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

8-21, 8-22]

FUNCTIONS OF LARGE ORDER

233

where G is the sign indicating a "generalised integral" (~ 6'4); and hence, by
integrating term-by-term and using Euler's formula, Meissel deduced that
(1) n 1(n), r ( 2I+? ) (4 i COS (- M+ 7).
7' m=0
Meissel also gave an approximation for X,,,, valid when m is large; and this approximation exhibits the divergent character of the expansion (1).
The approximation is obtainable by the theory developed in the memoir of Darboux,
"Sur lapproximation des fonctions de tres grands nombres," Journal de lMalth. (3) iv. (1878),
pp. 5-56, 377-416.
We consider the singularities of 0 qua function of t; the singularities (where 0 fails to
be monogenic) are the points at which d= 2r7r and t =(12r7r)~, where = +1, + 2, +3,...;
and near* t= + (127r) the dominant terms in the expansion of 0 are
~27r (367r)3 1 (l2 )~) A
By the theory of Darboux, an approximation to Xn is the sum of the coefficients of
t2m+1 in the expansions of the two functions comprised in the last formula; that is to say
that
X 2,,. (36rr) 3 33 (2m-)
(2snm+)! (I2T)-3m +
2 r (2m+ )
33 r (2) r (2n + 2). (127r)'
and so, by Stirling's formula,
(2) - -, 1 1
(18)~ r (2) (nt + ) 3 (127sr)t'
This is Meissel's approximation; an approximation of the same character was obtained
by Cauchy, loc. cit., p. 1106.
8'22. The application of Kelvin's principle to J, (v see /).
The principle of stationary phase has been applied by Rayleight to obtain
an approximate formula for Je (v sec /3) where / is a fixed positive acute angle,
and v is large+.
As in ~ 8'2 we have
J ( see /3) =1- (cos {v (0-sec/ sin 0)} dO + 0 (1/v),
and 0 - sec Pf sin 0 is stationary (a minimum) when 0 = /.
Write 0- sec /3 sin 0 = - tan / + b, so that b decreases to zero as 0 increases from 0 to /3 and then increases as 0 increases from / to 7r.
* These are the singularities which are nearest to the origin.
t Phil. MIag. (6) xx. (1910), p. 1004. [Scientific Papers, v. (1912), p. 620.]
+ See also Macdonald, Phil. Trans. of the Royal Soc. ccx. A (1910), pp. 131-144; and Proc.
Royal Soc. LxxI. (1903), pp. 251-258; LxxII. (1904), pp. 59-68.