The correlation functions of plane polygons

Abstract

The correlation function of a bounded plane figure of arbitrary shape and its derivatives are studied using their integral expressions. From these follows that the correlation function and its first derivative are continuous functions, while the second and the higher order derivatives can show finite and/or algebraic singularities [with exponents equal to −(n+1/2) and n∊Z¯+] in the presence of some well defined geometrical features of the figure boundary. The limit values of the first, second, and third derivatives at the origin are obtained. They are similar to those of the three dimensional case. In the case of a plane polygon of arbitrary shape the correlation function has a closed analytical form essentially equal to the sum of the values taken by two analytically known functions at appropriate sets of points. Two simple cases are explicitly worked out for illustration.