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

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BESSEL FUNCTIONS BEFORE 1826

7

It is easy to shew that An is a solution of the differential equation
C2 d2 An+ dA- _ n2 (1 -_ 2) An = 0.
Define ei by the formula An- 2n~-1 eftd/ln! and then
2 (+ u + eu-l2 (1- 2)=0.
Hence when n is large either u or U2 or du/de must be large.
If zc= 0 (na) we should expect u2 and dzl/de to be 0 (n2a) and 0 (na) respectively; and
on considering the highest powers of n in the various terms of the last differential equation,
we find that a= 1. It is consequently assumed that u admits of an expansion in descending
powers of n in the form
t= nuo + l + n2/n +...,
where uo, ul, lt2,... are independent of n.
On substituting this series in the differential equation of the first order and equating to
zero the coefficients of the various powers of n, we find that
2 =(1 _2)/e2, e (l '+ 2ui) + o = 0,...
where zto'=cduo/le; so that uo ~( )i= 2 o=, and therefore
ude= n log 1l +Jl-)+ /(1- 2)T+1 - 1log(1-2)+...
and, since the value of An shews that Jude - n log -e when e is small, the upper sign must
be taken and no constant of integration is to be added.
From Stirling's formula it now follows at once that
En exp {n '/(1-e2)}
" /(' T). n% (1 - 2)4 {1,(1 - E2)} n
and this is the result obtained by Carlini. This method of approximation has been carried
much further by Meissel (see ~ 8-11), while Cauchy* has also discussed approximate
formulae for An in the case of comets moving in nearly parabolic orbits (see ~ 8'42), for
which Carlini's approximation is obviously inadequate.
The investigation of which an account has just been given is much more
plausible than the arguments employed by Laplace to establish the corresponding approximation for B,.
The investigation given by Laplace is quite rigorous and the method which
he uses is of considerable importance when the value of Bn is modified by
taking all the coefficients in the series to be positive-or, alternatively, by
supposing that e is a pure imaginary. But Laplace goes on to argue that an
approximation established in the case of purely imaginary variables may be
used 'sans crainte' in the case of real variables. To anyone who is acquainted
with the modern theory of asymptotic series, the fallacious character of such
reasoning will be evident.
* Comptes Rezdus, xxxviii. (1854), pp. 990-993.
t Mecanique Celeste, supplement, t. v. [first published 1827]. Oeuvres, v. (Paris, 1882),
pp. 486-489.