Optimal Approximation of the Electrostatic Capacity Matrix of Two Conducting Spheres by a Short-Distance Asymptotic Expansion

An integral representation for the electrostatic capacity matrixC=[c_ij]_i,j=1,2 of two conducting spheres of radii R₁, and R₂is obtained. A short-distance asymptotic expansion is then derivedand its approximation properties for fixed (surface) distancer between the spheres are investigated. An error function is defined for c_ij(r) and its nthorder asymptotic approximant it has the property following from the divergence of the expansion, and thereby shows thatthe optimal approximation of c_ij(r) is achieved by an approximantof finite order n = n_ij(r) depending possibly on r and the indicesi,j. The value gives the quality of approximation of c_ij by the asymptotic expansion for a givendistance r between the spheres. The point sets and are introduced in order to describe the distance ranges where c_ij can be approximatedwithin a given error >0 by an asymptotic approximant of given order n, or at least by theoptimal approximant, respectively. The optimal order n_ij(r)and the -approximation sets and D() are investigated numerically.