Rigorous image-series expansions of quasi-static Green´s functions for regions with planar stratification

A novel image-series expansion scheme for quasi-static Green´s function in n+1 layered media is obtained by expanding the frequency-dependent Hertz potential in finite expansions and remainder terms. The expansions utilize a unique recursive representation for Green´s function, which is a generic characteristic of the stratification, and are explicitly constructed for n≤3. While results for 0≤n≤2 are given for reference only, the expansion scheme for a double-slab configuration, n=3, is quite general and outlines the procedure for n>3, without any increase in the complexity. The expansion-remainder terms can be made negligibly small for sufficiently large summation indices in the quasi-static limit, leading to rigorous image-series expansion. The image-series convergence is accelerated by including a collective image term, representing a closed-form asymptotic evaluation of the series-remainder integral. Thus, the proposed computational procedure can be used as a simple tool for producing analytical data for testing numerical subroutines applied to direct problems such as electrical simulation of muscles in the biomedical field and inverse problems, such as electromagnetic imaging.