potential theory and analytic properties of self-similar fractal and multifractal distributions

By the use of recursion relations and analytic techniques we deduce general analytic results pertaining to the electrostatic potential, moments, and Fourier transform of exactly self-similar fractal and multifractal charge distributions. Three specific examples are given: the binomial distribution on the middle-third Cantor set, which is a multifractal distribution, the uniform distribution on the Menger sponge, which illustrates the added complication of higher dimensionality, and the uniform distribution on the von Koch snowflake, which illustrates the effect of rotations in the defining transformations.