ABSTRACT
This constant can be evaluated by the methods of $3, but a simpler procedure here is to let z -, rfi icu in (4.01) and substitute the results in the reflection formula
This yields C = 4 ln(2a).
294 8 Sums and Sequences
The required expansion is obtained from (4.01) by integrating the last term repeatedly by parts, as in the construction of the Euler-Maclaurin formula. This gives
1 m - 1 lnz - z + - l n ( 2 ~ ) + 2 R2s 2 2s(2s-l)z2=-' + Rm(z), r= 1
where m is an arbitrary positive integer, and
To establish the asymptotic nature of this expansion for large Izl, write 0 = phz and assume that 101 G R-6, where6 is an arbitrary positive constant. Then using Theorem l.l(iii) and the substitution x = Izl r, we derive
4.2 Exponentiation of (4.03) gives
as z -, in the sector Iph z( G R-6. For positive z, this agrees with the result found in Chapter 3, $8.3 by Laplace's method. Comparing (4.03) and (4.04) we note that the former involves only alternate powers of 111.
