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The solution of differential equations through integral equations Swanson, Charles Andrew 1953

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THE SOLUTION OF DIFFERENTIAL EQUATIONS THROUGH INTEGRAL EQUATIONS by C h a r l e s Andrew Swanson  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of MATHEMATICS  We accept t h i s t h e s i s as conforming t o t h e standard r e q u i r e d from candidates f o r t h e degree of MASTER OF ARTS.  Members o f the Department o f Mathematics  THE UNIVERSITY OF BRITISH COLUMBIA April,  1953-  ABSTRACT  A method o f w r i t i n g the s o l u t i o n of a second d i f f e r e n t i a l e q u a t i o n through i s developed.  and t o bounded Quantum Mechanical  prob-  Some o f t h e r e s u l t s o b t a i n e d are o r i g i n a l , and other  r e s u l t s agree others.  Equation  The method i s a p p l i e d t o i n i t i a l value problems,  to special functions, lems.  a Volterra Integral  order  e s s e n t i a l l y w i t h t h e work done p r e v i o u s l y  by  ACKNOWLEDGEMENTS  We a r e deeply indebted t o Dr. T..E. H u l l o f t h e Department o f Mathematics a t the U n i v e r s i t y o f B r i t i s h Columbia f o r s u g g e s t i n g t h e t o p i c , and f o r r e n d e r i n g invaluable  assistance  i n d e v e l o p i n g the i d e a s .  Our  f u r t h e r thanks are due t o o t h e r members of t h e Department of Mathematics, e s p e c i a l l y t o Dr. R.D. James and Dr.  G.E. L a t t a , and t o members o f the Department o f  P h y s i c s at t h e U n i v e r s i t y o f B r i t i s h Columbia.  We a r e  a l s o pleased t o acknowledge t h e f i n a n c i a l support o f t h e National  Research C o u n c i l  of Canada.  TABLE OF CONTENTS  INTRODUCTION CHAPTER ONE.  (i) THE GENERAL METHOD  S e c t i o n 1.1.  (1)  D e r i v a t i o n Of The V o l t e r r a I n t e g r a l Equation  S e c t i o n 1.2.  (1)  D e t e r m i n a t i o n Of The ArbitraryConstants  S e c t i o n 1.3.  The V o l t e r r a I n t e g r a l E q u a t i o n F o r The  S e c t i o n 1.4.  (3)  General Cauchy Problem  The E q u a t i o n  With The F i r s t  (5) Derivative  Term M i s s i n g S e c t i o n 1.5.  S o l u t i o n Of The I n t e g r a l E q u a t i o n  S e c t i o n 1.6.  The General S o l u t i o n Of A Related  (10) ... (11)  Integral Equation  (16)  CHAPTER TWO. APPLICATION TO INITIAL VALUE PROBLEMS ... (20) S e c t i o n 2.1.  Introduction  S e c t i o n 2.2.  A Problem With A P e r t u r b a t i o n I n The F i r s t D e r i v a t i v e Term  (20)  (20)  S e c t i o n 2.3v  Another S i n g l e P e r t u r b a t i o n Problem . (22)  S e c t i o n 2.4.  M u l t i p l e P e r t u r b a t i o n Problems  S e c t i o n 2.5.  Another Treatment Of The Problem I n S e c t i o n 2.2  CHAPTER THREE.  APPLICATION TO SPECIAL FUNCTIONS  S e c t i o n 3.1.  Introduction  S e c t i o n 3.2.  The Expansion Of The S o l u t i o n Of The Confluent  (23)  (26) (23) (2$)  Hypergeometric Equation I n  S e r i e s o f B e s s e l Functions  (29)  TABLE OF CONTENTS (Continued) S e c t i o n 3.3.  A G e n e r a l i z a t i o n Of The  S e c t i o n 3.4.  