where the branch of the square root is continuous and satisfies as . Then
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
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
and are defined by
For numerical values of to 30D for and , where , see DiDonato and Morris (1986).
Special cases are given by
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
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).