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

266

THEORY OF BESSEL FUNCTIONS

[CHAP. VIII

and this is positive when u > a, so that the parts of the curve which go off to
infinity on the right must lie as shewn in the north-east corner of Fig. 23.
When 1-a tanh a + (7r - /) cot / > 0,
i.e. when (a, 3) lies in any of the domains numbered 1, 2 and 3 in Fig. 21, it is
found that the curve does not cross v = 27r - 3, and so the curve after crossing
v = tr passes off to - o + 7ri as shewn in Fig. 23 by a broken curve.

IT I i
2hz_27T2 I __ ___ _.
~., U'
(IT
Fig. 23.
We now have to consider what happens when (a, /) lies in the domain
numbered 6a in Fig. 21. In such circumstances
1 - a tanh a + (7r - /) cot / < 0;
and b (- a, v) has a maximum at v = 2r - /, the value of a (- a, 27r - /) being
negative. The curve, after crossing v = -r, consequently remains on the right of
=- a until it has got above v = 27r -.
Now q (-a, v) is increasing in the intervals
(3,27r-/3), (27r +, 47r-/), (47r +, 67r-/3),...;
let the first of these intervals in which it becomes positive be
(2Mi7r + 3, 2M7r + 27 - 3).
Then qb (z, 2Ml7r + 27r - /) has a minimum at i = - a, t which its value is
positive, and so the curve cannot cross the line v = 2M7r + 27r -; it must
therefore go off to infinity on the left, and consequently goes to
-c0 +(21 + 1) 7i;
it cannot go to infinity lower than this, for then the complete curve would
meet a horizontal line in more than two points.