@article{Elliott98a,
   author =   {David Elliott},
   title =    {The Euler-Maclaurin formula revisited},
   journal =  {J.~Austral.\ Math.\ Soc.~B},
   volume =   40,
   pages =    {E27--E76},
   year =     1998,
   number =   {E},
   month =    nov,
   note =     {[Online] \protect\url{http://jamsb.austms.org.au/V40/E005} [2 Nov 
               1998]},
   abstract = {The Euler-Maclaurin summation formula for the approximate 
               evaluation of $ I=\int_0^1f\left( x\right) dx$ comprises a sum of 
               the form $\left( 1/m\right) \sum_{j=0}^{m-1}f\left( \left( 
               j+t_\nu \right) /m\right) $, where $0<t_\nu \leq 1$, 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\left( 1/m^{n_1}\right) $, 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. \par This analysis is then extended 
               to Cauchy principal value and certain Hadamard finite-part 
               integrals over $\left( 0,1\right)$. In each case, the truncation 
               error is $O\left( 1/m^{n_1}\right)$. 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.},
}