Representations of complex functions, means on the regular n-gon and applications to gravitational potential

We present a method to analytically compute means of functions on regular n-gons and to study cyclic quantities of the complex variable. To achieve this, we construct representations of complex functions and compact expressions of their mean based on the use of a scalar product. Applied in the field of celestial mechanics, this method leads to results concerning gravitational potential and relative equilibrium composed by pested polygons.