A Further Generalization  S e c t i o n 3.5.  S o l u t i o n Of Related  Problem  (33) (35)  Differential  Equations Expressed In Terms Of  S e c t i o n 3.6. CHAPTER FOUR.  B e s s e l Functions  (37)  Appendix To Chapter Three  (39)  PHYSICAL APPLICATIONS  S e c t i o n 4»1«  Introduction  S e c t i o n 4.2.  The  I n t e g r a l Equation  (42) (42) For  Bounded  Quantum Mechanical Problems  (42)  S e c t i o n 4.3.  The  Bounded Hydrogen Atom Problem  S e c t i o n 4.4.  The  Bounded R i g i d R o t a t o r  S e c t i o n 4.5.  The  Ground L e v e l Of The Bounded R i g i d  Rotator S e c t i o n 4.6.  Higher L e v e l s Of The Rotator  BIBLIOGRAPHY  .(45) (49)  (51) Bounded R i g i d (54) (57)  (i)  INTRODUCTION The c e n t r a l theme of t h i s t h e s i s i s t h e use o f a Y o l t e r r a I n t e g r a l E q u a t i o n t o express t h e s o l u t i o n s of a second order d i f f e r e n t i a l e q u a t i o n i n terms of known functions.  By the procedure which i s f o l l o w e d , i t i s then  p o s s i b l e t o d e r i v e c e r t a i n p r o p e r t i e s of these systematically. ville tury.  (a)  solutions  The i d e a o r i g i n a t e d w i t h Cauchy, L i o u -  ( 7 ) , and contemporaries  i n the e a r l y n i n e t e e n t h cen-  I n p a r t i c u l a r , L i o u v i l l e transformed the e q u a t i o n  u V ; +• f -«&)  =  2  u&)  i n t o the i n t e g r a l e q u a t i o n of t h e second k i n d  x (b)  u&) =  u&) coyf*.+ jru (o)finf* f  + f-/ fr(t)s\*f(x<)u®tt.  The c l a s s i c a l approach was t o consider (a) as a nonhomogeneous d i f f e r e n t i a l e q u a t i o n whose homogeneous p a r t has known s o l u t i o n s  cos j>*  and  si^f-x  the method o f v a r i a t i o n of parameters  f  a  n d t o apply  or Laplace Trans-  form t h e o r y t o o b t a i n t h e i n t e g r a l e q u a t i o n (b) i n terms o f these known f u n c t i o n s .  The equation  (b) was used t o  study t h e asymptotic behaviour of the eigenvalues and t h e e i g e n f u n c t i o n s o f (a) f o r l a r g e f  •  More r e c e n t i n v e s t i g a t o r s , n o t a b l y Ikeda ( 4 ) , Fubini  ( 3 ) , and T r i c o m i (15) have changed the viewpoint  t o t h a t o f comparing t h e unknown s o l u t i o n s o f (a) w i t h the known s o l u t i o n s o f a d i s t i n c t d i f f e r e n t i a l  (c)  +• f^vC*) =  O,  o r , i n g e n e r a l , of comparing the s o l u t i o n s of  equation  (ii)  (d)  P&) ufo-h  uCr) = O  w i t h the supposed known s o l u t i o n s  (e)  ir"&) -f R&) ir'fr) +  through  &)  In  T h i s approach w i l l  the f i r s t  c h a p t e r , the i n t e g r a l e q u a t i o n asso-  w i l l be g i v e n f o r the d e t e r m i n a t i o n o f the a r b i t r a r y t a n t s i n order t h a t the i n i t i a l  n- -r*«  The  consThe  order l i n e a r  d i f f e r e n t i a l equation, g i v i n g a r e s u l t e n t i r e l y the second o r d e r case.  procedure  c o n d i t i o n s be s a t i s f i e d .  whole i d e a w i l l be g e n e r a l i z e d t o an  analogous  a p p r o p r i a t e e x i s t e n c e theorems  needed i n l a t e r chapters w i l l be The  be  the d i s c u s s i o n .  