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Summation methods in the two- and three-body problems Zelmer, Graham Keith 1967

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THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE  DEGREE OF  DOCTOR OF PHILOSOPHY of GRAHAM KEITH ZELMER B.Sc. Mech. Eng. U n i v e r s i t y of Manitoba, I 9 6 0 B.Sc. Hons. Math. U n i v e r s i t y of Manitoba, 1962 TUESDAY, May 9 , 1 9 6 7 , a t 1 0 : 3 0 a.m. I n Room 104, Mathematics B u i l d i n g . COMMITTEE IN CHARGE Chairman: K. C. Mann C. Clark P. B u l l e n A. H. Cayford G. White R. Westwick M. Sion E x t e r n a l Examiner: Dr. A. D e p r i t Boeing S c i e n t i f i c Research Laboratory S e a t t l e , Washington. Research Supervisor:  E. Leimanis.  SUMMATION METHODS IN THE TWO- AND THREE-BODY PROBLEMS Abstract Let  f(x)=  E f \ z be a f u n c t i o n which k=o i s a n a l y t i c a t the o r i g i n w i t h a r a d i u s of convergence R > 1 and w i t h f ( l ) =1 . k  K  f (z) ( f " 1 » ofc " °. k=o k > 0; f = f ) . The matrix F = ( f ) i s known as a Sonnenschein summation matrix. I f f ( z ) i s e n t i r e the summation method based on t h i s matrix sums the geometric s e r i e s i n the domain D = [z : | f ( z ) | < 1} . Let  n  f  0  l  k  0  k  n k  In t h i s t h e s i s we consider the methods corresponding to the f u n c t i o n s f ( z ) = ( l - r ) + r z , f ( z ) = (1-r) + r z and 2  f ( z ) = ( l - r ) z + rz- , 0 < r < 1 . The summation method f o r f ( z ) = (1-r) + r z i s the well-known Euler-Knopp or ( E , r ) method. Simple r e c u r s i o n r e l a t i o n s are developed f o r the various rows of the matrices F obtained from the f u n c t i o n s f ( z ) and the domain D i s described i n d e t a i l . More g e n e r a l l y , the (E,r,a,p) methods are defined corresponding to the-func-tions f-(z) = ( l - r ) z + r z , a,p p o s i t i v e i n t e g e r s . Some theorems holding f o r the ( E , r ) method <=(E,r,0,l) 5  a  a + P  method) are g e n e r a l i z e d to t h i s c l a s s . Let  a(z) =  E a, be an k=o a r b i t r a r y f u n c t i o n , a n a l y t i c a t the o r i g i n . A well-known theorem i n a n a l y t i c f u n c t i o n theory allows the c o n s t r u c t i o n of a domain D(a) i n which a g i v e n summation method sums t h i s s e r i e s , from a knowledge of the domain D i n which that method sums the geometric s e r i e s . I n Chapter I I a ( z ) i s taken as the s o l u t i o n to the two-body problem and the summation methods used are those corresponding to the three f u n c t i o n s f ( z ) above. I f the parameter r i s chosen p r o p e r l y , the domain D(a) w i l l c o n t a i n the i n t e r v a l [0,TT] . The above three methods are shown to be p r o g r e s s i v e l y more e f f e c t i v e i n o b t a i n i n g the s o l u t i o n on [0,7r] and, i n p a r t i c u l a r , are a l l much more e f f e c t i v e than a technique used by V. A. Brumberg (1964). K  I n Chapter I I I these methods are a p p l i e d w i t h more l i m i t e d success to the r e g u l a r i z e d three-body problem. An i n t e r e s t i n g numerical r e s u l t i s derived there concerning the width of the s t r i p about the r e a l a x i s I n which the s o l u t i o n to the r e g u l a r i z e d three-body problem i s a n a l y t i c . F i n a l l y , a theorem i s proved concerning the problem of the motion of a heavy r i g i d body  about a f i x e d p o i n t , showing that i t can be treated with summation techniques i n the same manner as the problems above. GRADUATE STUDIES P o i n t Set TopologyAlgebra Functional Analysis Theory of Functions of a Real V a r i a b l e D i f f e r e n t i a l Equations Theory o f Functions  P. S. B u l l e n H. A. Thurston c. W. Clark D. B r e s s l e r R. R. D. Kemp R. Cleveland.  PUBLICATIONS "On Q u a s i - M e t r i z a b i l i t y " , Canadian Math. J . (to appear) ( i n conjunction w i t h Dr. M. S i o n . ) .  SUMMATION METHODS IN THE TWO- AND THREE-BODY PROBLEMS  by  GRAHAM KEITH ZELMER B.Sc. (Eng.), U n i v e r s i t y of Manitoba, i960 B.Sc. (Hons. Math), U n i v e r s i t y of Manitoba, 1962  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS  We accept t h i s t h e s i s as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1967  In p r e s e n t i n g for  an a d v a n c e d  that  thesis  shall  I further  agree  for scholarly  Department  o r by h i s  publication  without  thesis  degree  the Library  study„  or  this  Department o f  at the U n i v e r s i t y make  AJU-S  it  that  freely  thesis  Mathematics  </,  /ft?  Columbia  for  It  requirements  Columbia,  I  reference  and  for extensive  for financial  permission.  o f the  of British  available  permission  representatives.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada Date  fulfilment  p u r p o s e s may be g r a n t e d  of this  my w r i t t e n  in p a r t i a l  copying of  agree  this  by t h e Head o f my  is understood gain  shall  that  copying  n o t be a l l o w e d  -iiABSTRACT Let  f ( z ) be a complex-valued f u n c t i o n of the  complex v a r i a b l e  z  which i s regular at the o r i g i n ,  has radius of convergence  R > 1 ,  and s a t i s f i e s the if  condition f (z) n  =  2  f(l)= 1 . f  n f c  z  k  ,  I f we w r i t e  f ( z ) = S f^z  n=0,l,2,..., the matrix  and  P =( f ^ )  leads t o a summation method g e n e r a l l y known as a Sonnenschein method.  The u t i l i t y of these methods l i e s  i n the f a c t that much can be s a i d about them simply from a knowledge of the f u n c t i o n  f(z).  I n the present  work we are concerned with the three methods associated with the three functions  f(z) = ( l - r ) + rz , f(z) = (l-r)+rz  -7.  and  f(z) = (l-r)z + r z  y  where  complex parameter ( g e n e r a l l y , based on the f u n c t i o n  r ^ 0  i s an a r b i t r a r y  0 < r < l).  (1-r) + r z  The method  leads t o the w e l l -  known Euler-Knopp method which has already been e x t e n s i v e l y studied.  We show that there e x i s t simple  recursion  r e l a t i o n s between the various rows of the matrix we make a d e t a i l e d study of the domain  D  these methods sum the geometric s e r i e s  2 z  F  and  on which  r k  » A  more general sub-class of Sonnenschein methods c a l l e d the  (E,r,a,p)  methods i s then defined and some of  the well-known theorems a p p l i c a b l e t o the are shown to hold f o r t h i s  sub-class.  (E,r) method  -iii-  The p r a c t i c a l a p p l i c a t i o n of the above three methods to the two- and three-body problems of c l a s s i c a l mechanics forms the major portions of Chapters I I and I I I . Much use i s made i n these chapters of a theorem, stated i n Chapter I , which allows us t o construct a domain D ( a ) r  on which one of the above methods sums an a r b i t r a r y function  a ( z ) regular a t the o r i g i n .  On the l i m i t e d I n t e r v a l s f o r which the above methods are a p p l i c a b l e , i t i s shown that they provide e f f e c t i v e methods f o r o b t a i n i n g the s o l u t i o n t o the two-and threebody problems.  Comparison i s always made with s i m i l a r  r e s u l t s obtained by  V. A. Brumberg and i t i s shown  that the methods used here have c e r t a i n advantages over his. The Sundman s e r i e s f o r the three-body problem are a l s o set up and u t i l i z e d .  Although the s e r i e s are not  very e f f e c t i v e , the convergence i s not as bad as i s g e n e r a l l y supposed.  An i n t e r e s t i n g argument based on  numerical evidence shows that the width of the s t r i p about the r e a l a x i s , i n which the s o l u t i o n t o the r e g u l a r i z e d three-body problem i s known t o be a n a l y t i c , i s not as narrow as Sundman's estimates give. F i n a l l y , a theorem i s proved f o r the problem  of the  motion of a heavy r i g i d body about a f i x e d point showing that t h i s problem can be treated i n the complex plane i n  -iv-  the same way as the two-body problem and r e g u l a r i z e d three-body problem.  -V-  TABLE OF CONTENTS INTRODUCTION CHAPTER  CHAPTER  CHAPTER  I  Sonnenschein Methods. 1.1  S u m m a b i l i t y and A n a l y t i c  Continuation.  1.2  S o n n e n s c h e i n M a t r i c e s and Construction.  1.3  Certain Explicit  1.4  The Methods  their  Methods.  (E,r,a,p).  A p p l i c a t i o n t o t h e Two-Body  II  Problem.  2.1  The Two-Body  2.2  The (E,r) Method i n t h e Two-Body Problem; Brumberg's R e s u l t s .  2.3  A p p l i c a t i o n o f t h e Methods  2.4  Analytic  III  and  Problem.  Continuation to  [0,7r].  A p p l i c a t i o n t o the Three-Body  3.1  The T h r e e - B o d y P r o b l e m ; Method.  3.-2  Limitations Methods.  3.3  The P a r a m e t e r s  BIBLIOGRAPHY  (E,r,0,2)  (E,r,l,2).  Brumberg's  and R e s u l t s f o r T  and  Problem.  8.  Sonnenschein  -vi-  A CKNOWLEDGEMENT The a u t h o r w o u l d l i k e to his  supervisor,  t o express h i s  D o c t o r E. L e i m a n i s , f o r  t h e p r o b l e m and f o r p r o v i d i n g e n u m e r a b l e and h e l p f u l  gratitude  suggestions.  suggesting  references  A debt of g r a t i t u d e  Is  o w i n g t o D o c t o r P. B u l l e n who t o o k t i m e t o a d v i s e some o f t h e a r g u m e n t s and t o p r o o f - r e a d t h e and t o M i s s F e r n F u l t o n who u n d e r t o o k t h e An N . R . C .  Studentship h e l d d u r i n g the l a s t  is also g r a t e f u l l y  acknowledged.  also on  manuscript,  typing. three  years  Summation Methods i n t h e Two-and T h r e e - B o d y  Problems  Introduction The m o t i v a t i o n f o r by V. A . B r u m b e r g [ l ] for  this  comes f r o m a r e c e n t  i n w h i c h he a t t e m p t e d t o g e t  the two-and three-body problems  valid  for  all  subject  to practical  technique  It  nk  o  f  a  matrix.  n  i n  limitations.  i n i - f c e  With t h i s  matrix  parameter.  the Taylor  of the s o l u t i o n functions ^  G = (g  n k  )  and  li)  Brumberg's  entries  known as a s u m m a t i o n  i n f o r m a t i o n one i s a b l e t o c a l c u l a t e  parameter used, converges  to the s o l u t i o n f o r  Independent  all  r e a l values •  w h i c h a r e more l i m i t e d  i n nature,  i.e.  to consider  matrices  f o r which the  o f p o l y n o m i a l s above c o n v e r g e s t o t h e s o l u t i o n f o r a l l of the parameter w i t h i n a c e r t a i n f i n i t e it  Is  shown t h a t  fairly  large  methods. f(z)  ,  matrix  a  parameter.  The i d e a b e h i n d t h e p r e s e n t w o r k i s G  course,  series the  sequence o f p o l y n o m i a l s w h i c h , d e p e n d i n g o n t h e  of t h i s  mechanics  In order to apply i)  paper  solutions  extremely ambitious and, of  i s n e c e s s a r y t o know  coefficients  of c l a s s i c a l  r e a l values of the independent  Such a p r o g r a m h o w e v e r i s  g  thesis  these matrices  class  interval.  o f s u m m a t i o n methods known as  r e g u l a r a t the o r i g i n , from which the is derived.  values  I n Chapter  correspond t o c e r t a i n of a  A s s o c i a t e d w i t h e a c h s u c h method i s a  G  sequence  Sonnenschein function  corresponding  F o r t h e most p a r t we s h a l l  take  I  -2f(z)  = (1-r)  where, It  + rz  ,  f(z)  i n general,  r  = (1-r)  + rz  + rz  or  f(z)  = (l-r)z  i s a l l o w e d t o assume any c o m p l e x  i s pointed out t h a t  f(z)  = (1-r)  is  t h e S o n n e n s c h e i n method  f(z)  the well-known Euler-Knopp or  do n o t seem t o h a v e b e e n d i s c u s s e d  i n the l i t e r a t u r e  to  and HE a r e i n v o l v e d w i t h t h e i r body problems r e s p e c t i v e l y . interval  effective  of  two anywhere  I , Chapters  a p p l i c a t i o n t o the two-and It  w i l l be shown t h a t o n  II three-  their  c o n v e r g e n c e t h e s e methods c a n be more  than Brumberg's.  t o apply since the m a t r i x Finally,  (E,r)  date.  H a v i n g o u t l i n e d t h e methods i n C h a p t e r  limited  value.  for  method w h i l e t h e methods c o r r e s p o n d i n g t o t h e o t h e r functions  + rz^  the p r a c t i c a l  Moreover, G  limit  o u r methods a r e  i s much s i m p l e r  to  evaluate.  t o B r u m b e r g ' s m e t h o d does  seem t o e x t e n d t o o f a r b e y o n d o u r l i m i t e d  easier  interval  of  not convergence.  -3CHAPTER I S o n n e n s c h e i n Methods 1.1  S u m m a b i l i t y and A n a l y t i c The n o t i o n s of generalized  Continuation  o f s u m m a b i l i t y and t h e r e l a t e d  sum and g e n e r a l i z e d l i m i t  detailed  treatment  i n Cooke  [l]  ,  as w e l l as  o f many s u m m a t i o n ^ m e t h o d s a r e  Dienes  [ l ] , Hardy  ideas  let  us b r i e f l y  {s )  an i n f i n i t e  known as t h e F - t r a n s f o r m  t  (1.1) (Assume t h e  n  =2f  index  n k  k  specified.)  divergent the l i m i t  is  main  F =  (i*^)  m a t r i x t h e sequence  {t J  of  s  s  n  s  called  -* s ( F ) x  .  (1^0,1,2,•••) .  conditions  be r e g u l a r  are  0  £ ) s  n  its  to  •  unless  converges whenever  n  J  n  f o r m a l l y d e f i n e d by  k  {t }  and i f w  e  c  a  n  t  -  s  generalized l i m i t  (s ) n  for  some  F  sequence  by  F  the  matrix  well-known necessary  on a m a t r i x  other-  assign to t h i s  I n such a c i r c u m s t a n c e  s a i d t o be r e g u l a r ;  sufficient  is  n  runs from  If  sequence  sequence and  {s J  d o e s , and t o t h e same l i m i t ,  F  as  For  r e v i e w some o f t h e  i s an a r b i t r a r y  n  (n,k=0,l,2,•••)  write  expounded  below. If  wise  the  [1] and Knopp [ l ]  w e l l as i n t h e w o r k s o f many o t h e r a u t h o r s . completeness,  concepts  and we  and  i n order that  it  -4i) (I)  E l nk' f  ii) f iii)  n k  Sf  < M  ' independent of n ,  f o r e a c h  n  wh<3re  M  i s  -* 0 as n -• » for each k , and n k  - 1 as n - °° .  Again, i f 2 u^ is an arbitrary sequence and G = (g^) (n,k=0,l,2,. ..) an infinite matrix, the sequence {t ) in this case is formally defined by n  (1.2)  t =2 n  u  (n=0,l,2,...) .  k  If Ct_} converges whenever E u. does having a limit equal to the sum of the series and i f t -» s n for some divergent' series 2 u , we can assign to this series the sum s called its generalized sum by G and write E u = s(G) . As above, G is called regular. Necessary and sufficient conditions for a matrix of this type to be regular are fc  fe  i) (II) ii)  £ |g - S  for each n , where M is independent of n , -» 1 as n.