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

240 THEORY OF BESSEL FUNCTIONS [CHAP. VIII
(III) The case in which v=x may be derived as a limiting case either
from (I) or from (II) by taking a or / equal to 0. The curves now to be considered are v = 0 and
(6) cosh u = v/sin v,
and they are shewn in Fig. 18.

Fig. 18.
We obtain information concerning H,(1) (v) and H,2) (v) by considering the
curves from - o to o + ri, while information concerning Jo (v) is obtained
from the curve which passes from cc - ri to so + vri. The detailed investigation will be given in ~~ 8-42, 8-53, 8-54.
8'32. Geometrical properties of Debye's contours.
An interesting result which will be found to be important in dealing with zeros of
Bessel functions (~ 15-8), and which is also used in proving certain approximate formulae
which will be stated in ~ 8-43, is associated with the second of the three contours just discussed (Fig. 17 of ~ 8'31). The theorem in question is that the slope* of the branch from
- o to oc + ri is positive and does not exceed ^/3.
It is evident that, for the curve in question,
du sin (v - ) - (v -,) cos v cos 3
sinh u -
dv sin2 v
But sin (v - 3) sec v - (v - 3) cos a has the positive derivative cos 3 tan2 v, and hence it
follows that
sin (v-, ) - (v - 3) cs os v cos
has the same sign+ as v -. Therefore since v- a and u are both positive or both negative
for the curve under consideration, dv/du is positive.
* Proc. Camb. Phil. Soc. xix. (1918), p. 105. Since, in the limiting case (Fig. 18) in which
=O0, the slope is 0 on the left of the origin and is ^/3 immediately on the right of the origin, no
better results of this type exist.
t This is obvious from a figure.