ABSTRACT
The discrepancy is entirely attributable to neglect of the error term. When (6.02) is truncated at the term for which s = n-1, the error is given by
The stationary points of this function are the zeros of q(')(t). Those in the interval of variation are t = 0, 1/ J3, and J3, and computations yield q(4)(0) = 24.00, q(4)(1/J3) = - 10.12, q(4)( 43) = 0.38, q(4)(n) = 0.06. Hence fi,,(q(4)) = 44.94, and (6.05) becomes
The size of this bound warns us that the series (6.03) may be grossly in error (although the actual error lies well within the bound). 6.2 A substantial improvement in the asymptotic expansion (6.02) is attainable by use of the identity
which is easily verifiable by contour integration. Addition to (6.01) produces
Applying the method of $5 to the last integral, we obtain n-I q(2r+l)(n)
~ ( m ) = lze-. + (-)HZ (-Y m2.+ + nZn(m). 2 r=O
where the new error term is bounded by
, I~zn(m)I d "Y,,w(q(2n')/m2"+'- (6.08) The representation (6.07) differs from (6.02) by the presence of the term +re-".
