Euler-Maclaurin Expansions for Integrals over Triangles and Squares of Functions Having Algebraic/Logarithmic Singularities along an Edge

We derwe and analyze the properties of Euler-Maclaurin expansions for the differences / ~ / s'(log.~) " /(.~. Qilfj is a combination of one-dimensional generalized trapezoidal rule approximations. 1. ~NlKOL)UC110N In this work we are intcrcstcd in deriving Euler-Maclaurin expansions for the singular double integrals where W(X) = x'(Iog x)'. s >-l.s'=O. 1. (1.3) and f(.~,~l) is as many times differentiable as needed. Specifically we are looking for asymptotic expansions. as h + O+, for the differences d,,lf'I = QISI ~ Q,lfl and di,lfl = Q'IJI-QAl./"l. where Q,,lfl and Q61fl are approximations to QlSl and Q'\f\, respectively, obtained as some combinations of one-dimensional generalized trapezoidal rule approximations with step size h. We now state some results which bear relevance to our derivation of the Euler-Maclaurin formulas for Q/f I and (2' [ .f 1.