The Lagrange -point formula is
and follows from the differentiated form of (3.3.4). The are the differentiated Lagrangian interpolation coefficients:
where is as in (3.3.10).
If is continuous on the interval defined in §3.3(i), then the remainder in (3.4.1) is given by
where and .
For the values of and used in the formulas below
where is defined by (3.3.12), with numerical values as in §3.3(ii).
If can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii))
where is a simple closed contour described in the positive rotational sense such that and its interior lie in the domain of analyticity of , and is interior to . Taking to be a circle of radius centered at , we obtain
The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2).
, . The integral (3.4.18) becomes
With the choice (which is crucial when is large because of numerical cancellation) the integrand equals at the dominant points , and in combination with the factor in front of the integral sign this gives a rough approximation to . The choice is motivated by saddle-point analysis; see §2.4(iv) or examples in §3.5(ix). As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands.
For partial derivatives we use the notation .
The results in this subsection for the partial derivatives follow from Panow (1955, Table 10). Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives.
For additional formulas involving values of and on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546).