ABSTRACT
The smallest term is given by s = [x], hence we are interested in the asymptotic approximation of Rn (n + 6) for large n and bounded [, especially [ E [0, I]. 4.2 With x replaced by n-ti, the integrand in (4.01) has the special feature that the saddle point and singularity coincide. This phenomenon was previously encountered in Chapter 4, Exercise 8.4. Following the methods of that chapter we take a new integration variable v, given by
The path transforms into a loop in the v plane: to-)
Rn(n+ 0 = e-.jm e-""q(v) dv, (4.02) where
Near v = 0 we have 2 2'12
where u1I2 is positive on the upper side of the positive real axis andnegative on the lower side; compare Chapter 3, $8.3. In consequence, q(v) can be expanded for small Iv) in the form
in which the coefficients q,([) are polynomials in [. In particular,
For the difference between q(u) and the first term, the path in (4.02) may be collapsed onto the positive real axis. Application of Watson's lemma then yieldst
To resolve the difficulty, we again turn to the methods of J. C. P. Miller (1952). In addition to (4.08), we employ the difference equation
which is easily obtained from (4.06) by increasing n by a unit. Setting
and substituting by means of (4.07), we find that
The factor I/(n+ I)'-' in the last sum can be expanded in descending powers of n. Then equating coefficients, we obtain
and so on. The first of (4.12) is satisfied by (4.10), whatever the value of c,. But on sub-
stituting in the second equation by means of (4.10) and (4.1 I), we find that the terms involving c, and [ disappear, leaving co = -+. Accordingly yo(() = (-+, in agreement with $4.2. The expression (4. 11) now becomes
and so on. 4.4 In applying the foregoing results to the calculation of e-"E1(xeix), it should be noticed that any possible improvement in accuracy resulting from computation of the converging factor ~jould be nullified ifthe term irre-" in (4.05) were neglected (on the plausible grounds that it is exponentially small compared with individual terms u,(x)). This is because
