessel functions. Thus it was stated by Todhunter in his Elementary Treatise on Laplace's
Functions, Lame's Functions and Bessel's Functions (London, 1875), p. vi, that "these
[i.e. Bessel functions] are not connected with the main subject of this book."]
5*72. Integrals associated with Mehler's fornula.
A completely different method of establishing the formulae of the last
section was given by Mehler and also, later, by Rayleigh; this method depends
on a use of Laplace's integral, thus
P, (cos 0) -= -F (cos 0 + i sin 0 cos 0b)' do
7T, 0
I r"
_ i n log (cos + i sin 0cos ) db.
r. o
Since
n log {cos (z/n) + i sin (z/n) cos - iz cos <
uniformly as n -- o when 0 - (b < r, we have at once
lim Pn (cos z/n) = e i cos o d = Jo ().
n — oo r'. 0
Heine t and de Ball+ have made similar passages to the limit with integrals
of Laplace's type for Legendre functions. In this way Heine has defined
Bessel functions of the second and third kinds; reference will be made to his
results in ~ 6'22 when we deal with integral representations of Y, (z).
Mehler has also given a proof of his formula by using the Mehler-Dirichlet integral
P (cos (n+ ) 0/d/
P (cos - I) 2 0/{2 (cos -cosO)}
If no=+4, it may be shewn that
P, (cos z/n) (- /,2)
but the passage to the limit presents some little difficulty because the integral is an improper integral.
Various formulae have been given recently which exhibit the way in which
Journal de M1ath. (3) I. (1875), pp. 384-385; the formula actually given by Laurent is
erroneous on account of an arithmetical error.
t Journal fir Mlath. LXIX. (1868), p. 131. See also Sharpe, Quarterly Journal, xxiv. (1890),
pp. 383-386.
T Astr. Nach. cxxviII. (1891), col. 1-4.