# §5.11 Asymptotic Expansions

## §5.11(i) Poincaré-Type Expansions

With the same conditions,

where

5.11.4

Also,

where and

5.11.6.

Wrench (1968) gives exact values of up to . Spira (1971) corrects errors in Wrench’s results and also supplies exact and 45D values of for . For an asymptotic expansion of as see Boyd (1994).

### ¶ Terminology

The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)).

Next, and again with the same conditions,

where and are both fixed, and

where is fixed, and is the Bernoulli polynomial defined in §24.2(i).

Lastly, as ,

uniformly for bounded real values of .

## §5.11(ii) Error Bounds and Exponential Improvement

If the sums in the expansions (5.11.1) and (5.11.2) are terminated at () and is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. If is complex, then the remainder terms are bounded in magnitude by for (5.11.1), and for (5.11.2), times the first neglected terms.

For the remainder term in (5.11.3) write

Then

where is as in Chapter 25. For this result and a similar bound for the sector see Boyd (1994).

For further information see Olver (1997b, pp. 293–295), and for other error bounds see Whittaker and Watson (1927, §12.33), Spira (1971), and Schäfke and Finsterer (1990).

For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4).

## §5.11(iii) Ratios

In this subsection , , and are real or complex constants.

If in the sector (), then

Also, with the added condition ,

Here

In terms of generalized Bernoulli polynomials 24.16(i)), we have for

Lastly, and again if in the sector (), then

For the error term in (5.11.19) in the case and , see Olver (1995).