Define to be the solution of the differential equation
that is infinitely differentiable on the interval , including . Then
all functions taking their principal values, with , corresponding to , respectively.
As through positive real values
uniformly for in all cases, where and are the Airy functions (§9.2).
The function given by (10.20.2) and (10.20.3) can be continued analytically to the -plane cut along the negative real axis. Corresponding points of the mapping are shown in Figures 10.20.1 and 10.20.2.
The equations of the curved boundaries and in the -plane are given parametrically by
The curves and in the -plane are the inverse maps of the line segments
respectively. They are given parametrically by
where is the positive root of the equation . The points where these curves intersect the imaginary axis are , where
The eye-shaped closed domain in the uncut -plane that is bounded by and is denoted by ; see Figure 10.20.3.
As through positive real values the expansions (10.20.4)–(10.20.9) apply uniformly for , the coefficients , , , and , being the analytic continuations of the functions defined in §10.20(i) when is real.
For proofs of the above results and for error bounds and extensions of the regions of validity see Olver (1997b, pp. 419–425). For extensions to complex see Olver (1954). For resurgence properties of the coefficients (§2.7(ii)) see Howls and Olde Daalhuis (1999). For further results see Dunster (2001a), Wang and Wong (2002), and Paris (2004).