# §8.12 Uniform Asymptotic Expansions for Large Parameter

Define

where the branch of the square root is continuous and satisfies as . Then

Also, denote

and

where is Dawson’s integral; see §7.2(ii). Then as in the sector ,

in each case uniformly with respect to in the sector ().

With , the coefficients are given by

8.12.9

where , , are the coefficients that appear in the asymptotic expansion (5.11.3) of . The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at , and the Maclaurin series expansion of is given by

where ,

and are defined by

8.12.13.

In particular,

8.12.14

For numerical values of to 30D for and , where , see DiDonato and Morris (1986).

For error bounds for (8.12.7) see Paris (2002a). For the asymptotic behavior of as see Dunster et al. (1998) and Olde Daalhuis (1998c). The last reference also includes an exponentially-improved version (§2.11(iii)) of the expansions (8.12.4) and (8.12.7) for .

A different type of uniform expansion with coefficients that do not possess a removable singularity at is given by

for in , with for and for . Here

and

8.12.20

Higher coefficients , , up to , are given in Paris (2002b).

Lastly, a uniform approximation for for large , with error bounds, can be found in Dunster (1996a).

For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function see Paris (2002b) and Dunster (1996a).

## ¶ Inverse Function

For asymptotic expansions, as , of the inverse function that satisfies the equation

see Temme (1992a). These expansions involve the inverse error function 7.17), and are uniform with respect to . As a special case,

8.12.22.