Sundman's series and related problems

Abstract

One of the topics of the international competition opened in 1885 for the prize of Sweden's King Oscar II was,as it is well known, to find the solution of the n-body problem of the celestial mechanics by means of power series uniformly convergent for all values of t, Later works and difficulties fastened the idea that the solution of this problem for n≥ 3 could not bo attained without the introduction of now and complicated trascendental functions. However,in 1912, Karl Sundman, Head of Helsingsfors's Observatory for many years, gave the exact solution of the three-body problem, not certainly in the sense of that of Euler, Lagrange and others, but in the sense of the theory of analytical functions, with the only tool of known theorems of mathematical analysis and without introducing any new special function.