The author bases this paper on the conventional Euler-Maclaurin formula. The thrust of this paper is the evaluation of Cauchy principal value (CPV) integrals and for certain Hadamard finite part integrals (FPI). To this end he includes the extra term, introduced originally by the reviewer, for the CPV and differentiates the expansion, with respect to an incidental parameter, to obtain corresponding results for an FPI. He shows that the sigmoidal transformations (aka periodising transformations) are helpful in this context, and obtains discretization error estimates valid for functions belonging to a specified Sobolev space. The author points out that these results should prove particularly useful in the context of the solution of integral equations. Although this paper appears in the journal immediately before a companion paper [the author, ibid. Ser. B 40, No. E, E77--E137 (1998; reviewed below)] on sigmoidal functions, it should clearly be read after the companion paper.