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

262

THEORY OF BESSEL FUNCTIONS

[CHAP. VIII

8'6. Asymptotic expansions of Bessel functions of large complex order.
The results obtained (~~ 831 —8-42) by Debye in connexion with J, (x)
and Y, (x) where v and x are large and positive were subsequently extended*
to the case of complex variables. In the following investigation, which is, in
some respects, more detailed than Debye's memoir, we shall obtain asymptotic
expansions associated with J, (z) when v and z are large and complex.
It will first be supposed that I arg z 1< 2 7r, and we shall write
v = z cosh y = z cosh (a + i/3),
where a and / are real and y is complex. There is a one-one correspondence
between a + i/ and v/z if we suppose that /3 is restricted to lie betwueent 0
and vr, while a may have any real value. This restriction prevents z/v from
lying between - 1 and 1, but this case has already (~ 8'4) been investigated.
The integrals to be investigated are
H^1 (z) = -. _ e-zf w) dw,
rtTJ _ -r
1 r-o-Tfi 1 w=
H (2) () e =; - ezf (w dw,
where f(w) _ w cosh - sinh zu.
A stationary point of the integrand is at y, and we shall therefore investigate the curve whose equation is
If(w) = I f(r).
If we replace w by u + iv, this equation may be written in the form
(v - /) cosh a cos 3 + (u - a) sinh a sin / - cosh u sin v + cosh a sin /3 = 0.
The shape of the curve near (a, /) is
{(r - a)2 - (v - /)2} cosh a sin / + 2 (u - a) (v - /) sinh a cos / = 0,
so the slopes of the two branches through that point are
X r + - arc tan (tanh a cot /3),
- v + I arc tan (tanh a cot /3),
where the arc tan denotes an. acute angle, positive or negative; Rf(w) increases as w moves away from y on the first branch, while it decreases as w
moves away from y on the second branch. The increase (or decrease) is steady,
and Rf(w) tends to + oo (or - o ) as w moves off to infinity unless the curve
has a second double-point+.
* Miinchener Sitzungsberichte, XL. [5], (1910); the asymptotic expansions of Iv (x) and K, (x)
were stated explicitly by Nicholson, Phil. Mag. (6) xx. (1910), pp. 938-943.
t That is to say 0< 3<7r.
+ As will be seen later, this is the exceptional case.