Euler-MacLaurin formula

In Eq. (4.15). pg is the areal density of the solid substrate. The fluid-fluid perturbation is just the attractive term of the LJ pair potential. Likewise, the (original) fluid-substrate perturbation is the L.T attraction -4efs (cTfs/r) . To be consistent with the smooth-wall approximation to the reference potential, we average the fluid-substrate attractions over the (x, y) positions of the substrate atoms in the planes in which they lie. We suppose each substrate to comprise an infinite half-space of atomic planes, separated successively by distance d. Approximating the sum over these planes by the Euler-MacLaurin formula [37] yields the expression in Eq. (4.15). 

For the radial integration, we have adopted the Euler-Maclaurin formula proposed by Murray et al. [47], in which the radial points and weights are given by... 

The Euler-Maclaurin numerical quadrature formula for such an integral (using n -F 1 equally spaced values of y yi = i/n, for i = 0, n) is ... 

---