Here denotes an arbitrary small positive constant and
Also and for
Numerical values of this function are given in Table 9.7.1 for to 2D. For large ,
As the following asymptotic expansions are valid uniformly in the stated sectors.
In (9.7.5) and (9.7.6) the th error term, that is, the error on truncating the expansion at terms, is bounded in magnitude by the first neglected term and has the same sign, provided that the following term is of opposite sign, that is, if for (9.7.5) and for (9.7.6).
In (9.7.9)–(9.7.12) the th error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign.
As special cases, when
For other error bounds see Boyd (1993).
with . Then
(For the notation see §8.2(i).) And as with fixed