Asymptotic Behavior of Two Series Used for the Evaluation of Kirchhoff Diffractals

Use of the Kirchhoff approximation allows expressing the electromagnetic power density scattered by fractal surfaces via two series expansions. Unfortunately, practical use of these two series may be problematic mainly due to the limitations imposed by the finite precision of the computers. In particular, it has been shown that to noticeably increase the region of practical applicability of such two series expansions, use of the theory of asymptotic expansions is strictly required. However, although in the literature it is usually assumed that the aforementioned two series can be seen as asymptotic expansions of the scattering integral, a rigorous proof of this has not been provided yet, and it represents the aim of this work.