ABSTRACT
An indirect approach, due to Ursell(1965), is to prove that the Laplace expansion is uniformly valid for a E [a,, a ) . A matching process then shows that the same is true of (12.20) and (12.23). In the present section we solve the problem in a direct manner by extending the mappings of $12.3. 13.2 Throughout the present subsection we suppose that a 2 a,. As integration contour Y in (12.17) we employ the circle
where p is independent of a. Provided that p < min(x, a,), this circle lies within the half-strip Re t>O, 1Im tl <x. The map of 9 in the v plane has the equation
v = sech a sinh (a +pel8) - a - pele. Hence from (12.08)
1v+3c3/21 = iP2 tanh a. Now consider the circle % in the w plane defined by
where a is independent of a , and a2 < $a,-$ tanha, (ensuring that Q lies in the right half-plane). The v map of % is given by
Since a 2 a,, C-3/4 is bounded. Hence for all sufficiently small a (independently of a) the v map of W lies in the disk (v+3C3I2(,< 2a2.
