By Boyd J.P.
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Additional info for Asymptotic, superasymptotic and hyperasymptotic series
The early numerical work on the porous channel was even more confusing , finding one or two solutions where there are actually three. Robinson resolved these uncertainties in a 1976 article that combined careful numerical work with the analytical calculation of the exponentially small terms which are the sole difference between the two physically interesting solutions. The reason that the exponential terms could be calculated without radical new technology is that the solution in the inviscid region (“outer” solution) is linear in Y plus terms exponentially small in : f (Y ) ∼ α( )Y + γ( ) −3 γ( ) = ± 1 6 2 π 7 1/4 exp − 1 1 exp − 4 4 α + Y 3 + ...
Stieltjes  showed that for an alternating series, one could considerably improve accuracy for both convergent and divergent series merely by multiplying the last retained term by a weight factor of 1/2. Airey developed an early (1937) hyperasymptotic method, restricted to alternating series for which the general term is known, which computed an improved, N -dependent replacement for Stieltjes’ 1/2 . Later studies have generally followed this convention. However, terminants are sometimes more convenient than error integrals as in the smoothing of Stokes phenomenon.
13). The contributions of the other two saddle points will be proportional to the integrand evaluated at these saddle points: exp(−zφ(t = 2)) = exp(−32z), exp(−zφ(t = 3)) = exp(−27z) (101) Because the saddle point at t = 2 controls convergence, the smallest term in the asymptotic series for a given z will be O(exp(−32z)), so we would expect this to be the magnitude of the error in the optimallytruncated series in inverse powers of z. In reality, the superasymptotic error is dominated by the contribution of the saddle point at t = 3, which is O(exp(−27z)) and therefore larger than the smallest term in the optimally- truncated series by O(exp(5z)).