Electrostatic image method for the anisotropic half space

Electrostatic image theory, well known for a point charge in front of an isotropic dielectric half space, is generalised to the case where the half space may be anisotropic, characterised by a symmetric and positive-definite permittivity dyadic. The theory is derived through Heaviside operational calculus involving manipulations with pseudodifferential operators. As a result, the image of a point charge is seen to consist of a point charge at the mirror image point plus a continuous distribution of surface charge filling an angular sector the apex of which lies at the image point. The surface charge is shown to obey a simple analytic law. The angle of the image sector depends on the amount of transverse anisotropy of the half space. For decreasing anisotropy, the angle becomes small and the angular image can be approximated by a quadrifilar line charge coinciding with the result recently obtained by these authors for the case of small anisotropy. Numerical tests confirm the validity of the image expression. The result can be readily applied to a wide variety of electrostatic problems involving sources and obstacles in front of an anisotropic half space, making a numerical development of a Green function for the anisotropic half space obsolete