TABLE OF CONTENTS Given in Sections 2 and 3 algorithms for calculating electrostatic fields and linear functionals of these fields, such as electrical capacitances, were reduced in these sections to calculating certain multiple integrals. Prom the point of view of numerical analysis one should integrate functions with at worst weak singularities. The numerical integration of such functions is a problem of independent interest. It has been discussed in detail for functions of one variable [21], [55], [54], but less is known about calculating multidimensional integrals of functions with weak singularities. The basic idea in the one-dimensional case is to integrate explicitly the singular part of the integer and thus to reduce the problem to the integration of a smooth function. This problem is well understood.
In the multidimensional case the first step in the above program was not discussed sufficiently.
In [10] optimal methods for calculating multidimensional integrals with weakly singular integrands are developed. These methods are presented in the Appendix.
In this chapter two problems of practical interest will be solved. First, the capacitances of circular metallic cylinders are tabulated. Secondly, the capacitances of metallic parallelepipeds of arbitrary dimensions are tabulated. In both cases there are no closed-form analytical solutions to the corresponding electrostatic problems, and the results are new. Special cases of these results, such as the capacitance of a cube, disk, or very long cylinder, will be compared with previously published results. The numerical results show that the formulas for calculating the capacitances, which have been derived in...