-» .» for each k . nk  l <  M  n k + 1  -5More fully the matrices F and G above are referred to as regular sequence-to-sequence and seriesto-sequence summation matrices respectively and the two notions are not unrelated. Thus from any regular sequenceto-sequence matrix F = ( f ^ ) one can obtain a regular series-to-sequence matrix G = (§k;) y putting b  n  g  00  nk  = S f m=k  , while from any regular matrix G = (g^)  the matrix F » ( f ^ )  with  = g^ - g ^ ^  will also  be regular i f , in addition, lim lim g , = 0 . If G provides a generalized sum for the series Eu^ , F will provide a generalized limit for the sequence {s } n  of partial sums of this series and vice-versa. The above ideas are utilized in analytic function theory as follows: Let a(z) be an arbitrary function of the complex variable z which is analytic at the origin so that (1.3)  a.(z) = 2 a z . k  k  If a regular series-to-sequence summation matrix G is applied to series (1.3) (here u = ) there results a  k  z k  k  a sequence of functions {t (z)} with n  -6-  (1.*)  t (z)  = E g  n  and, i f  the radius  we h a v e |z|  *  < R .  n  a z  k  (n=0,l,2,...)  k  f e  o f convergence o f  ( z ) -» a ( z )  at least for a l l  But i n f a c t  this  domain t h a n t h a t g i v e n by  limit |z|  circle  of  convergence,  < R .  o f a means o f a n a l y t i c a l l y It  that  is  z  R , satisfying  may h o l d on a  t o a representation of the f u n c t i o n its  (1.3)  ,  larger  Thus we a r e a(z)  led  outside  i s , we a r e i n  possession  c o n t i n u i n g the f u n c t i o n  a(z)  i s worthwhile n o t i n g t h a t very o f t e n the m a t r i x  G  w i l l be r o w - f i n i t e  ( e a c h r o w has f i n i t e l y many n o n - z e r o  entries).  case t h e e x p r e s s i o n s  In this  polynomials,  a r e s u l t which i s  We a l s o n o t e a t  w h i l e most m a t r i c e s  G  By t h i s  this is  is  this point  t h e r e do e x i s t m a t r i c e s  e x i s t s and c o n v e r g e s t o circle  a(z)  on a d o m a i n  To i l l u s t r a t e  Inside  [2],  the c i r c l e  of  [ 3 ] ; Agnew [ l ] )  .  G  series.  to  convergence.  t h e above i d e a s c o n s i d e r t h e (E,r)  are  (1.4)  o f convergence of the T a y l o r  known E u l e r - K n o p p , o r  that  extending  I n t h e s e c a s e s h o w e v e r t h e sequence may f a l l converge a t p o i n t s  for  restriction.  w h i c h a r e n o t r e g u l a r b u t f o r w h i c h sequence  beyond the  all  which are used i n p r a c t i c e  i n g e n e r a l an u n n e c e s s a r y  implied that  are  obviously desirable  p r a c t i c a l purposes.  regular,  (1.4)  s u m m a t i o n method  The c o r r e s p o n d i n g  well(Knopp  matrices  .  -7P  ( ) r  ( nk^ ^  =  f  r  parameter  (1.5)  r^O ^ ( r )  a  n  d  G  ( ) r  =  (g K:^ ^ r  n  depend on a  and a r e d e f i n e d b y = ( )  r (l-r) "  n  k  n  (k=0,l,... ,n)  k  =0  (k > n)  and  (1.6)  g  n  k  (r)  =  Q  E  r (l-r) m  n  (k=0,l,...,n)  m  m=k  =0 The m a t r i c e s For  r=l  (k > n)  are r e g u l a r  F(l)  Is the lower  is  if  and o n l y i f  0. < r  the i d e n t i t y matrix while  . < 1 ~G(l)  semi-matrix  /  (1.7)  10 0 1 1 0  \ and t h u s t h e method i s e q u i v a l e n t (usually For  considered a regular a(z)  so t h a t  series  E z  Then i f  k  F(r)  .  to i t s  sequence  let  becomes t h e g e o m e t r i c  G(r)  is  applied to this  sequence o f p a r t i a l n  where  convergence  method).  us c o n s i d e r t h e f u n c t i o n  (1.3)  {g (z)}  to ordinary  (l-z) "*" -  series series  sums) we g e t  the  (or  -8(1.8)  g (z) = l - « [ f c r } + » ]  n  n  which w i l l converge a t a l l p o i n t s Inside the c i r c u l a r domain (1.9)  D  r  to the f u n c t i o n  = iz  : Iz-tl-r" )! 1  (l-z)  < M" } 1  . Note that the boundary of  - 1  z=l and that f o r  D  r  always passes through  r$(0,l],  D  r  does not contain the u n i t d i s k and consequently  G(r) (and P(r) as w e l l ) cannot be r e g u l a r . We conclude t h i s s e c t i o n with a lemma and a theorem which r e l a t e the domain on which a given summation method with matrix  G  sums the geometric s e r i e s t o the a ( z ) . Let  domain on which i t sums an a r b i t r a r y f u n c t i o n the former domain be g (z) n  D . Thus i f G = ( g ^ ) and i f  i s defined by g (z) =  (1.10)  we have  n  2  g  S ( ) ~* ( l - ) ~ ^ * z  z  n f c  z  a s  (n=0,l,2,...)  k  n  n  "*  00  f ° each r  zeD .  Let us f u r t h e r assume i n a l l that f o l l o w s that s t a r - l i k e with respect t o the o r i g i n . function the domain  D is  Consider the  a ( z ) given by ( 1 . 3 ) and construct f o r a ( z ) D(a) as f o l l o w s :  -9-  Por each  9 ,  0 _< 8 < 2TT ,  let  r  Q  = lim inffr  : r  .  = »  > 0 ,  i9 z = re  a singular point  of  a(z))  Take  r^  i8 if  z = re  For a l l  i s always a r e g u l a r p o i n t  9  f o r which  D(  (1.11) and f i n a l l y  = {z  define  Is f i n i t e  0  :  z = z'-re  a(z)  define ,  1 9  easy t o  with respect  9  by  D(a)  check t h a t  star  principal  star  of  is  a(z)  half-lines  of  (l-z)"  A(a)  will  D(a)  t o the o r i g i n .  principal  D(a)  a(z)  .  < •}'  Q  a l s o be  Moreover, ,  1  of  if  will  f r o m t h e complex  the  outward t o i n f i n i t y  a(z)  are  deleted  [ l , p.  W i t h t h e n o t a t i o n as above l e t D(a).  Then t h e r e e x i s t s  closed r e c t i f i a b l e  containing  the  plane.)  a compact s u b s e t o f  A(a)  star  of  We now h a v e t h e f o l l o w i n g lemma ( H i l l e  (simple  the  (The p r i n c i p a l  originating at a singular point  LEMMA 1  is  D be  .  star-like  t h a t d o m a i n w h i c h r e m a i n s when a l l  and e x t e n d i n g r a d i a l l y  by  D( )  9  is  .  z>€D}  D(a) = n{D^ ^: 0 < 8 < 2ir , r  (1.12)  It  9)  r  of  0  and  orientable A  in its  be  A a  "scroc"  curve)  interior  70]):  in such  -10that fact  f o r each  zeA  and  CeC^  K = {uu : uu = z/C ,  subset of  zeA  the p o i n t , C ^}  D .  It  constitutes  t h a t found i n H i l l e THEOREM 1  i n a domain  [ l , p.  [ 1 , p.  t o the l i m i t  D which i s  while  190]  G  (l-z)"  sums  in  f o r every  1  a(z) z  on compact PROOF.  the  f o r each  ,  t o the  of function  uniformly with respect  F o r any compact s e t  A c D(a)  p r o v i d e d b y Lemma I  integral  zeA  D(a)  (1.10)  o n compact s u b s e t s (1.3)  the  to  subsets.  "scroc"  Cauchy's  z  a l s o sums t h e s e r i e s  i n the star  z  s t a r - l i k e w i t h respect to  is uniform with respect to G  error.  the  o r i g i n and t h a t t h e c o n v e r g e n c e o f t h e sequence  Then  that  the  seems s l i g h t l y  71]  Suppose t h e m a t r i x  geometric series  the work  a generalization of  c o r r e s p o n d i n g one f o u n d i n H a r d y  D .  In  i s a compact  6  The f o l l o w i n g t h e o r e m i s b a s i c f o r follows.  z/£eD .  .  formula  .  determine  Since  c  A(a),  gives  Consider the f u n c t i o n  t  (z)  d e f i n e d by  -11-  (1.14)  where  t ( z )=  J Mil g ( C  n  g  n  n  I s given by (1.10).  s i n c e b y t h e lemma  z/CeD  z€D ,  i t s radius  g r e a t as lies  Replacing  exists  c A(a) .  S ( ) "* ( l - ) z  z  Now ( 1 . 1 0 ' ) _  1  g (z/£) n  Thus  K  by series  (1.10) signs  and r e c a l l i n g t h a t  g (z/C). n  i n (1.14)  ,  inter-  ( w h i c h t h e above a, = &  d  f SJSl dC  r w  l  J n  f  fc+1  obtain t  i.e.  r  i n Lemma 1  o f convergence o f  c h a n g i n g s u m m a t i o n and i n t e g r a l  we f i n a l l y  o  f  n  : z€D} .  to the circle  argument a l l o w s )  (n=0,l,2,.-)  o f c o n v e r g e n c e must b e a t l e a s t as  l i m sup{|z|  interior  )dC  This i n t e g r a l  and  i s a T a y l o r s e r i e s and s i n c e  z / C  (z) = Z a  a z  we o b t a i n sequence ( 1 . 4 ) .  (n=0,l,2,...)  k  However,  from  (1.13)  and ( 1 . 1 4 ) we g e t t h e e s t i m a t e  |a(z)-t ( )| < 2 J j - f l ^ - ) I I (1-z/C) -8y;z/c)l<JC 5  (1.15)  n  _1  z  °A <  M.max  Kl-z/C)" -g (z/C)l  since  C  i s bounded away f r o m z e r o and  on  .  But the points  1  n  a(c)  i s bounded  z/C l i e i n t h e compact s e t K  -12-  and t h u s f o r  sufficiently  c a n be made a r b i t r a r i l y 1.2  large  n  s m a l l , which proves the  S o n n e n s c h e i n M a t r i c e s and t h e i r If that it  two p r o p e r t i e s . Individual  t o be u s e d  calculations  w o u l d be d e s i r a b l e  if  it  Firstly,  Secondly,  following  one w o u l d hope t h a t F  or  G  method c o u l d be e a s i l y c a l c u l a t e d , e i t h e r or recursively.  practically,  i n computing machines,  possessed the  rows o f t h e m a t r i x  it  without  t o o much d i f f i c u l t y ,  domain  D  theorem.  Construction  any s u m m a t i o n method i s  is for actual  t h e above maximum  the  used i n  the  explicitly  w o u l d be d e s i r a b l e one c o u l d o b t a i n  if,  the  on w h i c h t h e m e t h o d sums t h e g e o m e t r i c  Fortunately  these two p r o p e r t i e s  generally hold for  c l a s s o f methods known as S o n n e n s c h e i n m e t h o d s . methods t h e m s e l v e s w e r e o u t l i n e d b y J . in his doctoral thesis i n h i s papers subject of view.  [2]  i n 1946  and [J>].  series.  The  Sonnenschein  and w e r e l a t e r  In his  a  treatment  [1]  detailed of  the  Sonnenschein adopts a f u n c t i o n a l a n a l y t i c  point  Below we s h a l l a t t e m p t t o e s t a b l i s h h i s  ideas  more s p e c i f i c a l l y  i n the realm of a n a l y t i c  function  t h e o r y and t o d e v e l o p some s p e c i f i c methods w h i c h be a p p l i e d i n l a t e r  chapters.  We b e g i n b y r e m a r k i n g t h a t summation m a t r i x  G  will  the a p p l i c a t i o n of  t o the geometric s e r i e s  a  or a matrix  -13F  to its  sequence o f p a r t i a l  a sequence  {g (z)}  where  n  the closed  sums v e r y o f t e n l e a d s  S ( ) z  c  a  be o b t a i n e d  n  n  to in  form,  g (z) =  (1-16)  (z)  1-zf  n  x  .g  .  The s i m p l e s t example o f t h i s matrix or  G  i s the matrix  just  the n - t h p a r t i a l  that  is  i s when (1.7)  F  Is  so t h a t  the  identity  g ( )  is  z  n  sum o f t h e g e o m e t r i c  series,  g (z) =  (1.17)  n  F u r t h e r examples a r e g i v e n by i) where  the Euler-Knopp or g ( ) z  i  g i v e n b y (1.8)  s  n  ii) [1, p .  the Hausdorff  72])  or  methods  (Section  l.l)  ,  (H,q)  method ( s e e  Hille  where  l.-z  g ( ) =_  (1.18)  (E,r)  z  n  f "o  [l-(l-z)u] dq(u) n  1  1-z with  q(u)eBV[0,l]  t h a t when with  and  0 < r <_ 1 ,  q(u) = G ,  q(0) = 0 , i)  0 <_ u _< r  q(l)  = 1  is a particular ,  q(u) = 1 ,  (note  case o f r  < u £ l)  ii) ,  -14-  iii)  th e Borel or  (B) method [see Hardy [ l , p..79])  where  () = X  g  (1.19)  &  z  '  v n  V  z e  1-z  1  , 5  and more r e c e n t l y , the Lototsky or ( L )  lv)  method (Agnew, [2]) where a () n^ 7  g  z ;  - l-2(z+l)» • »(z+n-l)/ni ~ 1-z  The obvious conclusion to be drawn i n a l l these cases i s that  g ( z ) -* ( 1 - z )  - 1  n  f ^ ( z ) -» 0  on D  i f and only i f  on D . Comparing (1.10) and ( l . l 6 ) we obtain 1-zf (z) g (z) = Z g ^ z * = ITS"  (1.20)  •  n  In any u s e f u l method of a n a l y t i c c o n t i n u a t i o n the domain D  on which  S ( ) ~* ( l - z )  unit c i r c l e .  z  n  - 1  must extend beyond the  This implies that the s e r i e s i n (1.20)  must converge i n some c i r c u l a r domain which contains f^(z)  |z| < R , R > 1 ,  D . But t h i s i n turn implies that  must be a n a l y t i c i n the domain given by  |z| < R  and, i n order that the expression on the r i g h t hand side of (1.20) does not have a pole a t z = 1 , f (1) = 1  for a l l n .  that  -15One o f t h e s i m p l e s t ways o f o b t a i n i n g a sequence of functions  {f (z)}  s a t i s f y i n g the conditions  n  t h e l a s t p a r a g r a p h w o u l d be t o s t a r t w i t h a function  f(z)  (which, loosely t r a n s l a t i n g  w i l l be c a l l e d a d i s p l a c e m e n t glissement))  Let if  f(l)  0)  f(z)  f (z) n  =  is  is analytic  that  for  F  z = 1  t e n d i n g t o z e r o on  and  G  .21)  D .  let  us  f (z) n  in  .  It  is  de.  < R}  D  or,  Putting of  t o be of  no d e l e t e d  noted  D  and b y  neighbourhood  To o b t a i n t h e  matrices  write  = 2 f  z  n k  k  i s a p p l i e d t o the p a r t i a l  Iz|  ,  and f o r m t h e S o n n e n s c h e i n m a t r i x  we  < 1}  i s always a boundary p o i n t  could l i e  |z|  1,  o b t a i n a sequence  t h e maximum m o d u l u s p r i n c i p l e , of  |f(z)|<  Cz : | f ( z ) |  we do i n f a c t  n  z = 1  (fonction  |z| < R , R > 1 .  D = {z :  e n t i r e , by  = f (z)  functions  Sonnenschein,  1  be d e f i n e d b y  f(z)  fixed  satisfying  a)  D  function  of  < R  P = (f^)  •  1^  sums o f t h e g e o m e t r i c  F  series  obtain  _  1  ~ T^z  v r L  z  nk ~ T-I  2 t J  nk  7 z  k  -  ~  1  ~  z  f  n  ( ) z  1-z  -16since  f ( l ) =1  by  a) .  Consequently,  F  i s indeed  the type of matrix we wished to construct and the s e r i e s to-sequence matrix  G = (g ) n k  can be obtained from  F  00  by p u t t i n g  g  n k  =  S  f  .  Notice at t h i s p o i n t that  m=k examples i ) and i i i ) above are examples of Sonnenschein methods with displacement functions corresponding to f ( z ) = (1-r) + r z and f ( z ) = e ^ ^ r e s p e c t i v e l y . 1 - z  Let the displacement f u n c t i o n now be taken to be a polynomial  (1.22)  p(z) = p + Q  P l  z + ... + p z  X  x  which we s h a l l henceforth c a l l a displacement polynomial. The c o n d i t i o n f ( l ) = 1  here i m p l i e s  E i=o  ' i s now by  + °=  while  D  p. = 1 .  R  1  i s the l e m n i s c a t i c r e g i o n given  [z : j p ( z ) | < 1} (which may not be s t a r - l i k e with  respect to the o r i g i n ) .  The matrix  F = (i"^)  has the  form  (1.23) Since  f (z) = p ( z ) n  i s a polynomial of degree n\  n-th row of the matrix w i l l have are i n general non-zero.  nX + 1  the  e n t r i e s which  In f a c t i t i s not d i f f i c u l t  -17t o see t h a t t h e e n t r y  f  (1.2*)  f  c a n be g i v e n e x p l i c i t l y b y  fc  n k ' to ibJI.-V C P l ' - ' - V ' (° <"<>*> S  1  o  =0  X ( k > nX)  where t h e above sum i s t a k e n o v e r a l l p e r m u t a t i o n s the non-negative  Integers  b ,b.., • • • ,b. O X  x  2 b. = n  i=o the e n t r i e s  x  and  2 ib. = k .  i=o i s , however,  1  1  of  satisfying  K  Use o f (1.24) t o compute hot p r a c t i c a l .  A more  u s e f u l method o f c o m p u t i n g t h e s e e n t r i e s  is to  them r e c u r s i v e l y b y n o t i n g t h a t  = p (z).p(z)  so  p  n + 1  (z)  f  n  +  1  ,  k  =  £  Q  P  (0 <  l  This formula remains v a l i d f o r a l l n I s t a k e n t o be z e r o i f  k  be o b t a i n e d f r o m t h e r e l a t i o n s h i p G  i n general  (n l)X) +  k  provided integer.  