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Summation methods in the two- and three-body problems Zelmer, Graham Keith

Abstract

Let f(z) be a complex-valued function of the complex variable z which is regular at the origin, has radius of convergence R > 1 , and satisfies the condition f(1) = 1 . If we write [ equation omitted ]and [ equation omitted ], n=0,1,2,..., the matrix [ equation omitted ] leads to a summation method generally known as a Sonnenschein method. The utility of these methods lies in the fact that much can be said about them simply from a knowledge of the function f(z) . In the present work we are concerned with the three methods associated with the three functions f(z) = (1-r) + rz , f(z) = (1-r)+rz² and f(z) = (1-r)z + rz³ where r≠ 0 is an arbitrary complex parameter (generally, 0 < r < 1). The method based on the function (1-r) + rz leads to the well-known Euler-Knopp method which has already been extensively studied. We show that there exist simple recursion relations between the various rows of the matrix F and we make a detailed study of the domain Dᵣ on which these methods sum the geometric series [ equation omitted ].A more general sub-class of Sonnenschein methods called the (E,r,α,β) methods is then defined and some of the well-known theorems applicable to the (E,r) method are shown to hold for this sub-class. The practical application of the above three methods to the two- and three-body problems of classical mechanics forms the major portions of Chapters II and III . Much use is made in these chapters of a theorem, stated in Chapter I , which allows us to construct a domain Dr(a) on which one of the above methods sums an arbitrary function a(z) regular at the origin. On the limited Intervals for which the above methods are applicable, it is shown that they provide effective methods for obtaining the solution to the two-and three-body problems. Comparison is always made with similar results obtained by V. A. Brumberg and it is shown that the methods used here have certain advantages over his. The Sundman series for the three-body problem are also set up and utilized. Although the series are not very effective, the convergence is not as bad as is generally supposed. An interesting argument based on numerical evidence shows that the width of the strip about the real axis, in which the solution to the regularized three-body problem is known to be analytic, is not as narrow as Sundman's estimates give. Finally, a theorem is proved for the problem of the motion of a heavy rigid body about a fixed point showing that this problem can be treated in the complex plane in the same way as the two-body problem and regularized three-body problem.

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