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

490

THEORY OF BESSEL FUNCTIONS

[CHAP. XV

To prove that JO (x) has no zeros in the intervals (mrr + 7n, m7r + 7), write
x= (m + 1) 7?-,
and then Jo(x)=2(+' sin (0 - s) e2coto dO.
7r Jo sin 0 Vcos 0
The last integrand is negative or positive according as
0 < < 2 or 2 < 0< 2r.
Since S < -8r, the second of these intervals is the longer; and the function
e-2X cot 0
sin 0 /cos 0
is an increasing function* of 0 when x >- 7r and 0 is an acute angle.
Hence to each value of 0 between 0 and 29 there corresponds a value
between 20 and 1rr for which sin (-10 - 0) has the same numerical value, but
has the positive sign, and the cofactor of sin (a 0 - ) is greater for the second
set of values of 0 than for the first set. The integral under consideration is
consequently positive, and so J0 (x) cannot have a zero in any of the intervals
(mV7rw+~}, m27 +7r). Therefore the only positive zeros of J, (x) are in the
intervals (,mr + -wT, mr + — 7r).
15'33. Theorems of Schafheitlin's type, when - ~ < v <.
We shall now extend Schafheitlin's results to functions of the type
(x) - J, (x) cos a - Y, (x) sin a,
where 0 a < a r and - 2 < v.
We shall first prove the crude result that the only positive zeros of, (x)
lie in the intervals
(mnr + 4r + 7- -a, m7r + r - a)
where m = 0, 1, 2,.... This result follows at once from the formulae of ~ 612,
which shew that
~ 2+~1 x t'r cos-t0 sin (x ~+ -,0 + 0) e_ x cot dO,
(x ) r (v + ~1) r (~) ) 0 sin 0
for, when 7r - a < x < mzr + rT + V7r - a,
we have sgn [sin (x + a - vO + -)] = sgn (- 1)
and so, for such values of x, V, (x) is not zero. Consequently the only zeros
of, (x) lie in the specified intervals, and there are an odd number of zeros
in each interval, with the possible exception of the first if a > 3 r + I Vrr.
Next we obtain the more precise result that the only positive zeros of V (x)
lie in the intervals
(m7r7 + 37r + V1 r - a, nw- + 7- 7r + 1 7 _- a)
* Its logarithmic derivate is
(2x - sin 0 cos 0) cosec2 0 + tan 0.