G = (S k;) 00 g ^ = 2 f ^ c  a  n  n  o  w  n  .  i s a l s o r o w - f i n i t e w i t h t h e n - t h row having n\ + 1  above sum t o d e f i n e obtains  and  i s a negative  The s e r i e s - t o - s e q u e n c e m a t r i x  We see  k <  ( k > (n+l)x) .  =0  one  ,  n  that  (1.25)  f ^  obtain  non-zero e n t r i e s . g ^  f o r negative  I f one u s e s t h e values  of  k ,  -18-  (1.26)  g  n  = 1  k  With t h i s  (k  -  W  2 T )  ,  convention (n+l)X  ( 1  < 0)  m  =  !  (n+l)X X  W  k  X  E p. 1= o  =* m  (n+i:)x E m=k  f  k  £  P i '»..-!  _ .  X =  E  •  0  P j L  gn^.i  (0 < k <  (k > (n+l)x)  r i  which i s a r e c u r s i o n r e l a t i o n i d e n t i c a l w i t h the f , . We n o t i c e t h a t g = 1 for all nk no to i n i t i a t e  (1.28)  1.3  g,  k  the r e c u r s i o n  -  X Z  =  0  Certain Explicit In this  (n+l)X)  (1.25) for n and  relation  p  ( 0 < k < X ) ( k > X)  .  Methods  s e c t i o n we s h a l l  outline  several  Sonnenschein  methods o b t a i n e d by t a k i n g s p e c i a l d i s p l a c e m e n t and we s h a l l n o t e t h e d o m a i n  D r  polynomials  involved i n  all  us r e c o n s i d e r  the  cases. l)  As a f i r s t  example l e t  method w i t h d i s p l a c e m e n t  polynomial  -19p(z) = (1-r) + r z  (1.29) In this  case  p = 1-r *o  and  pr  (r^o) = r  l n  so t h a t  (^O)  S  n  +  1  ,  =  k  ( 0 < k < n)  + r g ^  = 0  D  0 < r < 1  (k > n)  the m a t r i x  i s the c i r c u l a r  r  G  i s r e g u l a r and t h e d o m a i n  r e g i o n g i v e n b y (1.9)  i n t e r e s t i n g t o o b s e r v e t h a t as approaches the l e f t  r -• o  half-plane  •  It  is  t h e domain  g i v e n by  D  r  Re(z) < 1 .  C o n s i d e r now a d i s p l a c e m e n t p o l y n o m i a l  2) the  recursion  (1.27) becomes  relation  For  the  of  form  P(z) = (1-r) + r z  (1.31) Here  p  Q  = 1-r  ,  = 0 ,  (r^o) .  2  and  p  2  = r  so t h a t  the  r e c u r s i o n r e l a t i o n becomes  C -^)  g  1  n  +  i,  k  -  (l-r)8nk + S n k - 2 r  we t a k e  points p(l) of  0 < r £ 1 ,  +iu  where  = p ( - l ) =1 D  y  .  When  z  then  p(z)  \x = { ( l - r ) / r } both  k  <  2  <  n + 1  (k > 2(n+l))  =0 If  <°<  z = +1  has r o o t s a t .  1 / / 2  lie  the  Since  on t h e b o u n d a r y  i s pure imaginary,  p(z)  is  real  .  »  -20-  < 1 , and we f i n d  and where  ?  + i§  L  = [(2-r)/r}  p(z) = -1 f o r z = + i ? « (2u +l)  1 / 2  2  > \x . Thus  1 / 2  are a l s o points on the boundary of D  L  we put  L  . If  r  z = § - e ^ , i t i s an easy matter t o c a l c u l a t e  that p o i n t s on the boundary of D • s a t i s f y the p o l a r r  equation (1.33)  §  2  - (n c o s 4  Hence f o r a given for  § and thus  2  2cp + a i + l ) 2  2  cp there i s only one p o s i t i v e s o l u t i o n D  r  i s s t a r - l i k e with respect t o the  We note (1.33) gives  origin.  - M cos 2cp .  1 / 2  f o r cp = + w/2 .  % - +?  L  We note a l s o that as r -• 0 IJ  and § -*  . I n an  00  L  attempt t o f i n d the l i m i t i n g domain as r -» 0 restrict take  cp t o the i n t e r v a l  u  l e t us  0 < cp <_ ir/2 and l e t us  Por 0 _< cp• <_ ir/k , (1.33) can be  large.  w r i t t e n as '' 9  2  %  2  =U  r /1 ,  o  C O S 2flp[(l+  1  j  2  -£r- + —2T  u cos 2cp  u cos  \ 1/2  ^—)  ,i  - IJ  2cp  or, a f t e r applying the binomial theorem, as (1.33)'  ?  2  = sec 2cp + o ( l ) ,  u - • .  Thus f o r 0 < cp < TT/4 , ? "* Vsec 2cp as | i - » ( r 0 ) . On the other hand, f o r TT/4 < cp _< i r / 2 ,  (1.33)  gives  -21?  2  = -u  cos 2p[(l +  2  +  2 g  g  u cos  2cp  ) + 1] u cos 2cp  «  1  1  /  2  g  or (1.33) "  ?  and therefore  2  = -3i  §'-*».  cos 2cp + 0(1) , u - •  2  The l i m i t i n g domain i s shown  cross-hatched i n F i g . 1 as w e l l as the domains for  u = 1  D  r  and u = 2 . The summation method based  on t h i s displacement polynomial w i l l be of use i n l a t t e r work. 3)  Continuing i n t h i s d i r e c t i o n we f i n d that  another displacement polynomial l e a d i n g t o a u s e f u l method i s the polynomial (1.34)  p(z) = ( l - r ) z + r z  For  0 < r <_ 1  and  z = + i u where  (r+0) .  5  t h i s polynomial has roots at z =* 0 |i = [ ( l - r ) / r }  1 / / 2  as above.  The r e c u r s i o n formula i s now S  - d- )g r  n + l j k  =0 The p o i n t s  z = +1  and i f z =• ? e i s given by  lcp  n k  -i + ^nk-3  <° < * < 5(n+D) (k > 3 ( n + l ) ) .  are again on the boundary of  D  r  the p o l a r equation of t h i s boundary  *4< -0  -23P  (1.36)  9 P  il  ll  P P  ? * ( ? > 2 s V c o s 2cp + ii^) => ( l - m * r  which f o r  cp = ir/2  (1.37)  becomes  §(? -H ) 2  It  suffices  2  of these.  /  u  r e l a t i v e maximum a t  \x  value. greater  provided  o f t h e domain  £  L  > \i w h i l e  § = -w/3  t h e c u b i c has a  \x  f  = 2.910684 .  r > r^ = ( l - H i ^ ) D  -  a  i s as f o l l o w s :  n  i t Is similar  As  n  increases,  f o r a l l values Thus  (Cassini ovals).  t o that  .065387 . Por s m a l l  r  that  The shape values  i n Fig. I with  \i = 1 .  t h e d o m a i n g e t s p i n c h e d i n above  D  breaks  r  When  u =» u ,  case where t h e t h r e e o v a l s D  equation  n < (i^ ,  and b e l o w t h e r e a l a x i s , w h i l e f o r s t i l l t h e domain  2  -  of  a  2^/3/3-I- -1  and a v a l u e  This value i s p o s i t i v e than  sign  2  r  of  taking  Hence t h e e q u a t i o n becomes  (1.38) h a s o n l y one r e a l r o o t p r o v i d e d is,  equation  - (1-na ) = 0  2  w h i c h h a s one r o o t a t  of  of this  sign since the roots f o r the negative  are the negative  at this  •  t o consider the roots  the p o s i t i v e  (1.38)  = ±(1-Hi )  2  is not star-like  larger  values  i n t o three d i s t i n c t  ovals  we have t h e l i m i t i n g  j u s t meet.  with respect  While t h e domain  t o the o r i g i n  even  -24-  for  values of  ,  u < u  It  largest  sub-domain o f  respect  t o the o r i g i n w i l l  the imaginary axis  D  star-like  contain that '  -15^  4) Consider f i n a l l y the  which i s  r  from  I s t r u e however t h a t  f c o  + i  the  with  segment  ?L  P  r  o  v  :  L  d  of e  d  displacement polynomials  of  form  (1.39)  p (z)  =  x  where  r  1-  r(l-z)  X  i s a n y c o m p l e x number  ji 0 .  "Let u s make  the  i  transformation  (1.40)  x  we now p u t  it  is E  ou = T e  satisfy T  Specifically,  if  P^( )  X loops X  and  i v  = |  the if  x  X  |q^(t»)|  on t h e  <  .  (-1.4.1)  is a lemniscate  i s a n e v e n i n t e g e r and  i s a n odd i n t e g e r .  Thus i f  we d e f i n e  1},  boundary  equation  cos Xv  curve  .  E = {uu :  t o show t h a t p o i n t s  the p o l a r X  let  becomes  z  x  not d i f f i c u l t  (1.41)  has  so t h a t  q (uo) = p ( l - t » ) = 1 - ru»  If  of  uu =» 1 - z  which  2X  loops D  by  -25-  (1.42)  D ={z: z = l  - uu ,' uu€E} ,  r  r e f e r r i n g to (1.40) we f i n d By (1.42)  D  for a l l  z = 1  (E,r)  For  X > 1  (X = 1  method) i t i s c l e a r that  never contains the u n i t d i s k and consequently F  r  TT anti t r a n s l a t e d to the  from thei o r i g i n .  reduces to the  matrices  zeD .  i s simply the lemniscate given by ( 1 . 4 l )  r  rotated through an angle point  |p^(z)| < 1  and  G  derived from t h i s  D  r  the  displacement  polynomial can never be r e g u l a r .  1.4  The Methods  (E,r,a,ft)  I t i s p o s s i b l e t o o b t a i n a g e n e r a l i z a t i o n of the (E,r)  method and p a r t s of some of Agnew's proofs [ l ]  r e l a t i n g to i t by c o n s i d e r i n g the sub-class of Sonnenschein methods f o r which the displacement polynomial i s given by (1.43)  p(z) = ( l - r ) z  Let us denote by (1.43).  a  (E,r,a,p)  + rz  a +  ^ ,  a,£ p o s i t i v e integers,  the method obtained from  Thus, i n t h i s context, the  (E,r)  method i s  equivalent to  ( E , r , 0 , l ) while our two previous methods corresponding to p(z) = (1-r) + r z 2 and p(z) = ( l - r ) z + r z 3  become that  (E,r,0,2)  and  (E,r,l,2) respectively.  (E,r,a,0) = (E,0,a,p) .  the matrix given by  F =-'( i ) f  n  c  Note  I t i s easy to see that  obtained from (1.43) w i l l be  -26f  n, k an " P  f Consequently,  a  and  in that  p j  for  r  and  (1.44) we 2|f  and t h a t  this  However, f o r =1  method i s  £  (otherwise). ,  n k  d =  and f o r  m a t r i x are independent determine only t h e i r  in  of  position  that  | = (|r|+|l-r|)  0  and  r =  to  If  (n=0,l,2,...)  n  and o n l y i f  0 £ r _< 1  0 , (1.43) becomes  such a d i s p l a c e m e n t  of checking I I ) ,  It ii)  function is  the  now o n l y a  and i i i )  of  obtain  THEOREM 2 is regular  see  c e r t a i n l y not regular.  S e c t i o n 1.1  For  a « 0 ,  and o n l y i f  and o n l y I f  sufficient  the  0 < r £ 1  (E,r,a,p)  method  and f o r  a > 0 ,  0 <_ r <_ 1 .  L e t us p o i n t  out t h a t  conditions  S o n n e n s c h e i n [3] has for  S o n n e n s c h e i n method a r e t h a t 3)  the non-zero e n t r i e s  sum i s b o u n d e d i f  simple matter  that  fixed  (E,r,a,p) a  (1.45)  if  0  (k=0,l,...n)  row.  From  p(z)  =  n k  any r o w o f a n  ,  +  ( s e e S e c t i o n 1.2)  and  the r e g u l a r i t y f(z)  satisfy  of a)  shown any and  -27Y)  k  Thus f o r any a=0 ,  (k=o,i,2,...) .  f >o (E,r,a,p)  r=>0) ,  method ( a s i d e f r o m t h e  the c o n d i t i o n s  case  a r e n e c e s s a r y as w e l l .  The n e x t t w o t h e o r e m s f o l l o w f r o m e q u i v a l e n t b y Agnew w i t h o n l y a change o f s u b s c r i p t and the proofs w i l l  n o t be g i v e n I n d e t a i l .  ones  consequently  The p r o d u c t  o f t w o s u m m a t i o n methods c a n be d e f i n e d as t h e  method  o b t a i n e d by a p p l y i n g t h e methods i n s u c c e s s i o n ;  in  of matrices  product  it  corresponds t o the ususal m a t r i x  o f t h e two m a t r i c e s . " " 8  such a p r o d u c t  i s denoted by  we can p r o v e THEOREM 3  (E,r)«(E,s,a,p)•-  For an a r b i t r a r y its  If  terms  t r a n s f o r m by  46)  t  n  sequence  (E,r,a,£)  -  (E,rs,a,0).  {s^}  yields  r^O  ME>(l-r)  47)  a  By t h e o r e m 3 , (E,l,a,0)  to  -  [a }  (E,r~  1  n  )  (1.44) ,  to  r"  k  {t 3 n  t  c a n be o b t a i n e d b y  {s J; but t h i s n  n  with  E ( ^ ( l - r - ) " ^ k=»o  {a )  £t 3  pk+an *  application of  t h e sequence  if  then, using  K=0 Assuming  ,  yields  applying  denotes  -28a,  .48) '  s - a = (a+3)n n a  S (^(l-r" ) k-o 1  s  o  , s  l ' 2 "' J  S  J  i m  P  n  .  elements i f summability of  summability of  l i e s  "^*^  Y , l e t us say a method  For any p o s i t i v e integer permits omission of Y  1 1  s Y  >  s Y +  i>  8  Y +  2 » ' *'  to the same value and permits, adjunction of Y i f summability of s j S ^ s , , , . . .  elements  implies summability of  ,c„> • • •-c ,s ,s_,-«. t o the same value where  1' 2  (For Agnew's d e f i n i t i o n s Y » 1 ) .  are a r b i t r a r y constants. THEOREM 4  If  3 - 0 ,  a / 0 ,  the  (E,r,a,B.)  method permits omission and a d j u n c t i o n of a If 3  c-.c-y  1' 2'  Y ° 1  a = 0 , p ^ 0  the (E,r,,a,3)  elements i f r ^ 0  elements.  permits omission of  and permits a d j u n c t i o n of 3  elements i f and only i f | l - r | < 1 . The f i r s t p a r t i s immediate f o r then from (1.43).  I f {s } n  {t^} where  adjunction of a sequence Thus  method i s then [ t J n  tn = san . The omission of a  the sequence  elements  it") where n  t =• t ' = t " , _ n n-1 n+1  a  i s an a r b i t r a r y sequence i t s  transform under the (E,r,a,0) where  p(z) = z  t ^ = a(n+l) s  c  i> 2 '** a c  ,  c  elements produces w h i x e  P  ^  r o d u c e s  e  the  to* => c, , tn* = s o^n-xj , \ , n=l,2,... ± and the proof holds,  -29-  The second p a r t f o l l o w s the corresponding theorem by Agnew.  I t i s e s s e n t i a l i n t h i s proof that one be  able t o o b t a i n  s^  k  f o l l o w s from (1.48) .  i n terms of  ^ Q ^ I * ***  >  this  -30-  GHAPTER  II  A p p l i c a t i o n t o t h e Two-Body 2.1  The Two-Body  Problem  Problem  I n the c l a s s i c a l  two-body problem of  celestial  m e c h a n i c s we a r e i n t e r e s t e d i n t h e m o t i o n o f t w o  particles  a b o u t one a n o t h e r when e a c h i s  gravita-  tional  field  o f t h e o t h e r and b y no o t h e r  Usually the equations relative  i n f l u e n c e d by the forces.  o f m o t i o n a r e s e t up f o r  m o t i o n o f one o f t h e p a r t i c l e s  about the  S i n c e s u c h m o t i o n I s known t o t a k e p l a c e i n a plane,  it  the other.  fixed  c a n be s u i t a b l y d e s c r i b e d i n r e l a t i o n t o  set of rectangular  C a r t e s i a n axes s i t u a t e d i n  a  this  p l a n e w i t h t h e o r i g i n a t one o f t h e t w o p a r t i c l e s . proper will  initial  conditions  exist,  t h e n move I n a n e l l i p t i c  (the f i r s t  particle)  at  t h e second  path w i t h the  one o f t h e  The s t a n d a r d e q u a t i o n s  Chapter  of motion f o r  origin  the  elliptic  following  6]) •2  (2.1) (2.2)  particle  foci.  t w o - b o d y p r o b l e m have b e e n r e d u c e d t o t h e ( s e e Dariby [1,  If  M => E - e s i n E = where t h e s y m b o l s u s e d a r e  n(t-T)  sin E  -31x,y  The l e f t  rectangular particle,  coordinates  a  length of semi-major axis  e  eccentricity  E  eccentric  M  mean a n o m a l y ,  n  mean m o t i o n ,  t  time,  T  t i m e o f p e r i h e l i o n passage  hand e q u a l i t y  gives  constant  t h e r e w i l l be no l o s s  If  we t a k e i t  equal t o  is  obtained:when  M  ( t h u s "by  x  hereafter.  and  "(2.2) known  has shown t h a t  o c c u r s as a  1 y  (2.3) where  Q  (2.4) rt  is  2'rnr +  in  generality  The  solution  ).  F. R. M o u l t o n [ l ]  singularities  (n=0,+l,+2,• • • )  g i v e n i n terms of the e c c e n t r i c i t y  i s not hard to v e r i f y n(e)  of  points  10  Q(e) = - . / T ^ e  and t h a t  well-known  multiplicative  of t  these s o l u t i o n s possess  M •  .  a r e known f u n c t i o n s  functions  i n t h e complex p l a n e a t t h e  ellipse,  ellipse,  the  Since  (2.1)  of  moving  anomaly,  I n (2.2) a  of the  Kepler equation. in  of the  2  + ln[(l+JT^e )/e]  that  2  Q(0) = + «  and  by  . Q(l)  i s a monotone d e c r e a s i n g f u n c t i o n  of  => 0  -32-  e  on t h e u n i t  n(e) .05  for to  interval.  i n st<?ps o f  Let the s o l u t i o n s  x - x(M)  (2.5)  Consideration of these f u n c t i o n s  of  i f  nearest  series  for  .  to  n  will  = 2  t h e n converge t o  at a l l  points.  evaluate  M  on  1  g  n  D(x)  x (M)  x  x(M)  -t-iQ(e)  and  .  ,  k  and  ,  f c  series-to-  D(y)  outlined  y (M)  n  y(M)  to d e f i n e d by  n  y (M)  - 2 g ^  y  k  M  k  respectively .  