# §11.6 Asymptotic Expansions

## §11.6(i) Large , Fixed

where is an arbitrary small positive constant. If the series on the right-hand side of (11.6.1) is truncated after terms, then the remainder term is . If is real, is positive, and , then is of the same sign and numerically less than the first neglected term.

For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445).

## §11.6(ii) Large , Fixed

More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)).

## §11.6(iii) Large , Fixed

For the corresponding result for use (11.2.5) and (10.19.6). See also Watson (1944, p. 336).

For fixed

and for an estimate of the relative error in this approximation see Watson (1944, p. 336).