§16.11 Asymptotic Expansions

§16.11(i) Formal Series

For subsequent use we define two formal infinite series, and , as follows:

16.11.1,

In (16.11.1)

and

16.11.4
,

where

and .

It may be observed that represents the sum of the residues of the poles of the integrand in (16.5.1) at , , provided that these poles are all simple, that is, no two of the differ by an integer. (If this condition is violated, then the definition of has to be modified so that the residues are those associated with the multiple poles. In consequence, logarithmic terms may appear. See (15.8.8) for an example.)

§16.11(ii) Expansions for Large Variable

In this subsection we assume that none of is a nonpositive integer.

¶ Case

As in ,

Here the upper or lower signs are chosen according as lies in the upper or lower half-plane; in consequence, in the fractional powers (§4.2(iv)) of its phases are , respectively. (Either sign may be used when since the first term on the right-hand side becomes exponentially small compared with the second term.)

For the special case , explicit representations for the right-hand side of (16.11.7) in terms of generalized hypergeometric functions are given in Kim (1972).

¶ Case

As in ,

16.11.8

with the same conventions on the phases of .

§16.11(iii) Expansions for Large Parameters

If is fixed and , then for each nonnegative integer

16.11.10

as . Here can have any integer value from 1 to . Also if , then

16.11.11

again as . For these and other results see Knottnerus (1960). See also Luke (1969a, §7.3).

Asymptotic expansions for the polynomials as through integer values are given in Fields and Luke (1963b, a) and Fields (1965).