With the notation of §§10.17(i) and 10.17(ii), as with fixed,
Corresponding expansions for , , , and for other ranges of are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). In particular, use of (10.34.3) with yields the following more general (and more accurate) version of (10.40.1):
For fixed ,
as in . Here and
In the expansion (10.40.2) assume that and the sum is truncated when . Then the remainder term does not exceed the first neglected term in absolute value and has the same sign provided that .
For the error term in (10.40.1) see §10.40(iii).
For (10.40.2) write
where denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that changes monotonically. Bounds for are given by
where ; see §9.7(i).