A geometric analysis of the Maxwell field in a vicinity of a multipole particle and a new family of special functions

A method of solving Maxwell equations in a vicinity of a multipole particle (moving along an arbitrary trajectory) is proposed. The method is based on a geometric construction of a novel trajectory-adapted coordinate system, which simplifies considerably the equations. The solution is given in terms of a series, where a new family of special functions arises in a natural way. Singular behaviour of the field near to the particle may be analyzed this way up to an arbitrary order. Application to the self-interaction problems in classical electrodynamics is discussed.