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

15-31, 15-32]

ZEROS OF BESSEL FUNCTIONS

489

provided that A (x) is chosen arbitrarily and that B (x) and C (x) are then
defined by the equations
2B (x) = x2A' (x)/(x2 - 2),
C (x) = 2 fB' (x) + A (x) - B (x)/4/(l2 - _2).
(I) If A (x) _ (x2 - v2) then
2D (x) (x2 - v2) - x3 (3x4 + 142 p2 + 4v4) > 0,
and so G (x) is an increasing function of x which is therefore less than
lim ~O (x) = 2/77.
X -— zr oe
Since e( (/,t) = (C2 - v2) 2 (,n) we see that when n assumes the values
1, 2,..., then the numbers ( 42 - v2)4 \ v (n) I form an increasing sequence
less than /(2/7r).
(II) If A (x) x, then
2D (x) (x2 - v2)4 = X- 2 t(4V2 - 3) X4 - 8v2 (72 + 1) w2 + V4 (4v2 - 1)}
<0,
provided that 4v2 > 3 and x exceeds the greatest root of the equation
(4v2 _ 3) 4 - 2 (v2 + 1) x2 + 4 (42 - 1) = 0.
In this case 0 (x) is a decreasing function and we can apply arguments, similar
to those used in theorem (I), to deduce the truth of theorem (II).
15'32. Schafheitlin's investigation of the zeros of Jo (x).
By means of the integrals which have been given in ~ 612, it has been
shewn by Schafheitlin* that the only positive zeros of J0 (x) lie in the intervals
(n7r + -r, ma r + 7r) and the only positive zeros of Y0 (x) lie in the intervals
(mr + ITr, m7r + 7w), where m= 0, 1, 2..
We shall first give Schafheitlin's investigation for J (x), with slight
modifications, and then we shall prove similar results for cylinder functions
of the type
J, (x) cos a - Yv (x) sin a
(where v lies between - - and ), by the methods used by Schafheitlin.
Schafheitlin's investigations were confined to the values 0 and ~7i- of a.
From an inspection of the formula of ~ 6'12 (7),
J 2 sin ( + 2) e-Cot 0 o,
VO Jo0 sin 0 Veos O
it is obvious that, when m7r < x < m7r + 4 r,
sgn {sin (x + I 0)} = sgn (- 1),
and so sgn J0 (x) = sgn (- 1).
Consequently J0 (x) has no zeros in the intervals (nw7r, m7r + t 7r).
* Journal filr Math. cxIv. (1894), pp. 31-44.