I n order  (2.7) it  need t o know t h e T a y l o r  for  Referring  D(x) = D(y)  occuring i n  as  using ( 2 . 3 )  and  n  k  of  w h i c h sums t h e  D ,  D(x) = D(y))  inside  the f u n c t i o n s  t h a t we w i l l  the s i n g u l a r i t i e s  G => ( g ^ )  the f u n c t i o n s  x (M)  (2.7)  .  ( 2 . 5 ) a c c o r d i n g t o t h e method  i n Section 1 . 1 (here Theorem 1 ,  be g i v e n b y  we a r e g i v e n a  a g a i n we c a n f o r m t h e domains the f u n c t i o n s  .  y(M) » 2 y  If  (1-z)"  from  these f u n c t i o n s , w r i t t e n  ,  k  of  varies  the o r i g i n are those a t  |M| < n(e)  series  e  y = y(M)  sequence s u m m a t i o n m a t r i x geometric  as  ( 2 . 1 ) and ( 2 . 2 )  ,  x(M) = 2 x  (2.6)  .05 )  ( 2 . 3 ) shows t h a t  the Taylor  converge  gives the values  e = .05 (.05) 1.00 ( i . e . 1.00  so t h a t  Table I  series  is  to  evident  coefficients  -33TABLE Values o f  0.(e)  e  ,05 .10  2.6895  1.3126  .25  1.0952  1.5959  for  e  nfe)  .30  .9199  1.9982  .15 .20  fl(e)  .40  e = .05(.05)1.00  0(e) .3698 .2986 .2361 .1814  e  .55 .60  .7741 •  I  .6503 .5437'  .65 .70  .45 , .50 . 4 5 0 9  • 75 .1339  e  .80  .85 .90  • 95 1.00  0(e)  .0931 .0589 ' .0313 .0108 .0000  TABLE I I Degree o f P o l y n o m i a l s i n t h e ( E , r ) Method G i v i n g A b o u t 8 D e c i m a l P l a c e s o f A c c u r a c y f o r e .-< .05, .25, .50, .75, .95  r/r  T  - .35(.05).65,  23  22  25  25 22  .95 20  21  19  17  23  21  19  22  19  .55  23  IP  .60 .65  24  20 20 22  16  .50  18 18 18,  21  19  17  27  24  23  21  19  .35  47  44  41  46  42  .45  69 60 53  41  .50  50  .55 .60 .65  49  43 43 44 45  37 34 33 32 33 33 36  ^7rj7~--^e_  M' S3  •5  a  M' Q  M' • .50 a n d Q .  .05  .35 .40  31  .45  .40  51  55  .25  48  .50  :  40 40  41 45  .75  39 38 37 37 38 42  15  16  -34-  i n (2.6).  x  For ease of computation a weight f a c t o r  i s introduced  so that  x  k = X  k  *  y  fc  » x  k * Y . k  The s e r i e s (2.6) can now be w r i t t e n (2.6)'  x(M) - 2 x * ( M )  k  X  , y(M) - 2 y * ( M )  k  X  .  From (2.1) and (2.2) i t i s an easy matter t o obtain the d i f f e r e n t i a l equations /-i  (1-e  % dx  2  -ex)  V  -ST?  -y  8 3  SL  75~  JT^  d M  (2.8)  (l-e -ex) % dM 2  - y i ^ e (e+x) 2  and the i n i t i a l conditions (M =» 0) x(0) = x* = 1 - e  (2.9)  When (2.6)'  , y(0) = y* = 0  i s s u b s t i t u t e d i n t o (2.8) we obtain the  recurrence r e l a t i o n s  X  ( 2  .  1 0  )  l  U  *  y  #  )  l X l-e l  ( l - e ) x n x - -JT^  o  n  '  ;  *  n  ~^/  \ *  y ^ + ex^Mn-k)^  *  x^_ , fe  (n^2,3,4,...) (l-e)xny  =  o  ^.  Vl-e^x _ n  1  n— 1 ^. £. + ex 2 (n-k)x Y _ k=l  fe  n  k  .  -35In this equal t o X  n" (e)  .  1  x (M) = Z g n  The  (E,r)  (2.7)  x^( M)  n k  the weight  G(r) = ( g j ( ) ) r  I  the e n t r i e s  g j ( )  ,  k  y (M) = E g n  c  f  o  ^  r  c  (1.30).  some s u i t a b l e  to construct  E  and a r e c u r s i v e  It  formula  and  y (M,r)  r  we t a k e  0 < r < »  are p e r i o d i c = x(-rr-M)  , l e t us c o n s i d e r the  interval  in  n  ,  .  r  which  (2.7)' c o n v e r g e .  M  M' <_ v ,  ,  the form E)( ^ 9  contain  n^O ,  [0,2TT] ,  r  so t h a t  inside  D ( x ) .'  D (x) r  as  [0,M']  r  outlined  I n (2.3)  the corresponding  and hence  and  (2.1) and (2.2)),  , f o r every s i n g u l a r p o i n t  2mr + iQ ,  functions  and s i n c e  (see  lies  and  is  x(M)  the problem of determining  Referring t o the construction of i n S e c t i o n 1.1  2ir  of period  y(ir+M) = - y ( 7 r - M )  [0,M' ]  to  Hereafter,  Since the f u n c t i o n s  in  (2.7)'  remains t h e r e f o r e  D (x) » D (y)  Results  method  r  value f o r the parameter : r  t h e domain  x (M,r)  X(TT+M)  k  ( * )  e  d e p e n d e n t o n t h i s p a r a m e t e r and i n w h i c h t h e  y(M)  y*(xM) .  n k  used i n t h e e x p r e s s i o n s  r  n  was g i v e n t h e r e b y  n  factor  now become  X  was d i s c u s s e d i n S e c t i o n 1.3  determine  taken  M e t h o d i n t h e Two-Body P r o b l e m ; Brumberg's  The m a t r i x  for  i s always  We n o t e t h a t w i t h  involved the expressions  (2.7)' 2.2  x  chapter the weight f a c t o r  .  of  domains On t h e  -56-  other hand, the domains to the s i n g u l a r i t i e s contain  D' '  '  71  Ifi and  w^-'  and  '  corresponding  -in  respectively w i l l  [0,M'] i f and only i f r  i s chosen so that  (see Figure 2) -I) / > ' , 1  Q(|  i.e.,  2  M  i f and only i f  )  0  < r < r  T  -  -  .  ¥e note i n passing that i f M' » Q , if  M' > 0 ,  the parameter if  r  then r  r  L  < 1 .  we r e q u i r e  then  r^ = 1  and  Consequently,  i n choosing  0 < r < r  However,  L  .  i s taken too s m a l l , while i t i s true that the  expressions ( 2 . T )  7  [0,M ] , the  s t i l l converge on  7  e f f e c t i v e n e s s of t h i s convergence w i l l be reduced. This w i l l be pointed out i n the f o l l o w i n g paragraphs where we s h a l l obtain the estimate  r =» . 5 r ^ f o r v  the most s u i t a b l e value f o r r . The b a s i s f o r t h i s estimate i s as f o l l o w s : Let K  be any compact subset of the z-plane which can be  contained i n some c i r c u l a r domain corresponding t o a given 0 < r < ro , K c  r .  (E,r ) Q  I f dr( z )  Dr  o  ,  method.  0 < ro < • , Then f o r  represents the  -37-  10  z-plane Figure 3  The domain  K  -58distance from the center of the c i r c l e arbitrary point  z ,  define  dL(K) I i d ( K ) »-IIm sup d ( z ) zeir  (2.12)  D  to  r  an  by  .  r  Finally,  If  this  quantity  i s d i v i d e d by t h e r a d i u s  t h e c i r c l e we o b t a i n t h e (2.13)  ? ( r , K ) •-  ratio  rd (K) r  w h i c h i s a measure o f t h e r e l a t i v e compact s e t Let  K  represent  n  which a r i s e s  from the  (1-z)  D  - 1  in  polynomial the Taylor i.e. after  distance  r  .  t h e sequence o f  (E,r)  Then i t  g (z,r) n  D  r  by t h e b i n o m i a l  the r e s u l t i n g double s e r i e s [g (z,r)3 n  .  r  polynomials  D  and  K  - 1  about the p o i n t  ,  truncating this  t h e o r e m , and i n powers o f  i s a sequence o f  lies  interior  ;  the  1 -  ,,  series  {z-(l-l/r)}  k  rearranging z .  Thus  polynomials  a r i s i n g from a truncated Taylor series w i t h c i r c l e convergence  to  constructing  n terms, expanding the expressions  t h e sequence  D  i s easy t o check t h a t  (1-z)  about the center o f  the  method and c o n v e r g e s  c a n be o b t a i n e d b y  series for  k=l,2,''',n,  of  from the center of the c i r c l e  {g (z,r)}  of  to this  of  circle.  ,  -39Consequently,  the best  r  value should be that f o r  which the r e l a t i v e distance from the c i r c l e i s a minimum,  K  to the center o f '  i . e . f o r which  §(r,K) i s  a minimum.. Returning to the two-body problem w i t h a given (E,r)  method, we must f i r s t reconsider (1.15) of  Chapter I where the f u n c t i o n a(z) by the f u n c t i o n x(M) functions  a  ( )  g (M/C,r) . n  l i n e sequent  of the two-body problem, the x  ( ? ) M  n  r  The compact set [0,M'] .  and A  g (z/C)  by  n  i s now taken t o be the  This equation can now be w r i t t e n  |x(M)-x (M,r)| - - ^ r I J 2p-£(l-H'«r -g (M/C,r))<lcl  .14) where  by  z  n  i s now replaced  1  n  C^  n  l i e s i n the p r i n c i p a l s t a r of  contains the o r i g i n and  A •  x(M)  The p o i n t set  K  and (C^)  defined by K(C ) = Cz : z = M/C  , M€A,  i s a compact subset of the z-plane and provided  0 < r < r  to prove that once  L  . C^  Furthermore,  CeC^} K  (^) c  D  r  i t i s not d i f f i c u l t  i s determined any other simple  closed r e c t i f i a b l e o r i e n t a b l e curve l y i n g i n the p r i n c i p a l s t a r of  x(M)  and c o n t a i n i n g  C  i n i t s i n t e r i o r can  -40-  replace  C  in  Lemma 1 .  Moreover,  the difference  (2.14) i s n o t a f f e c t e d b y u s i n g t h i s l a r g e r  in  H o w e v e r , as t h e c u r v e i s t a k e n l a r g e r point  set  K(C^)  .15)  shrinks  curve.  and l a r g e r t h e  onto the p o i n t  set  K = { z : z =• M/C, MeA,, C a s i n g u l a r point  U s i n g (2.3)  we see t h a t  K  of  x(M)} .  consists  of a l l radial  line  segments i s s u i n g f r o m t h e o r i g i n and e n d i n g a t a p o i n t  M'/(2mr + i f l ) , note f u r t h e r  n=0,+l,+2, • • •  that  K  the s i n g u l a r i t i e s  of  now assume t h a t ,  i s d e p e n d e n t o n l y on x(M)  bracketed quantity  paragraph, t h i s w i l l It  ,  L e t us  the l e f t - h a n d  the i n t e g r a l  being taken i n o c c u r when  is not d i f f i c u l t  small values of  r  .  and  i t s minimum v a l u e w h e n e v e r t h e  inside  M/C  M'  and n o t o n  as a f u n c t i o n o f  s i d e o f (2,l4) a t t a i n s  hand s i d e d o e s ,  ( s e e F i g u r e 3). We  on t h e K  §(r,K)  t o show t h a t ,  e ( < .0001) ,  right-  From t h e  last  i s a minimum. except f o r  the quantity  very  d (K) r  i n (2.12) i s o b t a i n e d b y t a k i n g t h e maximum o f t h e distances  t  h  4-  e  from the center  point X  l+4X  ?T) d  of the c i r c l e  or the point. where  \  =  v  "  .  ^  D  r  to  either  . f  The p e r p e n d i c u l a r b i s e c t o r  +  of  -41-  the l i n e j o i n i n g these two points i n t e r s e c t s the r e a l M' z-axls a t  d r  2  -—  X  . Thus  (K) - t ( l - i )  2  +  (|-)W  / 2  I  ,  X  - f  *  i  < x . | <  2  +  Ml x 2  1  r  1  IT  < 1 - i  r  < . —  f x  2  TT  ; i.e. 0 < r < 1+' ~ \ TT  I f we now form  §(r,K) » r d ( K )  c a l c u l u s t o show t h a t , since for  §(r,K)  0 < M' <_ TT , a. minimum  occurs when r=  ,16)  we can use elementary  r  — 1+(M'/G)  -  .5r_ .  2  The v a l i d i t y of t h i s estimate i s i l l u s t r a t e d very w e l l I n Table I I . I n the c a l c u l a t i o n f o r t h i s table M' =• .5Q(e) and respectively.  n(e) so that  The q u a n t i t i e s  r ^ » 1.6 x (M',r)  were c a l c u l a t e d r e c u r s i v e l y f o r e-values  and and  1.0  y (M',r) n  .05* .25,  .50, .75 and .95 and f o r r / r r a t i o s ,.35( .05) .65 L  -42-  8  u n t i l an n-value g i v i n g around accuracy was obtained. the polynomials  decimal places of  This n~value (the degree of  x (M',r)  and  n  y (M',r)) was  then  n  l i s t e d i n Table I I where i t Is noted that i t i s minimal f o r the If  r / r ^ ratio  M' = Q ,  .5 .  then (2.11) gives  r  L  » 1  and,  according to the l a s t paragraph, the best r-value i s .5.  With  r  set equal to .5 the q u a n t i t i e s  and  y(M)  were c a l c u l a t e d t o about 8 decimal places  of accuracy f o r e - .C*5( .05) .95  and  M - ,in(.lG)fi  and these values were l i s t e d i n Table I I I . corresponding degree of the polynomials y (M,r)  x(M)  The  x (M,r) and R  necessary to o b t a i n t h i s accuracy was  n  noted  as w e l l , and these r e s u l t s were compiled i n Table IV. We note from t h i s table that there i s an improvement i n the rate of convergence f o r l a r g e r values of  e .  I t i s c l e a r that the degrees of the polynomials as l i s t e d i n Table IV could be lowered i f the r-value was adjusted to each value of at .5 .  M  used above instead of being f i x e d  This e s s e n t i a l l y i n v o l v e s a r e c a l c u l a t i o n of  the matrix  G(r) » (§nk( ^ r  f o r  e a c h  M  value.  The  above program was i n s t i t u t e d , however, and, f o r each M  value,  r  was put equal to .5r  given by ( 2 . 1 l ) .  L  where  The values obtained f o r  r^ x(M)  is and  -45y(M)  were i d e n t i c a l w i t h t h o s e i n T a b l e I I I  7 d e c i m a l p l a c e s and o f t e n i n t h e 8-th.  up  The  to  corresponding  p o l y n o m i a l d e g r e e s were e n t e r e d i n T a b l e V where a c o m p a r i s o n shows t h e m t o be much l o w e r t h a n t h o s e Table IV f o r for  small values  M a Q .  of  and, of course,  of the polynomials the  G(r)  and t h e r e d u c e d  the computer time i n b o t h  methods and r e s u l t s  of  degrees  cases  principal  obvious advantages  His f i r s t  G = (s^)  a summation m a t r i x  us c o n s i d e r  V.. A., B r u m b e r g [1]  t o the two-body problem.  In its  of  same.  F o r t h e sake o f c o m p a r i s o n l e t  series  equal  Moreover, because o f the s i m p l i c i t y  c a l c u l a t i o n o f the m a t r i x  was r o u g h l y  M  In  star.  the,  pertaining  method i s b a s e d on  w h i c h sums t h e  geometric  While such a method has  o v e r t h e methods u s e d I n t h i s  chapter  when a p p l i e d t o t h e t h r e e - b o d y p r o b l e m ( s e e S e c t i o n it  i s n o t as e f f e c t i v e  problem.  when a p p l i e d t o t h e  two-body  As a n example we n o t e f r o m B r u m b e r g ' s  that  i n order t o o b t a i n the values  for  'M =  fi  and  e =  .05(.05).95  of  x(M)  and  y(M)  degree.  w h i c h exceeds the degree o f our p o l y n o m i a l s by 5  t o 5 times come). and  tables  w i t h t h e same a c c u r a c y  as o u r s a b o v e , h i s p o l y n o m i a l s a r e r o u g h l y o f 150  5.1)>  ( s e e T a b l e I V , a l s o T a b l e s V I I and I X  We n o t e i n p a s s i n g t h a t h i s v a l u e s f o r  y(Q)  agree w i t h ours a t l e a s t  and o f t e n i n t h e  8-th.  t o 7 decimal  to x(Q) places  TABLE I I I ( F i r s t p a r t ) Va]Lues of x(M) and Y(M) from the  (E,r)Method  .10  .05  f o r e = .05(.05)-95  and M - .10(1.0)1.Of  .20  . —JLJL ^  .25  .10  0.91024775E 00*  0.87549766E  00  0.83246209E 00  0.78659904E 00  .20  0.79477206E 00  0 . 8 0 3 6 9 9 4 7 E 00  0 . 7 8 0 8 8 0 6 8 E 00  0 . 7 4 7 1 0 2 1 6 E 00  .30  0 . 6 l 4 3 1 1 3 6 E 00  0.68947116E 00  0.6981968.4E 00  0.68352120E  V .40  0.38493987E 00  0.54013659E 00  0.59888934E 00  0 . 3 6 4 5 4 7 7 3 E 00  0.58884271E 00 0.45818577E 0 0  0.73938225E 00 0.70804574E 00 0.65746322E 00 0.58985574E' 00  0.49688247E  0.50791608E  0.172152Q0E  X  a  1  .50  I 0.12590363E  00  00  00  00  u . 6 0 [-0.14255Q14E 00 e .70 -0.40103392E 00  00  0.311Q718QE 00  0.38143954E 00  0.41453135E 00  -0.27770211E-01  0.15586757E 00  0.25645026E 00  0.31255458E 00  I - 0 . 