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

r.

608 THEORY OF BESSEL FUNCTIONS [CHAP. XVIII
We first deal with the factor Jv (tw)/Jv (w). We observe that*
(1) I < expt-(l)JV (tw) |(W
when w is on either contour; this follows from inequalities of the type
~ 18'21 (9) when w is not small, and from the ascending series when w I is
not large (i.e. less than ji).
We next consider 1t (w, x), which is equal to
-iw {IHv() (w) Hf(2) (xw)- H(1) (xw) H(2) (w)};
it is convenient to make two investigations concerning this function, the
former being valid when - < v < ~, the second when v > 1.
(I) The first investigation is quite simple. It follows from ~ 36 and
~ 7-33 that
]~ | eiXw kft e-ixw I
(2) H^() (xw) I < i I HV( (Xw) <
I \j W 1 'I I W I
for all the values of w and x under consideration when - I < v. Hence
J2
(3) <1 4(w, ) |< a exp {(1- x)! (w) }.
(II) When v> and Iw is not large, it is easy to deduce from the
ascending series for J, (w), YI (w), J, (xw) and Y, (xw) that
(4) l (w, x) 1< k3, IwX-^.
If I w is not small, we use the inequalities (deduced from ~ 7'33)
Ic | eiwIw | ( I l e-iw 1
(5) I1>(w)|I< ke l 'Ij
together with the inequalities
H,(1) (w) i<J4 {]w i- + I xw eixw",
(6) i H^(2) (xw)l < 4 {I X I- +lw I-} I e-W
It follows from ~ 36 and ~ 7'33 that the inequalities (6) are true whether
xw is large or not. Hence,
(7) I, (W, x) I < k2k {x4 + x- I w i-v} exp {(1 - x) I (w)1 },
when v ) and I w is large, whatever be the magnitudet of aw 1.
If we now combine the results contained in formulae (3), (4) and (7) we
deduce that, whether - v < I or v >,
(8) 1 iE (w, X) < k, (-i + x-) exp {( - x) j I (w) },
* It is supposed that the numbers kl, k2, k3,... are positive and independent of w, x and t;,
their values may, however, depend on the value, of v.
t Provided of course that 0 < x 1.