Finite-part Integrals and the Euler-maclaurin Expansion

The context of this note is the discretization error made by the m-panel trapezoidal rule when the integrand has an algebraic singularity at one end, say x = 0, of the nite integration interval. In the absence of a singularity, this error is described by the classical Euler-Maclaurin summation formula, which is an asymptotic expansion in inverse integer powers of m. When an integrable singularity (x with > 1) is present, Navot's generalization is valid. This introduces negative fractional powers of m into the expansion. Ninham has generalized this result to noninteger satisfying < 1. In this note, we extend these results to all by providing the nontrivial extension to negative integer. This expansion diiers from the previous expansions by the introduction of a term log m.