6 3 2 2 2 1 8 9 E 00  -0.22664578E 00  -0.48562583E-02  0.12553926E 00  0.20464656E 00  -0.416Q0203E  00  -0.16550232E 00  -0.80556318E-02  0.93186560E-01  -0.5Q20Q381E 00  -0.32201609E 00  -0.14153065E 00  -0.19762759E-01  0.18435278E 00  0.15986693E 00  0.14072168E 00  O.31444807E 00  0.27751411E 00  "S  .80 .90  -0.82172Q00E  00  1.00  -O.Q5846887E 00  .10  0_27880000E 00  0.2i8p06QQE  .20  0.534456Q7E 00  0.426|)2n0E 00  0.3611Q007E  y .30  n. 74.65?! O R E nn  n, fii nji noQE nn  n.5?36R833E nn  0.458Q2775E  00  n.4n67Q060E 0 0  ,4Q  n RQQ33322E 00  n  on  n.66fiP5364E nn  O.58Q3256QE  on  0.5255731QE  0.98317610E 00  n 88110922E 60  n. 782*81 37PR nn  n.7np6Q5P5E nn  0.63163716E 00  V  a .50  j&kiii  m  00  00  00  1 u .60 e .70  0.99446177E 00  0.95743065E 00  n.87686n?3F. nn  n.7Q7i5i77E 0 0  0.72353175E 00  0.93519156E 00  0.99238854E 00  n,'Q4i3nppQF. nn  n.87i77P34E nn  n.8oo5n707E 0 0  .80  0.81201301E 00  0.98697580E  00  n.Q7Rpi63RE nn  n.QP6i;p4Q7E no  0.86238Q3QE nn  .90  0.63515610E 00  0.94370407E 00  n 0885671 Q E nn  n.Q6t57739E 00  n.Q0Q44i6iE 0 0  LOO  0.41742975E 00  0 . 8 6 6 1 7 8 5 3 E 00  0.Q73Q4878E 00  O.Q7811Q66E 00  0.Q4223346E 0 0  s  E + nm i s equivalent t o m u l t i p l i c a t i o n by  T  !(>-  TABLE I I I (Second p a r t )  x: V  a 1  .in  .30 0.69139905E 0 0  .35 0.64293510E 0 0  .40 0.59414778E 0 0  .20  o.665qqo46E 0 0  0.62204943E 0 0  0.57683758E 0 0  .30 .40  0.62489809E 0 0  0.58822386E 0 0  0.56981718E 00  0.54278748E 0 0  u  „6n  0.50280374E 0 0 0.42606319E 0 0  s  .78  0.34177608E 0 0  e  0.25197741E 00 . QO 0.1*S8488l3E 00 .8n  1  y V  a 1  u  e s  .45  0.54513146E 0 0  .50 0.49594691E 0 0 0.48394879E 0 0  0.54877204E 0 0  0.53072499E 0 0 0.50734685E 0 0 0.47B85239E 0 0 0.43727978E 0 0  0.438I8981E 0 0  0.48735272E 0 0  0.51101151E 0 0 0.46484280E 0 0  OL40596513E 0 0  0.42364990E 0 0  0.41164741E 0 0  0.39274406E 0 0  0.36869629E 0 0  0.35339077E 0 0 0.27817380E 0 0  0.35279268E 0 0 O028955629E 0 0  0.34334942E 0 0 0.29012827E 0 0  0.32728006E 0 0 0.282555^4E 0 0  0.19943031E 0 0  0.22308314E 0 0  0.23400638E 0 0  0.23527532E 0 0  0.46446517E 0 0  .00  0.62888169E-01  0.11840318E 0 0  0.15436817E 0 0  0.17578898E 0 0  0.18609510E 0 0  .10  0.12484556E 0 0  0.11115346E 0 0  0.99009876E-01  0.88012771E-01 0.17449171E 0 0 0.25803273E 0 0 0.33746245E 0 0 0.41189300E 0 0 0.48073018E 0 0 0.54364590E 0 0 0.60053110E 0 0  O.77892983E-OI  0.42898144E 0 0 0.48652418E 0 0 0.53917088E 0 0  0.65144292E 0 0  0.58696477E 0 0  0.69655548E 0 0  0.63004375E 0 0  ;pn 0.24666738E 00 0.2I992662E 0 0 0.36270208E 0 0 0.32414244E 0 0 0.42198521E 0 0 • hn 0.47064896E 0 0 0.56877991E 0 0 0.51208638E 0 0 0.59353708E 0 0 .60 0.65595528E 0 0 0.66584484E 0 0 .70 0.73156841E 0 0 0.79545085E 0 0 0.72885971E 0 0 .80 QO 0.84776627E 0 0 0.78269141E 0 0 0.888Q1174E 0 0 0.82763226E 0 0 1.00  0*I96II782E 0 0  0.28958340E 0 0 0.37795336E 0 0 0.46012978E 0 0 0.53538017E 0 0  0.60330313E 0 0 0.66376968E 0 0 0.31685745E 0 0 0.76278947E 0 0  0.15454317E 0 0 0.22881261E 0 0 0.29974977E 0 0 0.36663386E 0 0  TABLE I I I ( T h i r d p a r t )  X  V  a 1 u e s  y  V a 1 u e s  .60  .65  .7S  .70  .in  .55 0 . 4 4 6 6 3 4 4 3 E 00  0..39722247E 00  0 . 3 4 7 7 3 1 5 3 E 00  0 . 2 9 8 1 7 6 7 2 E 00  0 . 2 4 8 5 6 9 I E 00  .20  0 . 4 3 6 6 6 8 5 4 E 00  0 . 3 8 8 9 9 5 8 2 E 00  0 . 3 4 1 0 1 1 1 6 E 00  0 . 2 9 2 7 7 4 2 2 E 00  0 . 2 4 4 3 3 0 2 1 E 00  .30  0.42047554E 00  0 . 3 7 5 6 2 2 1 7 E 00  0 . 3 3 0 0 8 1 5 7 E 00  0.28398472E 00  0 . 2 3 7 4 3 0 9 5 E 00  .4o  0 . 3 9 8 6 1 8 9 2 E 00  0 . 3 5 7 5 5 7 8 0 E 00  0 . 3 1 5 3 0 9 3 1 E 00  0 . 2 7 2 0 9 8 5 8 E 00  0 . 2 2 8 0 9 6 6 0 E 00  .50  0.37178313E 00  0 . 3 3 5 3 5 7 0 8 E 00  0 . 2 9 7 1 3 9 8 2 E 00  0 . 2 5 7 4 6 8 7 2 E 00  0 . 2 1 6 6 0 0 6 1 E 00  .6n  0 . 3 4 0 7 0 3 8 6 E 00  0.30961610E,00  0 . 2 7 6 0 5 2 2 3 E 00  0 . 2 4 0 4 7 4 9 2 E 00  0 . 2 0 3 2 3 7 2 7 E 00  .70 .80 .00  0 . 3 0 6 1 1 0 3 3 E 00  0 . 2 8 0 9 2 5 9 9 E 00  0 . 2 5 2 5 2 1 9 0 E 00  0 . 2 2 1 4 9 4 1 1 E 00  0 . 1 8 8 2 9 8 7 9 E 00  O . 2 6 8 6 8 5 2 3 E 00  0 . 2 4 9 8 4 0 4 3 E 00  0 . 2 2 6 9 9 4 3 3 E 00  0 . 2 0 0 8 7 9 7 2 E 00  0 . 1 7 2 0 5 9 3 1 E 00  0.22Q04l i SE 00  0 . 2 1 6 8 5 6 9 7 E 00  0 . 1 9 9 8 7 0 1 0 E 00  0 . 1 7 8 9 4 9 7 3 E 00  0 . 1 5 4 7 6 5 6 1 E 00  1.00  0 . 1 8 7 7 1 3 3 8 E 00  0 . 1 8 2 4 0 9 4 4 E 00  0.17149875E 00  0 . 1 5 5 9 8 1 8 3 E 00  0 . 1 3 6 6 3 3 3 6 E 00  .10  0.68462281E-01  0.59584387E-01  0.51157819E-01  0.43105218E-01  0.35366425E-01  .20  0.13591698E 00  0 . 1 1 8 3 5 4 7 8 E 00  0 . 1 0 I 6 6 5 4 3 E 00  0.85695209E-01  0.70334920E-01  .30  0 . 2 0 1 4 4 1 3 6 E 00  0.17556563E 00  0 . 1 5 0 9 1 9 4 0 E 00  0.1272Q798E 00  0.104^40Q-5E 00  .40  0 . 2 6 4 2 6 3 5 1 E 00  0 . 2 3 0 5 9 3 0 7 E 00  0 . 1 9 8 4 2 5 7 1 E 00  0 . 1 6 7 5 1 8 4 4 E 00  0.13767926E 00  .50  0 . 3 2 3 7 9 8 2 7 E 00  0 . 2 8 2 9 6 3 9 0 E 00  0 . 2 4 3 8 0 2 9 2 E 00  0 . 2 0 6 0 5 6 7 0 E 00  0 . l 6 Q 5 l 8 2 l E 00  .60  0.37965252E 00  0 . 3 3 2 3 5 9 8 6 E 00  0 . 2 8 6 7 9 5 6 7 E 00  0.24271078E 00  0.1QQQ0168E 0 0  .70  0 . 4 3 1 6 0 7 3 2 E 00  0 . 3 7 8 6 0 3 2 6 E 00  0.32726108E 00  0 . 2 7 7 3 6 7 6 0 E 00  o . 2 2 8 7 4 2 1 6 E 00  .80  0 . 4 7 9 5 8 7 0 9 E 00  0 . 4 2 1 6 3 2 0 7 E 00  0 . 3 6 5 1 4 8 9 4 E 00  0 . 3 0 9 9 8 7 1 3 E 00  '0.256008TOE 00  .90  0 . 5 2 3 6 2 5 1 0 E 00  0 . 4 6 1 4 7 2 2 1 E 00  0 . 4 0 0 4 7 9 4 2 E 00  0.34058502E 00  0 . 2 8 1 7 1 2 8 7 E 00  1.00  0 . 5 6 3 8 3 1 3 7 E 00  0 . 4 9 8 2 1 1 5 0 E 00  0 . 4 3 3 3 2 2 3 2 E 00  0.36921607E 00  0 . 3 0 5 8 9 7 1 5 E 00  c  l  (  TABLE I I I (Fourth p a r t )  .80 .10  .85 0.14923143E 00  .Q0 0.99513O17E-O1 0.98069264E-01  .95 0.49767996E-01  .20  0.19891877E 00 0.19571404E 00  X  .30  -  0.19049703E 00  0.14324329E 00  .40  0.13343576E 00  0.13822019E 00  V  .50 .60  0.17473462E 00  0.13202761E 00  0.1.646135QE 00  0.12482Q42E 00  0.95717847E-01 0.92532985E-01 0.88604970E-01 0.84031062E-01  .70 s .80 .90 1.00  0.15329124E 00  0.11675251E 00 0.107968.36E 00  0.78907956E-01 0.73326436E-01  O.Q85981C50E-01  0.67368422E-01 0.61105575E-01  0.34427210E-01  O.I36OI83OE-OI  0.67257832E-02  0.27073463E-01 0.40295595E-.01 .  0.13390234E-01 0.19937141E-01  0.53l68346E^Ol 0.656I5863E-01  0.26319451E-01 0.32501432E-01  0.77587009E-01  0.38458985E-01 0.44178585E-01  ci  1 u  y  V  a 1 u e s  0.14097267E 00 O.TOyRlp^R  00  0.14695311E 00  O.II4O6271E: 00  0.88.755944E-01  .10  0.27893862E.-01  .20 .30 .40 .50  0.55491238E-01  0.20649347E-01 0.410907QOE-01  .60  0.15820057E- 00 0.18123159E 00 0.20308951E 00  .70 .80 .90 1.0Q  0.82520489E-01 O.1O875420E 00 0.134Q1968E 00  0.22378313E 00 0/24334373F. 00  0.6ll33486E~01 0.80618.152E-01 0.99423540E-01 0.11746797E 00 0.13470564E 00 0.15112019E 00 0.16671775E 00 0.18152024E 00  0.89053088E-01 0.10000385E 00 0.11044313E 00 0.12038461E 00  0.49080102E-01  0.47959513E-01 0.46441295E-O1 0.44568105E-01 0.42385916E-01 0.39940445E-01 O.37274618E-OI 0.31432132E-01  0.49655415E-01 0.54891255E-01 0.59892559E-01  -48TABLE IV Degrees' of Polynomials i n (E,r) Method G i v i n g About 8 Dec. "Places of Accuracy f o r e' » .05(.05)95, M => .10( .lQ,)0, r=».5 .20 .40 .50 .60 .70 .80 .90 1.00 ..in .20 3Q 33 35 37 30 32 28 41 44 50 41 46 .10 26 28 36 20 26 21 22 22 .15 26 28 29 30 31 31 34 4o *5, 35 ;2ij 26 'm 28 20 27 22 25 26 29 44 26 .25 27' 27 29 34 38 43 27 29 • 32 25 28 . .30 26 26 22 28 42 27 29 24 26 26 28 28 25 • 27 .25 21 22 27 42 .40 25 25 26 26 30 36 4 1 33 27 27 25 25 25 32 36 41 .45 25 27 30 27 24 .50 20 22 26 40 27 25 25 27 25 24 24 25 26 26 32 25 .55 29 35: 39 24 .60 25 26 28 3 1 26 25 24 34 39 24 24 25 26 28 i 30 34 38 22 .65 25 24 .70 24 24 28 23 24 28 20 25 25 24 30 '24 25 28 23 23 33 37 , • 75 25 24 23 .80 22 24 '242 2? 22 26 29 22 24 24 22 23 26 28 32 3523 23 .85 ? 0 21 22 22 27 2324 •90 25 22 22 20 21 21 21 22 24 26 .95 29 32 22 1  1  TABLE V, Degrees of Polynomials Places of Accuracy f o r .10 .20 • 20 «M4 12 9 15 .05 .10 8 11 15 14 .15 8 11 .14 .20 8 11 8 11 12 .25 13 .30 8 11 8 10 13 • 25 8 10 13 .40 10 7 .45 13 .50 7 10 13 .55 10 7 12 12 7 10 ,60 12 .65 10 7 10 12 .70 7 10 12 7 • 75 .80 7 12 9 .85 7 9 12 9 11 7 .90 6 8 11 .95  i n (E,r) Method g i v i n g About 8 Dec. e » .05(.05)95, M - .1 ( . 1 0 ) 0 , r=.5r. 40 .50 .90 l . o o .60 .70 .8n 22 18 22 27 27' 42 50 . 18 21 30 35 41 46 26 21 29 34 17 25 39 45 , 24 21 44 17 29 22 29 20 24 • 28 17 33 37 42 24 17 20 28 32 37 42 16 20 27 23 31 37 42 16 19 31 36 4] 23 27 10 '26 19 31 22 35 41 , 16 22 26 19 31 35 40 ' 26 30 18 22 35 39 _ 15 18 22 26 15 34 39 29 18 21 25 15 29 32 28 24 18 21 33 38 15 29 24 14 18 21 28 33 37 14 ' 17 20 24 28 22 36 14 20 31 35 23 27 17 14 19 16 26 30 34 23 18 13 ' 21 24 28 32 15 :  -49-  A second more e f f e c t i v e involves  t h e s o - c a l l e d Sundman s e r i e s .  the s t r i p  of width  conformally  20  The s o l u t i o n s  on  9  series. w i t h the  (-l,+l)  under  (E,r)  functions  are t h e n g i v e n d i r e c t l y by t h e i r  method as i l l u s t r a t e d  b e i n g somewhat more e f f e c t i v e Of c o u r s e , w i t h t h e  (E,r)  of t h i s  the  (E,r)  method  method  f o r higher values of  method t h e r e  is  the  the m a t r i x  but,  the s o l u t i o n  on t h e o t h e r h a n d , i t  Taylor  i n T a b l e V shows  advantage o f h a v i n g t o c a l c u l a t e  Application  9-interval  o f t h e t w o - b o d y p r o b l e m as  them t o be a p p r o x i m a t e l y e q u i v a l e n t ,  3  mapped  i s mapped o n t o t h e r e a l  A comparison o f the e f f e c t i v e n e s s  I n terms o f  method  is  i n the 9-plane  Brumberg  transformation  i n which the r e a l M-axis (-l,+l).  In this  about the r e a l M-axis  onto the u n i t disk  the Poincare  of  method i n v e s t i g a t e d by  gives  e .  dis-  G(r)  ,  directly  M . o f t h e Methods  L e t us f i r s t  consider  (E,r,0,2) the  (E,r,0,2)  b a s e d on t h e d i s p l a c e m e n t p o l y n o m i a l where t h e m a t r i x  G ( r ) => ( g  ( )) r  n k  a i d of the r e c u r s i o n formula  and  i  p(z) s  (1.32).  (E,r,l,2) method = (l-r)  + rz  obtained with The domain  D  the r  -50-  is  illustrated  where it  I n Figure 1 f o r  ia = { ( l - r j / r } / 1  is clear  that  t h e domains  D ^  M  <_ TT ,  D (x) r  .  2  u » 1  As i n t h e  and (E,r)  one need o n l y d e t e r m i n e )  2  jjiO^/ ) 2  and  i n order f o r t h i s  » D (y) .  r  method so t h a t  [0,M'] ,  contain  interval  u = 2  to l i e i n  We n o t e t h a t f o r t h i s method D ^ ^ ^ » 2  r  ( 3TT/2 )  D  v  '  '  .  From S e c t i o n 1 . 3 , t h e above w i l l h o l d M' < n ?  or,  s i n c e ^L'[2^ +l) ^ 2  1  2  *  where we n o t i c e as i n t h e  that  (E,r)  1 /  = {(2-r )/r}  2  0 < r < r  18)  = 0(2u +l)  L  T  =»  L  r^  '  provided  2  1 / / 2  ,  provided  ir  1+(M / Q ) i s g i v e n b y t h e same  method.  Unfortunately,  expression  the problem  o f d e t e r m i n i n g t h e b e s t r - v a l u e b e t w e e n t h e s e bounds seems r a t h e r d i f f i c u l t .  I n d e e d , as opposed t o t h e  (E,r)  r/r^  method, the best  choice o f  M  is equivalent  .  This  M and  » .5Q r/r  L  the values  of  (E,r)  x(M)  t o around 8 decimal places and -  varies with the  i s p o i n t e d out i n Table V I which  t o Table I I f o r the  the former t a b l e calculated  ratio  M =- Q ,  .55(.05).85 .  method.  and  y(M)  In were  of accuracy f o r  e => .05, .25, .50, .75 Table V I c l e a r l y  shows  a n d .95 that  -51-  for  M » . 5 0 the best  M => 0  r / r r a t i o i s . 6 0 while f o r L  i t becomes . 7 0 . The table a l s o shows on  comparison with Table I I that the dependence of the rate of convergence on the parameter  r  i s more c r u c i a l  with the present method. There i s however a geometrically i n t u i t i v e reason f o r supposing that t h i s method w i l l provide b e t t e r convergence than the ( E , r ) method. condition K  cD  r  [0,M']  where  K  C  D ( x ) i s equivalent t o the c o n d i t i o n r  i s the p o i n t s e t given by  i l l u s t r a t e d i n Figure 3 . If  Note that the  (2.15)  and  From Figure 1 , we see t h a t \  0 .< 'r < r' , the domain L  D  r  of the present method  f i t s symmetrically about the point s e t K not the case i n the ( E , r ) method:  which i s  Consequently,  l i e s c l o s e r t o the "center" of P  K  On the strength  of t h i s argument, the values of x ( M ) and y ( M ) were c a l c u l a t e d with the same accuracy as I n the ( E , r ) method f o r e »  .05(.05).95  and  M -  .10(.10)0  . In  these c a l c u l a t i o n s , since  M = 0  r ^ = 1 , the value of r  was f i x e d at . 7 0 according  and, t h e r e f o r e ,  to the r e s u l t s of Table IV. The a c t u a l values f o r x(M)  and y ( M ) agreed with those i n Table I I I ; the  degrees of the polynomials  necessary t o obtain these  values were l i s t e d i n Table V I I .  These r e s u l t s are  very i n t e r e s t i n g i n that they show the rate of convergence  -52Degree of Polynomials G i v i n g ( E , r , 0 , 2 ) Method f o r ' e - . 0 5 , .85 and M-.50 and n . r7?L"— .05 .25 M S3  .5 n  M Il v  0  28 22 24  .60 .65 .70 .75 .80 .85  32 40 52 68  24 20 22 28 36 46 60  .55 .60 . 65 .70 .75 .80 .85  56 50 44 42 48 64 88  48 42 38 34 42 56 78:  . 55  TABLE VI About 8 Dec. Places of Accuracy i n . 2 5 , . 5 0 , .'75* . 9 5 > r / r = . 55( • 05) _ .50 .75 .95 22 24 24 18 18 18 18 16 20 22 24 26 28 32 34 42 40 36 46 56 52 T  44 38 34 28 36 48 66  40 36 32 32 40 52 72  40 36 32 28 32 42 56  TABLE V I I Degree of Polynomials G i v i n g About 8 Dec. Places of Accuracy i n ' ( E , r , 0 2 ) Method f o r e - . 0 5 ( .05) . 9 5 , M=..10(.m)o and r . 7 3  r  .in .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 •75 .80 .85 .90 .95  32 32 30 30 30 30 20 30 30 20 30 30 30 28 28 28 28 26 26  .20 32 30 30 30 30 30 30 30 30 20 30 30 30 30 30 30 28 28 26  .3" 34 32 30 28 30 30 20 30 30 20 30 30 30 30 20 30 28 28 28  An  1  34 34 32 32 28 28 30 30 30 20 30 30 30 30 20 20 30 28 28  • 50 ..6a 32 36 34 34 34 34 32 34 32 32 30 32 28 30 28 30 28 30 30 3.0 30 30 30 30 30 30 30 30 30 30 30 30 30 30 28 28 28 28  .70 38 32 34 34 22 32 32 30 28 28 20 30 30 30 30 30 30 28 28  .80 38 36 32 34 34 32 32 30 30 .28 30 20 30 30 30 30 30 28 28  .90 28 38 32; 34 34 22 32. 32 30 28 28 20 30 30 30 30 30 28 28  1.00 42 38 36 34 24 2* 34 34 32 32 30 20 30 28 .28 28 28 28 28  -53-  for a fixed [0,Q]  .  r  Indeed, f o r  e => .60  of the polynomial i s 30  .65  and  that f o r smaller values of  M  the degree  M-value.  f o r each  t h i s table with Table IV f o r the  p a r t i c u l a r f o r M => Q ,  the  (E,r)  method i s M ,  (E,r,l,2)  based on the displacement polynomial  method  p(z) = ( l - r ) z + r z  whose matrix i s determined from (1.35)-  For such a M = Q§  i s given as the s o l u t i o n of  T  and i n  the l a t t e r method i s .  F i n a l l y , l e t us consider the  r  Comparing  (E,r) method we note  more e f f e c t i v e while f o r l a r g e r values of  polynomial,  M in  to be almost independent of  whe^e  Li  §  s a t i s f i e s the cubic equation (1.37) with  Consequently,  r  M ' = (I ,  x(o)  and  y(0)  e = .05,  were c a l c u l a t e d .25,  .50,  .75 , .95  and  The degrees of the polynomials  necessary to o b t a i n these r e s u l t s are l i s t e d i n Table V I I I from which we see  r = .60  i s the best value.  