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 Summation methods in the two and threebody problems
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Summation methods in the two and threebody problems Zelmer, Graham Keith
Abstract
Let f(z) be a complexvalued 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) = (1r) + rz , f(z) = (1r)+rz² and f(z) = (1r)z + rz³ where r≠ 0 is an arbitrary complex parameter (generally, 0 < r < 1). The method based on the function (1r) + rz leads to the wellknown EulerKnopp 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 subclass of Sonnenschein methods called the (E,r,α,β) methods is then defined and some of the wellknown theorems applicable to the (E,r) method are shown to hold for this subclass. The practical application of the above three methods to the two and threebody 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 twoand threebody 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 threebody 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 threebody 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 twobody problem and regularized threebody problem.
Item Metadata
Title 
Summation methods in the two and threebody problems

Creator  
Publisher 
University of British Columbia

Date Issued 
1967

Description 
Let f(z) be a complexvalued 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) = (1r) + rz , f(z) = (1r)+rz²
and f(z) = (1r)z + rz³ where r≠ 0 is an arbitrary complex parameter (generally, 0 < r < 1). The method based on the function (1r) + rz leads to the wellknown EulerKnopp 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 subclass of Sonnenschein methods called the (E,r,α,β) methods is then defined and some of the wellknown theorems applicable to the (E,r) method are shown to hold for this subclass. The practical application of the above three methods to the two and threebody 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 twoand threebody 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 threebody 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 threebody 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 twobody problem and regularized threebody problem.

Genre  
Type  
Language 
eng

Date Available 
20110919

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0302278

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.