Associated homogeneous distributions (AHDs) with support in the line R are the distributional generalizations of one-dimensional power-log functions. In this paper, we derive a number of practical structure theorems for AHDs based on R and being complex analytic with respect to their degree of homogeneity in some region of the complex plane. Each theorem gives a representation that is designed to have a distinct advantage for calculating either convolution products, multiplication products, generalized derivatives and primitives, Fourier transforms or Hilbert transforms of AHDs. Copyright © 2008 John Wiley & Sons, Ltd.