# §33.12 Asymptotic Expansions for Large

## §33.12(i) Transition Region

When and , the outer turning point is given by ; compare (33.2.2). Define

33.12.1

Then as ,

For derivations and additional terms in the expansions in this subsection see Abramowitz and Rabinowitz (1954) and Fröberg (1955).

## §33.12(ii) Uniform Expansions

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 .

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).