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

214

THEORY OF BESSEL FUNCTIONS

[CHAP. VII

The domain of integration becomes the whole of the (~, V) plane; and it is
found that
R 2x e-2a X J -p ~ a2o frs} doo,
_Rp -3 e-pl+~h a a,,, 4q dcdq,
7r2 -oo 00 r, s = O
where
a0, 0=1, 2,0= 22 - -1 0,2-=
by some rather tedious arithmetic. It follows* that the dominant terms of
the asymptotic expansion of Rp for large values of p are
2 ()eX2 a o, + a+, o 2+
so that
(2) 4
(2) _Rp2($) e- 2+ 2f +.2
It is easy to verify by Stirling's theorem that
(_ )(-)P. 2 () e-,
(2x) ' ) p
so that the error due to stopping at one of the smallest terms is roughly half of the first
term omitted.
In like manner Stieltjes proved that, if P (x, 0) and Q (x, 0) are defined as
in ~ 7'3, then
P-l(_y (o, 2m)
(3) P (x, 0) = E () m) + (-) R),
m=O (2x)2"
(4) Q (x. o) = -' (2)2M+ l) + (-)P Rw(Q)
=o0 (x)2
where
/\ I e-2a; T1 2 + 1~ 7_
(5) R(P) - - + 4 +.,
x \ -2a rI l2 - 1 1
(6) Rp(Q ( - -+ 2T -6 +
xv e J 1-/T22 P J
provided that p is chosen so as to be nearly equal to x, and 7 is defined to
be x - p.
Results of this character are useful for tabulating Bessel functions in the critical range;
some similar formulae have been actually used for that purpose by Airey, Archiv der Math.
und Phys. (3) xx. (1913), pp. 240-244; (3) xxII. (1914), pp. 30-43; and British Association
Reports, 1913, 1914.
It would be of some interest to extend the results, which Stieltjes has established for
Bessel functions of zero order (as well as for the logarithmic integral and some other
functions), to Bessel functions of arbitrary order.

* Cf. Bromwich, Theory of Infinite Series, ~~ 133, 137, and 174, or the lemma which will be
proved in ~ 8'3.