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ACTA UNIVERSITATIS OULUENSIS Applying the trapezoidal assumptions asymptotic error consider the eigenvalues continuously differentiable denote derivatives have left differential equation 57 discontinuous weight function eigen eigenfunction eigenvalue problem eigenvectors equation 72 equation 84 equation b Kyix error term Euler-Maclaurin sum formula exists a positive exists an eigenvalue finite FINLAND follows from equation four are uniformly four derivatives four times continuously four times differentiable function Klx.t function pit group N streptococcus hence inner point integral equation integral operator interval a,b inverse matrix K(tjc Ky(x Let the number lim n2 lt follows matrix c(y(n natural numbers notations number k satisfies numerical integration obtain the following operator Kn partial derivatives piecewise differentiable positive constant Professor SAKARI PIHA right-hand limits satisfies the equation SCIENTIAE RERUM NATURALIUM SERIES A SCIENTIAE singular points slime forming group Studies on slime subintervals theorem Tienari trapezoidal rule uniformly continuous UNIVERSITATIS OULUENSIS SERIES University of Oulu VESA