ABSTRACT
In the first case it is supposed that v and z are real or complex numbers, v being fixed and lzl large. The saddle points are located at the zeros of cosh f, that is, at I = ++ni, +$xi, ... . The integration path can be deformed to pass through any number of these points, but it is not obvious how to choose a path on which Re(z sinh I) attains its minimum at one or more of the saddle points. Accordingly, we follow the suggestion made in $7.2 and begin by mapping the strip 0 < Im f < n (which contains one of the saddle points) on the plane of
$9 Bessel Functions of Large Argument and Order 131
The map is quickly determined by the following considerations: (a) The positive real t axis corresponds to the line segment Im v = - 1, Rev 2 0. (b) u-$etas Rer++m. (c) Increasing t by ni changes the sign of v + i. (d) dvfdr is real on the imaginary axis and changes sign at r = fni. (e) u - +i(t - f xi)' as t + +xi. (f) Images in the imaginary axis correspond.
