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

68

THEORY OF BESSEL FUNCTIONS

[CHAP. III

3'571. The integral of Poisson's type for y(O) (z).
It was shewn by Poisson* that
f""
feixcosw log (x sin2 o) dwo
is a solution of Bessel's equation for functions of order zero and argument x;
and subsequently Stokes obtained an expression of the integral in the form of
an ascending series (see ~ 3'572).
The associated integral
2 ft^
- cos (z sin ). log (4z cos2 0) dO
7r 0
was identified by Neumannt with the function yO) (z); and the analysis by
which he obtained this result is of sufficient interest to be given here, with
some slight modifications in matters of detail.
From ~ 2'2 (9) we have
_ 2n = 2 cos (z cos 0) cos 2?nOcO,
and so, if we assume that the order of summation and integration can be
changed, we deduce that
(-y1Jn^z) 4 [ie os os 2n 0
2 -2 J2n () = 4 cos (z cos 0) cos 2 dO
n=1 n 7r Jo n=l n
2 [7
_ - - f cos (z cos 0). log (4 sin2 0) dO;
from this result combined with Parseval's integral (~ 22) and the definition of
Y(O) (z), we at once obtain the formula
(1) y(~) (z) = 2 fcos (z cos 0). log (4z sin2 0) dO,
7TJ0
from which Neumann's result is obvious.
The change of the order of summation and integration has now to be examined,
because 2n-l cos 2nO is non-uniformly convergent near 0=0. To overcome this difficulty
we observe that, since 2 (- )n J2n (z)/n is convergent, it follows from Abel's theorem t that
(-) Jn()/- lim 2 (- ) a J,(z)/n= lim 2 cos(os )aco2ndO
n=l a —1-0 n=1 a-p-l-0 7T n=l n
* Journal de l'Vcole R. Polytechnique, xII. (cahier 19), (1823), p. 476. The solution of an
associated partial differential equation had been given earlier (ibid. p. 227). See also Duhamel,
Cours d'Analyse, II. (Paris, 1840), pp. 122-124, and Spitzer, Zeitschrift filr Math. und Phys. II.
(1857), pp. 165-170.
- Theorie der Bessel'schen Functionen (Leipzig, 1867), pp. 45-49. See also Niemiller, Zeitschriftfiir Math. und Phys. xxv. (1880), pp. 65-71.
+ Cf. Bromwich, Theory of Infinite Series, ~ 51.