c i a t e d w i t h the e q u a t i o n (d) w i l l be d e r i v e d , and a  to  of  SfrJ *rfr) _- o  a V o l t e r r a Integral Equation.  used throughout  i / ^ &?  and  proved.  very nature of the method suggests t h a t i t be  used t o get expansions  of s o l u t i o n s of c e r t a i n  differential  equations i n terms o f b e t t e r known s o l u t i o n s of other equations.  I n the second  c h a p t e r , we use t h i s i d e a t o expand the  s o l u t i o n s o f the Confluent Hypergeometric F u n c t i o n s o f the f i r s t  and  second  kind.  Equation i n Bessel The  computational  value o f such an expansion has been d i s c u s s e d by K a r l i n ( 9 ) . In  the f o u r t h c h a p t e r , we  value problems as w e l l as i n i t i a l handled  by adapting the method.  s h a l l show t h a t boundary value problems can be  In p a r t i c u l a r , the bounded  Quantum Mechanical problems are d i s c u s s e d , and for  the Hydrogen atom problem and f o r the r i g i d  problem are  calculated.  eigenvalues rotator  (1)  CHAPTER ONE THE  1.1  GENERAL METHOD  D e r i v a t i o n o f t h e V o l t e r r a I n t e g r a l Equation. The  o b j e c t i s t o express t h e s o l u t i o n s o f the  second o r d e r d i f f e r e n t i a l (1.1.1)  equation  " V ? + P(*) u'?) -h p G O u&) = O  i n terms o f the known s o l u t i o n s (1.1.2)  nr &) + u  and  R (x) *rV; +  o f t h e equation  S&)*r&)^o  by a V o l t e r r a I n t e g r a l E q u a t i o n .  I t i s assumed t h a t t h e  e q u a t i o n has t h e same s i n g u l a r i t i e s as the 'V  equation.  U The  r e s u l t w i l l be o b t a i n e d by adapting the method o f v a r i a t i o n o f parameters f o r s o l v i n g non-homogeneous d i f f e r e n t i a l l e t ( 1 . 1 . 1 ) be r e w r i t t e n i n the  equations: (1.1.3)  +  +  = CW-PCrlJ  form u &? -4- Ls&l-Q&ijufr) r  w i t h supposed s o l u t i o n (1.1.4) where  u&) =  <ri &)*ri&)+  v,6) and  (1.1.5)  satisfy  C^rJ^r) (1.1.2).  From e q u a t i o n  (1.1.4),  c^i^z)  w V ; •= ^ £ ? * / < - w - f  provided that (1.1.6)  c/@1 ir &)+•  &) «^&?  {  =0  }  and (1.1.7) If  u &) = c &)<yr "(x) + ^ &) *r u  (1.1.4),  t  {  ( 1 . 1 . 5 ) , and ( 1 . 1 . 7 )  + c/&7 ^'(rf + c/fr)  are s u b s t i t u t e d i n t o  (1.1.3),  the r e s u l t i s (1.1.S)  cfc)^?)  <±(*)*'Gr>  =  LZ(zl-PQr)]  </&)+  [?p)-QvJu0rl.  (2)  o  (.1.1.8) i s  o  §  (1.1.9)  Via)  where Hence, (1.1.10) and  ***>- * + /  (1.1.11) where  and  fi  }  are constants of i n t e g r a t i o n ,  and t h e Wronskian o f ^  stant , (1.1.12) If  U&)  i s a con-  i s g i v e n by  'a? I AT,  (1.1.10) and (1.1.11)  that  and  b  are put i n t o  must s a t i s f y t h e i n t e g r a l  (1.1.4), i t i s seen equation  or (1.1.13) U&) - ci  -f /?, *rj& +  f NCt x)(^lt  ffrl}(/e)Jx + f *M¥0(S-tp} u  where (1.1.14)  1  If b  M l v ) J W .  i s not an o r d i n a r y p o i n t of the d i f f e r e n t i a l equation  (1.1.1), t h e r e r e s u l t s apply only f o r the s o l u t i o n  U(2j  of  (1.1.1) which i s f i n i t e at b ; the d e f i n i t e i n t e g r a l s i n (1.1.10) and  (1.1.11) do not e x i s t i n g e n e r a l f o r the other  However, by d e l e t i n g the constant the i n t e g r a l s , t h e r e s u l t  b  solutions.  