With  r  f i x e d at .60 , Table XI was then prepared g i v i n g the degrees of the polynomials  x (M,r) N  to obtain the usual accuracy f o r M » .10(.lQ)n .  1 /  L  In order to determine a best  with the usual accuracy f o r r => .45( .05) .85 .  L  does not have the simple form of the  previous two methods. r-value f o r  p. » { ( l - r ) / r }  and  y ( M , r ) necessary  e = .05(.05).95  and  Comparing t h i s table with Table V I I  we note a f u r t h e r general improvement i n the rate of  -54TABLE V I I I Degree of Polynomials G i v i n g About 8 Dec. Places of Accuracy i n ( E , r , l , 2 ) Method f o r e = . 0 5 , . 2 5 , . 5 0 , . 7 5 * . 9 5 * - r = . 4 5 ( . 0 5 ) . 8 5 ; a n d M=o .45 • 50 .55 .60 .65 .70 .75 .80 .85  .05  .25  .50  48  42  39  42  3Q  39 36  36 .  "56 33 30 36 45 57  42  30 39  54 72  63  93 129  81 114  48  ...  75  105  .75 39  .95 33  33  30  30  27  30  27 30  33  54  36 45  6Q Q6  60 81  42  TABLE IX Degrees of Polynomials G i v i n g About 8 Dec. Places of Accuracy i n (E,r,1,2) Method f o r e=.05( .05) .95* M=.10( .10)0 and r=.60 .20 .10 .38 .40 .50 .60 .78 .80 .00 1 . 0 0 .05 21 33 24 30 33 33 33 36 30 -tf .10 21 24 27 30 33 33 22 33 33 33 18 24 .15 30 33 30 33 33 3 3 33 27 24 .20 18 30 30 30 27 33 22 33 33 J 24 18 30 30 20 .25 27 20 27 33 30 18 24 30 30 30 30 30 30 .30 27 27 18 21 30 24 30 30 30 30 27 30 .40 30 3D 18 30 30 21 27 24 30 30 30 30 30 18 27 21 24 30 30 30 .45 .50 18 21 24 27 27 30 30 30 30 30 21 24 18 .55 30 30 27 30 30 30 27 18 21 .60 24 30 30 30 27 27 30 27 21 24 .65 18 20 30 30 20 27 27 27 18 .70 21 27 30 30 24 27 27 27 27 18 21 ?7 .75 24 27 27 27 27 24 30 .80 18 21 24 24 27 27 27 27 27 27 21 21 27 27 27 24 27 27 27 .85 15 .Q0 21 15 3.8 27 27 27 27 24 24 27 ..05 15 1,8 24 24 24 24 21 27 27 27  -55-  convergence i n the present method over the  (E,r,0,2)  method. In summary, we can conclude that the methods (E,r,0,2)  and  (E,r,l,2)  (E,r),  become p r o g r e s s i v e l y more  e f f e c t i v e i n o b t a i n i n g the s o l u t i o n t o the two-body problem on the I n t e r v a l  [0,fi]  i n the sense that the  maximum degree of polynomial necessary t o o b t a i n a f i x e d accuracy i s p r o g r e s s i v e l y reduced.  However, the  methods themselves become p r o g r e s s i v e l y more complicated and, i n p a r t i c u l a r , the best value of the parameter becomes harder t o evaluate.  Since i n the  t h i s value can be computed t o be f o r any  M  e[0,Tr]  ,  r  (E,r) method  , 5 r = l/{l+(M/fi) } L  t h i s method i s p a r t i c u l a r l y u s e f u l .  The usefulness of the " s t a b i l i t y " of the r / r  L  ratio  i s f u r t h e r i l l u s t r a t e d i n the next s e c t i o n . 2A  A n a l y t i c Continuation to  [0,TT]  I d e a l l y , as f a r as the two-body problem i s concerned one would l i k e a method which would sum t o the s o l u t i o n s on  [0,TT3  f o r a l l values of e ; as pointed out i n  Section 2.1 symmetry and p e r i o d i c i t y would then give t h e i r values f o r a l l values of M . From Table I we note that f o r a l l values of e Q(e) < TT . accuracy on  t r e a t e d thus f a r ,  Consequently, i n order to obtain the same [0,TT]  as on  [0,Q] one would expect the  -56-  degrees of the polynomials  x (M,r)  increase.  Q(e) -» 0  Moreover, since  n  and y (M,r) t o n  as e - » l , the  degree o f these polynomials w i l l a l s o Increase as e increases. Table X i l l u s t r a t e s the r e s u l t s obtained i n attempting to apply the above three methods t o o b t a i n the s o l u t i o n s at  M  • TT . Prom ( 2 . 1 ) and ( 2 . 2 ) i t i s easy t o see that  the exact values of x(M) .and y(M) a t M =» TT are -(1+e)  and 0  respectively.  For Table X these r e s u l t s  were achieved t o a t l e a s t 8 decimal places of accuracy. The table i t s e l f l i s t s the degrees of the polynomials necessary t o o b t a i n t h i s accuracy f o r various r a t i o s f o r the  ( E , r ) and  (E,r,0,2)  various r-values f o r the ( E , r , l , 2 ) the polynomials  x (M,r)  methods and f o r  method. However,  and y (M,r)  n  n  r/r^  were a r b i t r a r i l y  l i m i t e d t o degree 1 5 0 and i n the cases where t h i s degree was reached before a s u f f i c i e n t accuracy was obtained a dash i s r e g i s t e r e d i n the t a b l e . r e s u l t s obtained f o r note that the best  I n no case were  e > . 1 5 . Prom the table we  r / r r a t i o f o r the ( E , r ) method L  i s . 5 5 which s t i l l agrees quite w e l l with our estimate of . 5 . Note that f o r  e => . 0 5 and e =» . 1 0 the  maximum degree i s again p r o g r e s s i v e l y lowered with each method.  However, f o r e = . 1 5 the ( E , r ) method  alone gave a r e s u l t w i t h i n the a l l o t t e d range.  This may  -57be due t o the f a c t that not enough r  r/i*  L  r a t i o s or  values were tested f o r the other two methods . In concluding t h i s chapter l e t us note that the  l i m i t a t i o n above to polynomials of degree 150 completely  arbitrary.  is  The only p r a c t i c a l c o n s i d e r a t i o n  i n t h i s d i r e c t i o n i s the time used on the computer which i s increased I f the degree i s l a r g e r .  Moreover, the  time taken to evaluate the n-th polynomial with as  n n  varies  and therefore I t grows d i s p r o p o r t i o n a t e l y large Increases.  Aside from any academic I n t e r e s t In  the above methods, i t Is f o r t h i s reason that greater effectiveness i s desired.  -58-  TABLE X Degree o f P o l y n o m i a l s G i v i n g 8 Dec. Places o f Accuracy Various r/r R a t i o s and r V a l u e s , e=.05,.10,.15 and T  (E,r) • Q5  ,10  .3R  92  137  .40 .45  80 71 64  120  .55  60  84  .60  6?  ,65  67  .50  (E,r,0,2)  Method  106  .05  .10  .55  76  114  .60  68  112  .65  62  .70  .15  _  94 121  Method  (E,r,l,2) .05  .15 .10  _  .20  123  94  .?o  72  56  86  .40  54  42  -*  50  80  03  -7* .80  64  74  •50 .60  97  .85  88  82  .70  _  69  for M»TT  Method .10  .15 _  72 _ —  —  mm  -59-  CHAPTER I I I Application 3.1  to the Three-Body  Problem  The Three-Body Problem; Brumbergs Method The three-body problem i s concerned with the motion of three p a r t i c l e s with masses  m , m^  and  o  m  g  a t t r a c t i n g one another according t o Newton's law of gravitation.  A s u i t a b l e o r i g i n f o r an i n e r t i a !  reference frame i n which t o describe t h i s motion can be taken a t the center of mass of the three p a r t i c l e s f o r / i n any i n e r t i a l frame, t h i s center of mass moves with a constant v e l o c i t y .  Often however the  equations are not s e t up using the coordinates of the p a r t i c l e s as the dependent v a r i a b l e s , but rather t h e i r r e l a t i v e coordinates.  Thus i f the  vector distances between the p a r t i c l e s are given by , r"o » m.1 m_2 *, 1 r", => m2mo 0  arid  r"_ » mom, , the 2 1 '  equations of motion i n r e l a t i v e coordinates can be written r, r r. r r , - -fM - i , + fm.(-S + - i + -§) (1=0,1,2) i o l 3 0  (3.1)  r  where  f  r  r  r  i s the g r a v i t a t i o n a l constant,  M  i s the  cl  sum of the masses r.^0)  and  and  r ^ = |r~^| (* *»  .  Given  r ^ O ) , the s o l u t i o n of the three-body  problem would then e n t a i l s o l v i n g the system of  -60d i f f e r e n t i a l equations (3.1) subject t o these  initial  conditions. In an exhaustive paper on the subject K. Sundman [ l ] showed that the only s i n g u l a r i t i e s i n the three-body problem f o r r e a l time occur when two or a l l three particles collide. momentum  He a l s o proved that i f the angular  (which i s constant during the motion) I s  not zero, t r i p l e c o l l i s i o n s cannot take place.  Finally  he showed that by i n t r o d u c i n g a new r e g u l a r i z i n g parameter  uu i n place of the parameter  t ,  the motion  can be a n a l y t i c a l l y continued past any double c o l l i s i o n which might occur and consequently that the r e l a t i v e coordinates  r~^ ,  i n a s t r i p of width constant  n  i-0,1,2 2Q  and the time  t  are a n a l y t i c  about the r e a l co-axis.  The  depends on the I n i t i a l conditions and  the masses of the p a r t i c l e s .  The choice of uu I t s e l f  may vary; a usual choice i s given by (3.2)  duo » U dt where  (3.3)  ,  ou - 0  when  t - 0 ,  U • i s the f o r c e f u n c t i o n U -  f(4-i o  +  -§-°- - p l 2 +  z  ).  r  I f we Introduce a t t h i s p o i n t a weight f a c t o r that i n s t e a d of (3.2) we use  x  s o  -61-  (3.2)'  duo - U d t , X  then (3.1) can be transformed  (3.4)  into  r' + i U'r< - T ^ s r t - M ^  +  ±  x  u  r  + 3 3 l 2 +  i  o  r  r  (1=0,1,2) w h  ere  ' »  .  .  In order t o apply any summation methods i t i s necessary t o f i n d the Taylor s e r i e s c o e f f i c i e n t s of the vector f u n c t i o n s  r ^ t u ) . This problem Was n e a t l y  solved by Steffanson [ l ] as f o l l o w s :  L e t A.^ and  be defined by (3.5)  A  i  " i  >  r  CT  i  =  r  ~l  (1=0,1,2) .  Then system (3.4) can be reduced to the f o l l o w i n g system: A 2A  i  T  (3.6)  = a[  ,  (i-0,1,2)  + 3CT A^ - 0  (1=0,1,2)  ±  V r ^ -(- £v'r£ . - i f a r + ^ K ^ + a ^ - K r ^ ) X  X  (1=0,1,2) U - f(m ib^^ 4in m a A 4« m a 4 ) , 1  2 V - U  ,  XUt' - 1 .  o  2  o  1  1  o  1  2  2  -62-  The Importance of t h i s system i s that i t i s of second degree only i n the v a r i a b l e s concerned.  Consequently,  i f one assumes power s e r i e s s o l u t i o n s f o r these v a r i a b l e s of the form - 2  ~r  ±  (3.7)  r  i k  uo  U » 2 u uok  - 2  h  ,  k  ±  i k  uo  ±  cr <» l k  (1=0,1,2)  k  t » 2 t a>k  ,  k  - 2  a  ,  k  V = 2 a cok  ,  k  A  (t »0) n  v  the f i r s t c o e f f i c i e n t s are given by ,  r ^ - r ^ O )  (3.8)  u v  Q  -  n ^  2  o  2 = u_ o 'o  0  A  ^  0  Q  l  O  1^(0) |  -  o  W  +  l  o  4  ,  2  o  - 1^(0) |  i o  o  i— ( 0—) , r._ » — ' i l xu 7  A  c  (1-0,1,2)  + . V2<^2o ' m  l  - 3  )  0  , .t - 1 ' i X%  *  w  The s u b s t i t u t i o n of (3.7) i n t o (3.6) gives the recurrence relations k A  i k -  *  _ T Q  U  7  ik-J  '  k-1  1  ik»^A77 .f (? -JKj * i k - ; k  CT  c  (3.9)  u  k  k - f 2 (m m a  v.K -  1  k E  j_  2  uJ , u k  J  A ^ j - r m^a^A^.,^  o  oV2j 2k-J * )  A  ,  0  k—1 i k + l ' k(k+l)v , " ^ J  ^.j^  T  r  m  S  n  O  [  n  .J^O  + 1  )(  k + 1  > k - J i J + l - "T I J i k - j - l p  a  x  r  -65-  oj ok-J-l lj lk-j-l" 2J 2k-j-l r  t. k+1  + a  r  h 7  r  £ (j+l)t (k+l)u o j»o x  Using the above Ideas V. A. Brumberg [1] o u t l i n e d a method which can t h e o r e t i c a l l y be used t o obtain the s o l u t i o n of the three-body problem a t each p o i n t of the r e a l tw-axis, a t l e a s t i n those cases where the t o t a l angular momentum i s not zero. Section 2.2  I t was noted i n  that t h i s method i s not the most e f f e c t i v e  when a p p l i e d t o the two-body problem. idea i s t o construct a matrix  Brumberg's b a s i c  G - (g ) n k  which sums  the geometric s e r i e s In i t s p r i n c i p a l s t a r .  According  to Section 1.1 such a method could then be used t o sum the f u n c t i o n s  r ^ - (x^,y^,z^) ,  1-0,1,2,  and t  of the three-body problem i n t h e i r p r i n c i p a l s t a r s which, i n the r e g u l a r i z e d problem, contain r e a l oo-axls.  the e n t i r e  I n applying h i s method to the three-body  problem Brumberg gives r e s u l t s correpsonding t o three p a r t i c u l a r problems which were worked out numerically by Zumkley [ l ] , Stromgren [ l ] and Burrau [1] . I n each of these examples the choice of the weight f a c t o r X  i n (3.2)'  Is p r a c t i c a l l y important f o r i t determines  the rate of growth of the c o e f f i c i e n t s obtained r e c u r s i v e l y from (3.9); i f t h i s rate i s too high or too low one i s  -64-  l i m i t e d to the number of c o e f f i c i e n t s the computer handle.  Brumberg uses  x  _1  will  • 3 . 2 5 , 1 6 . 2 5 and 40  r e s p e c t i v e l y i n the above three examples and gives i n these examples the r e s u l t s of employing h i s method for 3.2  ou -  1  .  L i m i t a t i o n s and Results f o r Sonnenschein Methods Because of the s i m p l i c i t y of the methods used i n Chapter n and t h e i r greater e f f e c t i v e n e s s i n the two-body problem as opposed t o Brumberg's method o u t l i n e d above i t would be of i n t e r e s t t o t r y and apply them to the three-body problem.  Of course one  eould never get a s o l u t i o n v a l i d f o r a l l r e a l values of  uo because i n a l l cases the domains  and consequently so are the domains where  r^ » ( x ^ y ^ z ^ ) ,  D (t). r  I»0,l,2,  D  r  are f i n i t e ,  D ( x ) , D^y^), etc. r  1  and the domain  What then i s the l a r g e s t i n t e r v a l on which  convergence w i l l occur? To solve t h i s problem l e t us assume that every p o i n t of the form  u> + i n ,, 0  00  < uu < +• 0  is a  s i n g u l a r p o i n t of the three-body problem where the let  n  is  constant r e f e r r e d t o e a r l i e r i n t h i s s e c t i o n , and (-W(r), W(r)) denote the l a r g e s t obtainable i n t e r v a l  f o r a s p e c i f i c method and a f i x e d value of  r .  -65Considering f i r s t the  (E,r)  method one f i n d s that  according to the c o n s t r u c t i o n of Section 1 . 1  the  I n t e r v a l (-W(r), ( r ) ) must he I n t e r i o r to every w  (a \  domain D ' , -TT < 8 <_ ir where we have put u> o +— ifl «=> re i9 . Here D (Q) i s the i n t e r i o r of v  v  1  i9 the c i r c l e passing through n(l - 1 )  re  and centered at  (cot 9 + i ) (see Figure 4 ) .  I t i s clear  0 < 9 < n/2  that we need only consider  .  I f the  above c i r c l e i n t e r s e c t s the p o s i t i v e r e a l m-axis at  W(r,9)  then from the equation of t h i s c i r c l e  we obtain  (3.11)  W(r,9)  -  Q[£  \  csc 9-(i-l) } 2  2  1 / 2  -(|-l)cot9] .  r  Minimizing  W(r,9)  (3.12)  9  we  find  cot 8 - 1 - r .  Consequently, as from  as a f u n c t i o n of  T T / 2 to  r  v a r i e s from .  ir/k  Further,  1  to  0 ,  (3.11)  and  9  varies  (3.12)  combine to y i e l d  (3.13)  W(r) -  and therefore as  (2-r)n  r -» 0 ,  W(r) -* 2 Q  r e g u l a r i z e d three-body problem the  .  Thus i n the  (E,r)  method can  only be used with confidence on the I n t e r v a l  (-2Q  , 20).  -66-  m  -in  uo-p].ane  Figure 5  /  \  /  -67-  I f the same n o t a t i o n i s adopted f o r the method, use of equation  (E,r,0,2)  (i.33) e a s i l y gives  W ( r , 9 ) - n c s c 9 [,(|i cos 2e+2a +I) -u cos29]  (3.14)  2  where as usual  2  2  u - (i-l) / 1  2  lf  .  2  Putting  :l/2  2  2  A(u,9) - [ W ( r , 9 ) / n ]  2  (3.14) becomes (3.15)  s i n 9 A (u,9) + 2 s i n 9 ( i - 2 s i n 9 )A(u,9 ) - ( 2 u + l ) - 0 . 