Formal representations of complex functions, means on regular n-gons and relative equilibria.
by
Dominique Bang
CFTC, Lisboa (Portugal)

Consider n vortices (with same vorticity) at the vertices of a regular n-gon. The previous system generates a potential that can be easily written in a compact way, using the special morphism structure of the logarithm. In the case of n equal masses with same location than previous vortices, Lindow built in 1924 an integral formula in order to express the corresponding Newtonian potential. To generalize those formulas to a larger class of potential, we give new representations of functions of the complex variable (especially modulus of holomorphic functions) and use them to express in a formal way (using a scalar product) their mean on the regular unit n-gon. In the case of homogeneous potential, explicit representations are given and we conclude this talk with some applications to relative equilibria composed by nested regular n-gons.

Date received: June 12, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calt-05.