§8.18 Asymptotic Expansions of

§8.18(i) Large Parameters, Fixed

If and are fixed, with and , then as

for each . If and , then the -term can be omitted and the result is exact.

If and and are fixed, with and , then (8.18.1), with and interchanged and replaced by , can be combined with (8.17.4).

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

¶ Large , Fixed

Compare also §24.16(i).

¶ Symmetric Case

Let

8.18.8

Then as ,

uniformly for and , , where again denotes an arbitrary small positive constant. For see §7.2(i). Also,

with , and

8.18.11

with limiting value

8.18.12

For this result, and for higher coefficients see Temme (1996b, §11.3.3.2). All of the are analytic at .

¶ General Case

Let denote the scaled gamma function

, and again be as in (8.18.8). Then as

uniformly for and . Here

with , and

8.18.16

with limiting value

8.18.17

For this result and higher coefficients see Temme (1996b, §11.3.3.3). All of the are analytic at (corresponding to ).

¶ Inverse Function

For asymptotic expansions for large values of and/or of the -solution of the equation

see Temme (1992b).