When and , the outer turning point is given by ; compare (33.2.2). Define
Then as ,
uniformly for bounded values of . Here and are the Airy functions (§9.2), and
With the substitution , Equation (33.2.1) becomes
Then, by application of the results given in §§2.8(iii) and 2.8(iv), two sets of asymptotic expansions can be constructed for and when . See Temme (2015, §31.7).
The first set is in terms of Airy functions and the expansions are uniform for fixed and , where is an arbitrary small positive constant. They would include the results of §33.12(i) as a special case.
The second set is in terms of Bessel functions of orders and , and they are uniform for fixed and , where again denotes an arbitrary small positive constant.
Compare also §33.20(iv).