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

446 THEORY OF BESSEL FUNCTIONS [CHAP. XIII
13'74. Deductions from Nicholson's integrals.
Since Ko (:) is a decreasing* function of:, it is clear from ~13'73 (1) that
J;2 (X) + Y2 (x)
is a decreasing function of x for any real fixed value of v, when x is positive.
Since this function is approximately equal to 2/(7rx), when x is large, we
shall investigate
x {J.2 (x) + Y2 (x)}
and prove that it is a decreasing function of x when v >, and that it is an
increasing function of x when v < 1.
It is clear that
dx
d-L [X TJ2 () +~ Y (x)}]
= [K, (2x sinh T) + 2x sinh TKo' (2x sinh T)}cosh 2vTdT
r= LIKo (2x si ) tnh )tanh cosh 2
+ 2 d Ko(2x (sih2tnhT) cosh2tanhcosh 2vT} dT,
on integrating the second term in the integral by parts. Hence
d
d [ {jJV2 (x) + Yp2 (X)}]
= -8 K0 (2x sinh T) tanh T cosh 2vT {tanh T - 2v tanh 2vT} dT.
77/". 0
Now X tanh XT is an increasing function of X when X > 0, and so the last
integrand is negative or positive according as 2v > 1 or 0 < 2v < 1; and this
establishes the result.
Next we prove that, when x > v > 0,
(X2 _ 2) { JV2 (X) +- Y2 (x)}
is an increasing function of x.
If we omit the positive factor 8 (x2- w 2)- /Vr2 from the derivate of the expression under consideration we get
x {Ko (2x sinh t) + 2 (x2 - v2) sinh t. Ko' (2x sinh t)} cosh 2vt dt,
and to establish the theorem stated it is sufficient to prove that this integral
is positive.
* This is obvious from the formula
Ko ()= e cosh t dt.
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