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

254

THEORY OF BESSEL FUNCTIONS

[CHAP. VI1I

It is clear that
g (0, x) = V(1 - x2) < /(1 + X2),
g (7vr, x) = 1 < (1 +2 X),
so that, if g (0, x), qua function of 0, attained its greatest value at 0 or Tr,
that value would be less than \/(1+x2). If, however, g(O,x) attained its
greatest value when 0 had a value 00 between 0 and r', then
1 - 2 cos 200 (o- x2 sin 00 cos 0,)2
(002 - X2 sin2 00)~ (0o2 _ x2 sin2 0o)
and therefore
g (0, x) < g (0o, X) = V(1 - _ 2 cos 20o) < V(1 + X2),
so that, no matter where g (0, x) attains its greatest value, that value does
not exceed /(1 + X2).
Hence
a (0, x) 2- X2 sin 0 cos 0
>~ \/(o - x~ sin' 0) >
0 ~V/(1 + X2)
and so
F(o,.)-F(O,x)= a (0 x) dO 0 0 sin 8 cos 0 d,
Jo ~(1 + x2)
whence (7) follows at once.
Another, but simpler, inequality of the same type is
(8) F(0, x) > F(0, x) + 2 02 V(1 - X2).
To prove this, observe that
F(0, >)) V(02 - X2 sin2 0) > 0 /(1 - C2),
and integrate; then the inequality is obvious.
From these results we are now in a position to obtain theorems concerning
J, (vx) and J,' (vx) qua functions of v.
Thus, since
aDJ (c -c) 1 fwOF(o, x) e- F(o x) dO O0,
v r 0Jo
the integrand being positive by (5), it follows that Jv(vxc) is a positive decreasing function of v; in like manner, Jv' (vx) is a positive decreasing function
of c.
Also, since
a{e.F(~,x' j.{()}v_ = F(0, x) - F (0, x)} eF(O, )-Fo, e O< O,
the integrand being positive by (5), it follows that e"^(~,x) J, (vx) is a decreasing
function of v; and so also, similarly, is e"'F(~,) J' (vx).