2  2  2  2  2  M  f  Noting that W(r,9 )  A(u,9)  takes on a minimum value  whenever  does, d i f f e r e n t i a t i o n of (3.15) to determine  t h i s minimum gives cos 9 . o , But f o r 9 - ir/2  or  A(u,9) - u ( 4 - c s c 9 ) . 2  (3.14) y i e l d s  which tends t o i n f i n i t y with Using the d e f i n i t i o n of  2  W(r,ir/2) - Q ( 2 n + l ) 2  u  (therefore as  A(u,9)  1//2  r -» 0 ) .  and the second c o n d i t i o n  above, (3.14) now gives 2 (3.16) and therefore  sin 9 - i 9 - TT/6  ^ +  as  a -* » .  I f p. i s large  (3.14) gives W(r) «  Da[(l+ -  2  + \)  1 / 2  -l)  1 / 2  -68or, using the binomial expansion twice,  (3.17)  W(r) - 2/2 Q ( l + o ( l ) ) ,  Consequently as  r -» o ,  r - o .  ¥(r) -» 2/2" Cl and the  present  (-2/2~Q , 2/2~0) .  method i s r e s t r i c t e d to the i n t e r v a l  The above estimates can be Improved I f the radius of convergence t  R  of the s e r i e s i n (3.7)  are taken i n t o account (R  for  and  may be taken as the  radius of a s p e c i f i c one of these s e r i e s or perhaps as the minimum of the 10 p o s s i b l e v a l u e s ) . R > 0 the  and that (E,r)  9" - s i n ( n / R ) (see Figure 5).  ir/4  9 "*  +  as  the above a n a l y s i s , we see that f o r (-20,  20)  I f on the other hand W(r) of  For  -1  method, since  the i n t e r v a l  Suppose  r -•  0  in  TT/4 <_ 6~ <_..ir/2  remains the l i m i t i n g i n t e r v a l .  0 < F < ir/4 ,  the value of  w i l l be obtained s o l e l y from the c o n s t r u c t i o n B^ \  and  that i s  W  ¥(r) - W(r,9").  cot 9" » ( R - Q ) 2  (3.18)  2  1 / / 2  /fi ,  (3.13)  Lg^/g  W(r)•-  Since  esc 9 - R/Cl  gives  +  o(l)  r-0  ,  [i-(n/R) ] /2 2  1  »  so that as note f o r  0  ,  W(r) -  8" - ir/4  ,  R -  r -  l i m i t i n g value of Q » 0  and  2Q  W(r) -• R  R/[l-(a/R) ] 2  and  1/2  (3.l8)  as before, while f o r as expected.  .  We  gives a W » 0 ,  (E,r,0,2) method a s i m i l a r argument to  For the  that above shows that the l i m i t i n g i n t e r v a l remains unchanged f o r ir/6 j< 9~  v/2 , while f o r 0 < 9" < ir/6 ,  '(5.1*0 gives 19)  W(r) (-W(r,T))  ^  -~  1 / 2  + o(l),  r -  [1-2(07*0 ] ' and therefore  W(r) - R/[l-2(n/R) ] '  Again, i f 8 =  TT/6 , R - 20  for  F - 0  2  Q - 0  and  and  1/  2  as  r - o .  W(r) - 2/20  while  W(r) - R . We note that no  estimates s i m i l a r t o the above were attempted f o r the  (E,r,l,2)  method.  Bearing these l i m i t a t i o n s i n mind the above methods were a p p l i e d t o the s p e c i a l case of three-body problem considered by Zumkley and the r e s u l t s were l i s t e d i n Table XI and X I I .  In Table XI the degrees of the  polynomials g i v i n g s i x decimal places of accuracy are l i s t e d f o r ID - . l ( . l ) 1.2 and f o r various methods. Those spaces with a dash i n d i c a t e i n s u f f i c i e n t accuracy with polynomials of degree 150 . For the  (E,rj0,2) and r  (E,r) ,  (E,r,l,2) methods, although several  values were t r i e d , the values  give representative r e s u l t s .  r - .5  and  r = .2  We note that f o r small  values of uu the Taylor series i s s t i l l the most e f f e c t i v e but that the range of uu values f o r which i t provides convergence i s the smallest.  Also, f o r  -70TABLE X I Degree o f P o l y n o m i a l s f o r V a r i o u s Methods i n t h e T h r e e - B o d y P r o b l e m Zumkley's E x a m p l e . P a r a m e t e r w, x - 3 . 2 5 , A c c u r a c y A p p r o x . 6 Dec.Places _1  TqyLor (E . ^ M e t h o d E.r.a2)M3thod(E.r.l.2 Mettiod r». ? r». 5 r=. P  Series r - . 5 r - . P  .3 .4  6 8 11 14  21 22 24  •5 .6 .7 .8  19 25 39 77  27 29 31 34  59 64 68 74 80 84  36  92  .1 .2  .9  1.0 1.1 1.2  _ _  -  19  40 42  -  51 56  100  107  -  38. 40 40 42 42 44 44 44  104 110 112 114 116 118 122 124  15  18 21 24 27 30 30 33  46 124 52 128 _  _  —  —  27 33 39 48 54 63  33  69 72  _  _  —  _  32  ;  18 24  TABLE X I I V a l u e s o f t h e Time and t h e C o o r d i n a t e s i n t h e T h r e e - B o d y P r o b l e m Zumkley's E x a m p l e , f o r uo-1 . Zumkleys (E,r) Brumbergs Values Method Interpolatec R e s u l t s r-. 5 t Xg yo xi yi  y2 y?  1.810016 1.810016 1.810016 O.616 O.616707 O.616708 3.306862 3.306862 3.305 -1.124 -I.123587 -I.123587 -2.114 -2.112465 -2.112464 0.506880 O.506879 0.507 -1.191 -1.194397 -I.194398  -71t h e o t h e r methods t h e t a b l e does n o t length of the values of that  r  i n the  interval .  It  r  r » .5  = .2  o f convergence f o r  (E,r)  method t h e r e s u l t s  were  o f d e g r e e 149 and 150  printed  at  uu -  t h e y were c o n v e r g i n g b u t a g r e e d o n l y t o L e t us f i n a l l y  note t h a t ,  t h e above r e s t r i c t i o n o n t h e d e g r e e o f t h e (E,r)  method p r o v i d e s  Interval while on  regard  [0,1]  .  the  convergence  (E,r,l,2)  on t h e  method i s most  largest  effective above  t  relative  1-0,1,2,  g i v e n i n Table X I I f o r  uu -  (x^y^z^) 1 .  ,  to  polynomials,  methods t h e computed v a l u e s o f t h e t i m e r^  for  four  subject  As a c h e c k on t h e a c c u r a c y o f t h e  coordinates  .  1.2  t h e v a l u e s were c l e a r l y d i v e r g i n g w h i l e  decimal places.  the  the  smaller  s h o u l d be p o i n t e d o u t i n t h i s  out f o r polynomials For  show a g a i n i n  Here we a r e  Zumkleys example where t h e masses and i n i t i a l  and t h e are  using conditions  are  r (0)- (2.5, 0, 0) , r (0) - (0, 2.5, 0) , Q  .20)  Q  r (o)-..'(-1.5, 0, 0), F ^ O ) - (0,-1.0, 0) , 1  r (0)- (-1.0, 0, 0), r" (0) - (0,-1.5, 0) . 2  2  We n o t e t h e a l m o s t e x a c t a g r e e m e n t w i t h results.  Brumbergs  The same p r o c e d u r e was t h e n a p p l i e d  to  -72S t r ' o m g r e n ' s example i n w h i c h t h e masses and conditions  are  1 , m^ 7 (o) - (-10,0,0)  2  r (0) - (-7,0,0)  ^(o) - (o,V37r, o)  r (0) - (17,0,0)  7 (o) - ( o j j m +JJ77,o)  m  Q  • m^ «•  o  (3.21)  x  2  The r e s u l t s example 3.3  closely followed  r (0) - (o,V57io, 0) o  2  the p a t t e r n i n  T  and  One c o u l d t r y the l a s t  W ( r ) -* <*> however, plausible  8  t o avoid the l i m i t a t i o n s  as  displacement polynomials  r'-» 0 .  seems r e m o t e  (if  to f i n d a  or a p o r t i o n of i t s e l f , of width  20,  i s mapped I n t o  suitable  into  and t r a n s f o r m i n g t h e  I n the  itself above  itself  case where t h e  l e t us n o t e t h a t  on t h e b o u n d a r y o f t h e above s t r i p n o t on t h e i m a g i n a r y a x i s ,  is again l i m i t e d  A more  i n t o a d o m a i n more s u i t a b l y  t o the p r e s e n t methods.  since the l i n e  which  problem,  i n d e e d one e x i s t s ) .  method w o u l d be t o t r y  In  Firstly  for  The s o l u t i o n t o t h i s  c o n f o r m a l mapping t a k i n g the r e a l a x i s  point  posed  s e c t i o n i n one o f t w o p o s s i b l e w a y s .  one c o u l d l o o k f o r  axis  Zumkley's  above.  The P a r a m e t e r s  strip  Initial  if  adaptable real any  point  i s mapped i n t o a  the  (E,r)  method  passing through  this  -73-  l a t t e r p o i n t and normal t o the r a d i a l l i n e i n t e r s e c t s the r e a l a x i s .  This p o i n t of i n t e r s e c t i o n w i l l be  an upper or lower bound f o r W(r). Consider therefore the mapping (3.22)  T  - sinh '(g)  , a, 20  which maps the s t r i p of width  -2£  ^n(T (l T ) / ) 2  +  1  2  +  in"the uu-plane  conformally i n t o the e n t i r e complex T-plane with the exception of the imaginary l i n e segments  [i,i°°) and  [ - i , - i« ) . The r e a l a x i s i s mapped i n t o i t s e l f and +10. are mapped Into  the points  + i . From ( 3 . 2 ) and  the above transformation can be represented  (3.22)  d i r e c t l y by (3.23)  dT  (1+T ) ^ 2  -  1  2  u dt ,  We note that f o r t h i s new v a r i a b l e D (x ), r  1  D (y )-, r  1  the  the domains  e t c . , are a l l given by  for any of our three methods. for  T  (E,r) and  (E,r,0,2)  when t - 0 .  T-0  D^ ^n / 2  Cdnsequently, methods where  the same meaning with respect t o the v a r i a b l e with the v a r i a b l e uu . Thus the  ( E , r ) and  (E,r,0,2)  W(r) "*  08  methods.  as  D^  Is bounded f o r the ( E , r , l , 2 )  W(r) has T  as  r -* 0 f o r  However, from  method.  ^  W(r) - ( 2 / r  the arguments i n Section 1 . 3 , I t i s easy t o see that W(r)  5 i r / 2  -7*-  Without a c t u a l l y w r i t i n g down the system of equations f o r the parameter for  T  corresponding t o (3.6)  oo and the r e c u r s i o n formulas f o r the Taylor  s e r i e s c o e f f i c i e n t s corresponding t o (3.9), l e t us only remark that the above steps were c a r r i e d out and that the s o l u t i o n s f o r example were found.  t and r ^ ' (i-0,1,2)  f o r Zumkley's  The method, however, turned out  to be of l i t t l e value since the degrees of the polynomials r e q u i r i n g a c e r t a i n accuracy were very high. r e s u l t s can be a t t r i b u t e d t o two causes.  Such poor  Firstly,  the exponential nature of the h y p e r b o l i c sine f u n c t i o n i n (3.22) r e q u i r e s large values of T  corresponding  to only moderate values of uu . Secondly, I f n i s small the above problem Is aggravated.  Let us note  that f o r Z u m k l e y ' s " i n i t i a l conditions (3.20) the estimate o f G  given I n Sundman's paper Is Indeed  very s m a l l ; i n f a c t  0 « 1.1 x 10"^ . (However, i t  should be pointed out i n t h i s instance that Sundman uses a somewhat d i f f e r e n t f u n c t i o n UJ .) The smallness of t h i s estimate has caused B e l o r i z k y [ l ] t o despair using the Sundman s e r i e s f o r any p r a c t i c a l s o l u t i o n of the three-body problem i n the same way that Brumberg used such s e r i e s i n the two-body problem (see Section 2.4). On the other hand, i f Sundman's estimates are poor, i t may be p o s s i b l e t o show that the Sundman s e r i e s are usable.  fl i s not small and that  -75Let us therefore reconsider the parameter  (2.18)  0 of  redefined here i n terms of the parameter  ID  by fx  oin  20  (3.24)  »  -  fl  T  exp(7rou/2Q)-l '  9  "  exvlZyinUi >.  keeping i n mind t h a t , as opposed t o the two-body problem, we do not have an exact knowledge of fl . R e c a l l that the transformation (3.24) defines a conformal mapping of the s t r i p of width  20  i n the co-plane onto the  u n i t d i s k i n the 9-plane t a k i n g the r e a l to-axis i n t o the 9 - i n t e r v a l , ( - l j + l ) . Combining (3.2) and.(3.24) 10  the parameter  can be eliminated y i e l d i n g the  transformation  (3.25)  d9 -  (1-9 )U dt , 2  9-0  when  Let us now introduce an a r b i t r a r y weight f a c t o r use i n place of (3.25)'  d9  (3.25) the  ,  I f we put  (3.26) (3.25)'  \ and  transformation  (1-9 )U dt 2  t-0 .  can be r e w r i t t e n In the form  9-0  when  t-0 .  -76(3.25)"  d9 - K(1-9 )U dt  ,  2  Since  x  9-0  i s a r b i t r a r y , (3.26) shows  K  when  t-0 .  i s also  a r b i t r a r y and hence the transformation (3.25)" i s independent of Q . On the other hand, the radius of convergence of the r e s u l t a n t s e r i e s i n terms of the v a r i a b l e  9  But from (3.26)  of (3.25)" i s no longer  x=  40K/7T  1 but x •  and since fi i s unknown  so i s x • However, using transformation (3.25)" , (3.6) can be r e w r i t t e n as A  t  - r  ,  2  (1-0,1,2)  2AJCT* 4- 3 a A j - 0 ,  (1-0,1,2)  t  ( l - S ) ^ * * 2  (3.27)  2  +  [ ( ^ ) V - 2 9 ( i - 9 ) v ] 7^ 1  2  2  - - - f M K ^ ^ + f K (cT r +a r +a r ) m  2  1  o  o  1  1  2  2  U - ft^m^^^m^A^m^o^g), 2 V - U , (l-9 )Ut* - K , 2  * where  d $Q •  W  e  note again that t h i s system i s of  second degree i n t h e dependent v a r i a b l e s and t h e i r derivatives.  Proceeding therefore as i n Section 3.1  we consider power s e r i e s s o l u t i o n s of the form  (1-0,1,2),  -77-  (3.28)  r - 2 r ±  U - 2 u  Prom the i n i t i a l m  u v  o  ^  9 ,  U  /  V - 2 v  k  k  -  0  >  )  A  , 9 ,  A  - u  2  3  o  '  + m  m  a  A  4  K  o  a  i  Q  •  4.,  m  a  ( 3 k  *  3  (3.28)  A i k  V  k -  f  .f( l m  j  * " |o J U  U  K  r a  2 o/ok-j a  '^  + m  2%  C T  lj lk-j ^ A  +  '  " J  (k-2)(k4)(-v 2v )7. _ (k-l) (-v 2v )7. _ ^ 2  +  3 +  1  k  2 +  2 +  + 2^j4-2)(j l)(-v _._ t'2v _._ -v _ ._ k  +  J—o  k  1  k  3  k  1  —  into  k  u  j  Q  - |r (0)|  o  J ) a i J  (t -0) .  A  _ _ r . . r., . ,  Ik" ^T^.X  k  o l 2o 2o^  we o b t a i n the r e c u r s i o n r e l a t i o n s k 2  (i-0,1  k  9  k  7 i,l -u— 71 , '( 0 )1, .ut , -  F i n a l l y , s u b s t i t u t i n g the power s e r i e s  (3.27)  + r n  9  k  values  2  2 o lo lo  i  t - 2 t  k  k  - 2 a  CTi  |F.(0)| ,  -  l Q  f(^Voo oo  =  Q  e \  k  c o n d i t i o n s we o b t a i n the i n i t i a l  *lo \(  (5-29)  l  o  )7..  k  1  + 2  4-  .  _ : 5  ( 1  pk-l +  f  m  i  K  (  C  T  r  r  K  2k-j-l 2j^  T  r  <  7.  where  ok-j-i o^lk-J-l l j-  and £  H  a  r  e  z  e  r  o  i  Using the r e c u r s i o n formulas  f  4  <  0  (3.30)  >  •  the Sundman s e r i e s  were c a l c u l a t e d f o r Zumkley's and Stromgren's cases corresponding t o the i n i t i a l conditions (3.21)  respectively.  (3.20)  and  (We note that Burrau's example  mentioned e a r l i e r i s never used since i n h i s case the angular momentum i s zero and thus the p o s s i b i l i t y of t r i p l e c o l l i s i o n s i s not eliminated.) examples  K  was taken as  1  I n both these  and with t h i s choice  there was no d i f f i c u l t y i n o b t a i n i n g as many Taylor s e r i e s c o e f f i c i e n t s as we wished. are shown In Table X I I I .  The r e s u l t s of our c a l c u l a t i o n s To begin with we note that  the method Is not as e f f e c t i v e as Brumberg's method discussed i n S e c t i o n 3 . 1 .  For example, with polynomials  of degree 1 5 0 Brumberg reaches a t-value of about 1 . 8 l In Zumkley's example and  33.29  i n Stromgren's example.  From Table X I I I we see these values are r e s p e c t i v e l y about 2 and 1 0 times our values f o r the same degree of polynomial.  On the other hand i t should be noted  TABLE X I I I ( F i r s t Part) Values of the Sundman Series f o r Zumkley's Example and Stromgren's Example of the Three Body Problem. 9 = . l ( . l ) . 9 , .91(.Ol).95, X = 1 , Accuracy of 6 Dec. Places. Zumkley's Example 9 .1 ,2  •2 .4 .5 •  .6  t 6 0.048558 8 0.098167 10 0.150018 12 0.205633 16 0 . 