The Euler-Maclaurin formula revisited

Abstract

The Euler-Maclaurin summation formula for the approximate evaluation of I = \int 0 1 f ( x ) d x comprises a sum of the form (1/ m )\sum j =0 m -1 f (( j + t ? )/ m ), where 0? t ? ? 1, a second sum whose terms involve the difference between the derivatives of f at the end-points 0 and 1 and a truncation error term expressed as an integral. By introducing an appropriate change of variable of integration using a sigmoidal transformation of order r ?1, (other authors call it a periodizing transformation) it is possible to express I as a sum of m terms involving the new integrand with the second sum being zero. We show that for all functions in a certain weighted Sobolev space, the truncation error is of order O (1/ m n 1 ) , for some integer n 1 which depends on r . In principle we may choose n 1 to be arbitrarily large thereby giving a good rate of convergence to zero of the truncation error. This analysis is then extended to Cauchy principal value and certain Hadamard finite-part integrals over (0,1). In each case, the truncation error is O (1/ m n 1 ). This result should prove particularly useful in the context of the approximate solution of integral equations although such discussion is beyond the scope of this paper.