Bengt Fornberg, Department of Applied Mathematics, University of Colorado Boulder
Euler-Maclaurin without analytic derivatives
We consider here the Euler-Maclaurin (EM) formulas in the context of approximating infinite sums. If the function to be summed can be integrated analytically, these formulas provide highly accurate asymptotic expansions for the difference between the sum and the corresponding integral.
Even modern text books routinely tell that the EM formulas usually become impractical if a large number of terms are needed in the expansions since these terms require a correspondingly large number of analytic derivatives. For most non-trivial functions, the algebraic complexity grows prohibitively fast with repeated differentiations. We will see that, with virtually no loss of accuracy, all the analytic differentiations can be entirely eliminated. This is without making any ‘excursions’ into the complex plane, in which case the Abel-Plana formulas offer additional opportunities.