2 6 7 1 8 7 21 0.338183 2Q  .8 .9  .91 .92 ,93 ,94 .95  0.425092  43 •0.542510 8o' 0.738018 86 94  0.767644 0.800808  100 0.838477 110 0.882093 148 0.933837  y2 -0.072782 -0.146800 -0.223428 -0.304357 -O.391887 -0.489473 -0.602931 -0.743616 2.068842 1.711626 -1.468896 -0,772634 -O.599945 -0.938992  x y yi 2.497921 0.121354 -1.499942 -0.048572 -0.997979 2.491515 0.245084 -1.499757 -0.098284 -0.991758 2.480237 0.373861 -1.499410 -O.150434 -O.980828 2.463018 0.511049 -1.498826 -0.206692 -O.964192 2.437930 0.661361 -1.497860 -0.269473 -0.940070 2.401412 0.832181 -1.496216 -0.342708 -o.905196 2.346254 1.036662 -1.493205 -0.433731 -O.853050 2.255092 1.302810 -1.486875 -0.559194 -0.768218 2  Q  2.037820  1.768859 2.002505 1.831171 1.961798 1.899557 1.914161 1.975280 1.857397.. 2.059882  -O.805067 -0.572562 - 0 . 8 4 1 2 4 5 -O.541614 -0.882079 -O.506236 -0.928827 -0.465245 - 1 . 4 4 0 4 0 0 -0.983257 -0.416997  -1.465258 -1.460891 -1.455562 -1.448916  -0.963792 -0.989825 -1.017478 -1.046453 -1.076625  TABLE X I I I (Second  Part)  Stromgren's Example  9 .1  .2 .3  .4 ,5 .6  .7 .8  .9 .91 .92 .93 ,94 -Q5  JSg, 01 Polv.  t  x  O  yo  xi  2.809647 -9.948496 2.923153 -9.944254 3.051216 -9.939268 3.198367 -9.933277 3.371614 -9.925866  x  2  y  2  -0.120620 16.998850 -0.243689 16.995306 -0.371969 16.989059 -O.508955 16.979504 -0.659595 16.965548 -0.831723 16.945154 -I.039565 l6.9l4l6o -1.314414 16.862342 -I.754478 16.753113  0.221543 0.447613 0.683319 0.935135 1.212231 1.529158 1.912394 2.420562 3.257559  -I.819122 16.734280 -I.890836 16.712534 - l _ 9 7 l 4 8 l 16.687000 -2.063776 16.565358 -1.845945 -6.692647 -2.171897 16.618513  5.557856 5.491650 5.642555 5.815074  0.184259 -9.999778 -0.100923 -6.999072 8 0.372312 -9.999095 -0.203924 -6.996210 10 0.568445 -9.997891 -0.311351 -6.991167 12 0.778092 -9.996049 -0.426l8o -6.983456 16 1.008965 -9.993356 -0.552636 -6.972192 ' 20 1.273326 -9.989419 -0.697435 -6.955735 28 1.593547 -9.983428 -0.872829 -6.930732 44 2.019180 -9.973394 -1.105948 -6.888949 86 2.707618 -9.952166 -1.482881 -6.800947 6  94 106 118 138 162  yi  -1.538714 -6.785784 -I.600814 -6.7682B0 -1.670854 -6.747732 -1.751298 -6.723081  4.017841  -8i-  that the use of Sundman s e r i e s requires no summation matrix  G  directly.  since we are e v a l u a t i n g the Taylor s e r i e s A more i n t e r e s t i n g r e s u l t i s the f o l l o w i n g :  We note from the above argument that the radius of convergence of the Sundman s e r i e s f o r the parameter of (3.25)" i s x • The numerical evidence x » 1  suggests that example.  Since  K  0  of Table X I I I  i n both Zumkley's and Stromgren's  was chosen equal t o 1  i n both  these examples, (3.26) gives the r a t h e r s u r p r i s i n g estimate  n« J  .31)  which i s i n marked contrast t o the values u s u a l l y computed from Sundman's estimate. Let us now note the f o l l o w i n g i n t e r e s t i n g p o s s i b i l i t y . On the b a s i s of the r e s u l t i n the l a s t paragraph, the poor r e s u l t s obtained.with the' transformation (3.22) can then be a t t r i b u t e d s o l e l y t o the exponential nature of the s i n h f u n c t i o n . (3.22) takes the l i n e s v  /  E s s e n t i a l l y , the transformation ouo — + Ifi , -°° < too < +  c0  , of  the ou-plane and f o l d s them onto the h a l f - l i n e s [I,i») and  [ - i , - i»)  i n the T-plane.  I t was pointed out  e a r l i e r that such a "severe" mapping i s necessary f o r the as  ( E , r ) method i f ' W(r)  i s t o tend to I n f i n i t y  r -* 0 . However, such i s not the case f o r the  (E,r,0,2)  method.  L e t us assume i t i s p o s s i b l e t o  -82-  f i n d a conformal mapping .32)  z = h(tt))  which takes the l i n e s  w  Q  + if} of the to-plane Into two  curves which l i e above and below the two l i n e s passing through the o r i g i n and i n c l i n e d at 45° to the p o s i t i v e and negative z-axes. s t r i p of width  2Q  L e t us f u r t h e r assume that the i s mapped i n t o the r e g i o n between  the two curves and that the r e a l tu-axis i s mapped onto the r e a l z - a x i s . of  D  r  f o r the  I f we now consider the l i m i t i n g shape (E,r,0,2)  method as i l l u s t r a t e d i n  Pigure 1 and r e f e r t o the method ,ot c o n s t r u c t i n g D ( x ) , D ( y ^ ) , e t c . , as o u t l i n e d i n Section 1.1, i t r  1  r  i s not hard t o see that i n t h i s case  W(r)  *  as  r -• 0 . Unfortunately, the author has not y e t been able t o f i n d a s u i t a b l e mapping (3.32). I n concluding t h i s chapter, l e t us make the general remark t h a t the methods used i n t h i s work do provide e f f e c t i v e me^ns of o b t a i n i n g the s o l u t i o n to,the twoand three-body problems w i t h i n a . l i m i t e d i n t e r v a l . Moreover, w i t h i n t h i s I n t e r v a l they are more e f f e c t i v e than Brumberg's method as w e l i as being much simpler to use.  Furthermore, I t i s not c l e a r from Brumberg's  paper [ l ] that the p r a c t i c a l l i m i t to the I n t e r v a l of convergence on i\rhich h i s method w i l l work extends much  -83beyond ours.  I t should be pointed out, however, that  our methods require some knowledge of the d i s t r i b u t i o n of the s i n g u l a r p o i n t s of the s o l u t i o n while i n Brumberg method, one need only know whether or not the r e a l a x i s l i e s i n the p r i n c i p a l s t a r of the s o l u t i o n f u n c t i o n . A l s o , the e n t r i e s i n the matrix  G  of Brumberg's  method are of a u n i v e r s a l character and thus, once c a l c u l a t e d , they can be stored and used f o r any future problem. One could p o s s i b l y broaden and improve our r e s u l t s by considering other displacement, polynomials; perhaps polynomials l e a d i n g t o methods outside the c l a s s  (E,r,a  In a d d i t i o n one might be able to obtain good r e s u l t s by an appropriate combination and conformal mappings.  of summation techniques  Other problems might a l s o be  t r i e d ; f o r instance l e t us consider the problem of the motion of a heavy r i g i d body about a f i x e d p o i n t . The equations of motion f o r t h i s problem are (see Lelmanis,  [ l , Chapter I ] ) A£ = (B-C)qr + m g O z ^ v y J , •  Bq = (C-A)rp + mg(YX -az ) , o  .33)  Q  Cr = (A-B)pq + mg(ay -Px ) , o  a = pr - y q  ,  m.  3 = yP - a r > Y = aq - 3p  o  -84where  A, B, C,rag,X , y q  p o s i t i v e constants,  and z^ are c e r t a i n r e a l  Q  p,q and r are the components of  the angular v e l o c i t y vector with respect t o a conveniently chosen body-centered set of axes, and a, B and y the d i r e c t i o n cosines of the f i x e d t o these axes. (3.34)  are  OZ a x i s with respect  Such a system admits the i n t e g r a l s  Ap + Bq + C r = 2 ^ - x ^ - y ^ - z j ) 2  2  2  and  (3.35) where  a + B +Y = 1 2  2  Prom (3.34) and (3.35)  h^ i s a l s o a constant.  we immediately  (3.36)  2  obtain  |a|,  | Y !< 1  and  (3.37)  |p|, |q|, | r |  < | (|h |-K/3k) - K /  1  where K' = min(A,B,C) > 0 , k = max( | X | , |y |, | z j ). q  Q  Consider now the f o l l o w i n g theorem due t o Cauchy on the existence of s o l u t i o n s of d i f f e r e n t i a l equations (see P i c a r d , [ l , Chapter X I ] ) .  -85THEOREM _5_  L e t Q j ( q q , . . . ,q ) l5  2  which do not contain the time  n  t e x p l i c i t l y and which  are developable i n powers of q^ - q (3.38)  i Q  \q - q | < q^ ±  where the  whenever (1=1,2,..,,n)  l Q  q^ are c e r t a i n p o s i t i v e constants.  there e x i s t s p o s i t i v e f i n i t e constants (3.39)  be functions  iQjU-^qg/---^)!  whenever the v a r i a b l e s  < Qj  Assume  Qj such that (J=l,2,...,n)  (3.38).  q^ s a t i s f y  Under these c o n d i t i o n s the system of d i f f e r e n t i a l equations  (3.40)  q.y= Q j U ! * ^ " * n ^  (j=l,2,. • • ,n)  , q  admits one and only one s o l u t i o n which has the property that  q. tends t o q. ^•I io  as t tends t o a f i n i t e value  t o • In t h i s s o l u t i o n , the functions developed i n power s e r i e s i n t - t (3.41)  q. i can be provided  |t-t | < T Q  where *2 , -..,-"-) n , T = m i n ( l4 , -f. Q; Q4 o; q  q  -86-.  s a t i s f i e s ( 3 . 4 l ) the functions  Moreover, i f t  q^  satisfy (3.38). Let us now w r i t e the d i f f e r e n t i a l equation ( 3 . 3 3 ) i n the form P =  + T r O V  0  0  ^  =  Q (  p , q  1  '  r , a , P , Y  ^  q = -^g^-rp + M ( x - a z ) = Q (p,q,r,a,B, h Y  o  o  2  Y  r = - ^ ^ - p q + x r ( a y - P * ) = Q ^ p ^ r ^ B ^ ), o  (3.33)'  o  a = Br - yq = Q (p,q,r,a,B, ) , if  Y  B = Y P - ar = Q ( p , q , r a , 3 , ) > 5  Y  }  y = aq - Bp = ^ ( p , q , r , a , p , Y ) • Assume we are given r e a l constants &  with  a  Q  ,6  and Y  q  satisfying (3.34).  choose a r b i t r a r y p o s i t i v e constants (3.43)  k-q l> |r-r | < a  |a-a |,  |P-P |,  o  Y (3-44)  Q  r  (3.35)  and  ro  Let us now and  IY-Y I Q  b  and assume  < b .  s a t i s f y ( 3 . 3 4 ) and v  we obtain |p|,  |q|, | r | < K + a  |a|,  |p|, | | < 1 + b Y  and Y „o  Q  p  Q  po > 1 o  a  |p-P L 0  Since  p ^ ,q o o, ro ,a o ,o3  ao , 3o  and  -87-  and  thus, from  (3.33)'  |Q I,  ,  |-Qg|, |Q_|  1  < L(K+a)  + 2tk{l+h)  2  ,  (3."*5) \%\> l Q l » l % l < 2(K+a)(l+h) 5  where  I f we now define  T by  T = T(a,b) = min( ^ , — ) L(K+a) +2tk(1+b) 2(K+a)(1+b) b  (3.47)  we can prove the f o l l o w i n g THEOREM 6 _  I f a t some r e a l instant  given the r e a l constants satisfying a(t),  3(t)  P(°) =  X> > 0  (3.35),  p ,q ,r Q  Q  then the solutions  and y ( t ) of equations q(°) = Q  the complex v a r i a b l e  0  to  ct , 3 Q  we are and Y  q  O  p(t), q(t), r ( t ) , (3.33)  satisfying  , etc. are a n a l y t i c functions of t  i n a s t r i p of width  2T about  the r e a l t - a x i s . Proof: and  b  Since f o r a r b i t r a r y p o s i t i v e numbers  i n ( 3 . ^ 3 ) the conditions of Theorem  5  a  are s a t i s f i e d ,  we are therefore quaranteed the existence of s o l u t i o n s which are a n a l y t i c i n the d i s k  | t - t | _< T where  T is  -88given by (3.47). 't  < t' < t  L e t t ' by any value of t  + T . The f u n c t i o n s  satisfying  p(t), q(t), etc.  w i l l be a n a l y t i c a t t ' and t h e i r values there w i l l s a t i s f y (3.36) and (3.37) since f o r r e a l functions are r e a l valued.  t the  Taking t h e i r values a t t '  as i n i t i a l values and using the same  a  and b  as i n  (3.43), the estimates (3.44) and (3.45) are s t i i r v a l i d and thus  T  remains unchanged.  Applying Theorem 5  again we are able t o conclude t h a t the functions p(t),  q ( t ) , e t c . are a n a l y t i c i n the d i s k | t - t ' j £ T .  I t f o l l o w s that f o r any r e a l value  t * of t the  functions are a n a l y t i c i n the d i s k  | t - t * | < T and  thus they are a n a l y t i c i n the s t r i p of width  2T  about  the r e a l t - a x i s . R e c a l l that  a  and b  p o s i t i v e r e a l numbers. .48)  used i n (3.43) are a r b i t r a r y  Let us therefore define  Cl by  Q = max T(a,b). a>0,b>0  Determination of the maximum involves the s o l u t i o n of a q u a r t i c equation i n b when  which reduces t o a quadratic,  L = 2 . Theorem 6 can now be r e s t a t e d with  replaced by  Q .  T  -89Thus we see t h a t , from the p o i n t of view of the d i s t r i b u t i o n of the s i n g u l a r i t i e s , the problem of the motion of a heavy r i g i d body about a f i x e d p o i n t bears some resemblance t o that of the r e g u l a r i z e d three-body problem.  Consequently, the methods which we have  used i n t h i s work and that a l s o of Brumberg's can be a p p l i e d t o t h i s problem as w e l l .  -90-  BIBLIOGRAPHY Agnew, R. P.  E u l e r transformations.. Amer. J . Math. 66 (19^4), 3 1 3 - 3 3 8 . The Lototsky method f o r the e v a l u a t i o n pf s e r i e s . Michigan Math. J . 4 ( 1 9 5 7 ) / IO5-128.  BajSanski, B. M.  Sur une classe generale de procedes de sommations du type d ' E u l e r - B o r e l . Acad* Serbe. S c i . Publ. I n s t . Math. 10 (1956),  B e l o r i z k y , D.  131-152.  Recherches stir 1 ' a p p l i c a t i o n pratique des s o l u t i o n s generales du probleme des t r o i s corps. J . des observateurs  A6  (1933),  109-132,  149-172,  189-211.  Brumberg, V. A.  Polynomial s e r i e s i n the three-body problem (Russian). B y u l l . I n s t . Teor. Astron. 9 no. 4 ( 1 0 7 ) ( 1 9 6 3 ) , 2 3 4 - 2 5 6 .  Burrau, C.  Numerische Berechnung eines S p e c i a l f a l l e s des Dreikorperproblems. Astron. Nachr. 195 ( 1 9 1 3 ) ,  113-118.  Cooke, R. G.  I n f i n i t e Matrices and Sequence Spaces. London : MacMillan, 1 9 5 0 .  Danby, J . M.  Fundamentals of C e l e s t i a l Mechanics. New York : MacMillan, 1 9 6 2 .  Dienes, P.  The Taylor S e r i e s . Press, 1 9 3 1 .  Hardy, G. H.  Divergent S e r i e s . Press, 1 9 4 9 .  H i l l e , E.  A n a l y t i c Function Theory, V o l . I I . Boston, New York: Ginn and Co. i 9 6 2 .  Knopp, K.  Theory and A p p l i c a t i o n of I n f i n i t e S e r i e s . 2nd Eng. e d i t . New York: Hafner, 1 9 4 ? .  Oxford: Oxford:  Clarendon Clarendon  Uber das Eulersche Summierungsverfahren I . Math. Z. 15 ( 1 9 2 2 ) , 2 2 6 - 2 5 3 . Uber das Eulersche Summierungsverfahren I I . Math. Z. 18 ( 1 9 2 3 ) , 1 2 5 - 1 5 6 .  -91-  Leimanis,  E.  [1] M o t i o n o f C o u p l e d R i g i d B o d i e s about a Fixed P o i n t . Berlin. Heidelberg-New York: SpringerV e r l a g , 1965. [l]  M o u l t o n , F. R.  The t r u e r a d i i o f c o n v e r g e n c e o f the expressions f o r the r a t i o s o f t h e t r i a n g l e s , when d e v e l o p e d as p o w e r s e r i e s i n t h e t i m e intervals. A s t r o n . J . 23 ( 1 9 0 3 ) , 93-102).  Picard,  [1] T r a i t e d ' A n a l y s e , Gauthler-Villars,  E.  Sonnenschein,  J.  [l]  Vol. I I , Paris: 1905.  Sur l e s s e r i e s d i v e r g e n t e s . T h e s i s , B r u s s e l s , 1946.  [2] Sur l e s s e r i e s d i v e r g e n t e s . Bull. A c a d . R o y a l e de B e l g i q u e 35 ( 1 9 4 9 ) , 594-601.  [3] Sur une c l a s s e de p r e c e d e s de sommation. Colloque sur les T h e o r i e des s u i t e s ( B r u s s e l s , 1957). P a r i s : G a u t h i e r - V i l l a r s , pp.  Steffanson,  Stromgren,  J.  E.  F.  [l]  119-130.  On t h e p r o b l e m o f t h r e e b o d i e s i n the plane. M a t . - f y s . Medd. Danske V i d . S e l s k . 31 No. 3 ( 1 9 5 7 ) , 18 p a g e s .  [1] E i n n u m e r i s c h e g e r c h n e t e r S p e c i a l f a l l des D r e i k b r p e p r o b l e m s m i t Massen untf D I s t a n z e n v o n d e r s e l b e n Gr'o3enordnung. A s t r o n . Nachr.  Sundman, K.  F,  [l]  1§2  (1909),  l8l-192.  Memoire s u r l e p r o b l e m e des t r o i s c o r p s . A c t a . M a t h . 36 ( 1 9 1 2 ) , 105-179.  Zumkley,  J.  [l]  E i n numerische gerechneter S p e c i a l f a l l des a l l g e m e i n e n Dreikorperproblems i n vereinfachter B e h a n d l u n g . A s t r o n . N a c h r . 272 (1941),  66-76.  

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