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

15-33, 15-34]

ZEROS OF BESSEL FUNCTIONS

491

where m = 0, 1, 2,..., except that, if a is sufficiently near to 'r, there may be
zeros* in the interval
(7r + -~v - a, T -a).
We shall prove this result by proving that v (x) has a fixed sign throughout
each of the intervalst
(m7r + 4 1 + PV - a, mn7 + v - a).
Write x = (mn + 1) 7 - a - (1 - 2v) 0,
where b is an angle between 0 and -8tr.
With this value of x,
Y ( _) (_) +1 _ 2+ C - sin {(-2v) ( -)} e otO d
( ~, W) 17 (V + s n) r_ (2_)_ ___ ___ o __ __
^^ r +^ fo - Sin --- u
To each value of 0 between 0 and 20 there corresponds a value between 20
and v7r for which sin {(1 - 2v)(0- 0)} has the same numerical value, but
has the positive sign.
Again e-e cot COS- 0/sinv+l 0
is an increasing function of 0 provided that
I sin30]
2x > max \(2v + 1) sin 0 cos 0 + (7 - 1) c os ]
2 Cos j
and this condition is satisfied when x > I since < 1.
Hence, if x > ~ and
m7T + 77r + V7T - a < x < mV7 + 7 - a,
we have sgn W (') = sgn (- 1) +s,
and this proves the more precise theorem.
15'34. Theorems of Schafheitlin's type, when 2 < v < -.
We next consider the function
WY (x) - J (x) cos a - Y, (x) sin a,
where 0 a< - r, as before, in which it is now supposed that 1 < v < -.
We shall first prove the crude result that the only positive zeros of W, (x)
lie in the intervals
(mwr - a, DVr - I 7r + 2 w- a)
where- m = 0, 1, 2,.... This result follows at once from the formulae of ~ 6'12,
which shew that
" ().-= r (V 2+ ) r (1)ov r cos- 0 sin (x + a - vO + 2) e'-xcot dO;
v+f 12 (21) -0sin2v+l 0
for when mr - 1 - r + v- a < x < (m + 1) r - a,
* By taking as an alternative function J v I (x) and applying the theorem of ~ 15'24, we see that
there cannot be more than one such zero.
+ If x < and m = 0, the reasoning fails when -,r + i vr - a<.
+ If a>(v- ) r, the interval for which m=0 is, of course, to be omitted.