Matrix of moments of the Legendre polynomials and its application to problems of electrostatics

• Published in: Computational mathematics and mathematical physics Vol. 57; no. 1; pp. 175 - 187
• Main Author: Savchenko, A. O.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 2017; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Electric fields; Hypotheses; Integral equations; Linear equations; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrices (mathematics); Matrix; Matrix methods; Nonuniform; Physics; Polynomials; Studies; Theorems; ball; charge; electric field; Legendre polynomials; matrix of moments; dipole moment; potential
• ISSN: 0965-5425; 1555-6662
• DOI: 10.1134/S0965542517010134

In this work, properties of the matrix of moments of the Legendre polynomials are presented and proven. In particular, the explicit form of the elements of the matrix inverse to the matrix of moments is found and theorems of the linear combination and orthogonality are proven. On the basis of these properties, the total charge and the dipole moment of a conducting ball in a nonuniform electric field, the charge distribution over the surface of the conducting ball, its multipole moments, and the force acting on a conducting ball situated on the axis of a nonuniform axisymmetric electric field are determined. All assertions are formulated in theorems, the proofs of which are based on the properties of the matrix of moments of the Legendre polynomials.