f r o m t h e lower l i m i t of  (1.1.15) i s expressed  i n terms o f an  i n d e f i n i t e i n t e g r a l , and the above r e s t r i c t i o n i s removed.  (3)  I n order t o get  i n the form of a V o l t e r r a  (1.1.13)  E q u a t i o n , we remove the  u'fr) term by performing  Integral  a partial  i n t e g r a t i o n : with  and  we o b t a i n  since  %  /yf.  from  •  (1.1.14)  Thus,  b  if where  ^  and  /%, are c o n s t a n t s .  Putting t h i s into  ( 1 . 1 . 1 3 ) ,  we o b t a i n f i n a l l y  U&  (1.1.15)  =  /?ntj&  -4-  /"  K(*,x^) utvlJx  y  where  K C*,TC) =  (1.1.16)  and  1.2  ^  c < J  'i+  e /  7.  Wa.x;  and  {s  /3^/J,-f /?_. are c o n s t a n t s .  D e t e r m i n a t i o n of the A r b i t r a r y For i n i t i a l  Constants.  value problems, the c o n s t a n t s  can be c a l c u l a t e d e x p l i c i t l y . we f i r s t  | i { /VMfe-P«))}  In order t o a r r i v e at t h e r e s u l t ,  need t o develop a p r o p e r t y of t h e f u n c t i o n  s t a t e d i n the Lemma: Proof: (1.2.1)  A l s o from  K  —  /Z&)-Pgr) .  From (1.1.14) ^ 7 ( T a ( 1 . 1 . 1 4 ) ,  (  * ; = o  and ft  y  /  V  M =  o.  fCC^z)  ,  (4)  Hence, (1.2.2)  |Ofe*>  „  _ K * ;  941 foe?  f W^;  =  (1.2.3)  'd'iLift)  ^ . » | . u/^;  _  =  , or  n c e ^ ^ ^ l  7  where use i s made o f (1.2.1) and (1.2.2).  From  (1.1.16),  The r e s u l t now f o l l o w s because of (1.2.1) and (1.2.3). From (1.2.4)  (1.1.15),  =  t>}'+  fl^W  and (1.2.5)  ufe =  f* ?£ >* Jx-f(9  7  K(**)u&).  Upon use of the Lemma,  Hence, f o r (1.2.6)  t = i>  uV ~ teQ,)-Pto}nQ = <«i'<i>) + /nr^O-) .  The s o l u t i o n of t h e l i n e a r a l g e b r a i c equations (1.2.6)  u(i)  then gives  and  u'ii?  and (3  (1.2.4) and  i n terms of the i n i t i a l  values  :  where (1.2.8)  <%)=  u ^ ;-  { zto-wy  T h i s i s t h e r e s u l t t h a t Ikeda method•  (4)  uQ,) . obtained by a d i f f e r e n t  (5) 1.3  The V o l t e r r a I n t e g r a l E q u a t i o n For The.General Cauchy  Problem. The s o l u t i o n of t h e n-th order l i n e a r  differential  equation  (1.3.1)  u  assuming t h a t  P J5t? u m  i n i t i a l values  U  Q.) (4=/,Zj • - -  i s t o be expressed i n terms of the l i n e a r l y independent solutions  ^(xl  (1.3.2)  C^=/ i .--•*,_) }  -v h?-<- Z  *  (H  W  C  " \ x ) = o  through a V o l t e r r a I n t e g r a l E q u a t i o n . functions  P  of  .  ^  (1.3.3)  and K &) Let  We  suppose t h a t the  are a n a l y t i c f o r a l l r e q u i r e d  value  (1.3.1) be r e w r i t t e n i n t h e form  " " (*)-h C  o f the equation  Z  }  Z^)*"*)-  Z f ^ K  ^  1  ^  ;  w i t h supposed s o l u t i o n  (1.3.4)  u(*) =  Z  Cr&)«rr*).  >--=/  F o l l o w i n g the method of V a r i a t i o n o f Parameters, we  have  « 6r? = IE C Qr? 4/^V>, g  r  (1.3.5)  fi-i?  p r o v i d e d t h a t the  (1.3.6)  Putting  *  c«V  c s  S,  vJ&=o  9  (I.3.5) i n t o (I.3.3), we get  (1.3.7) %<^*>V""^ Let  satisfy  }  r  =  Z  { *~&>-rjfr)}u"'i)=  iV^/) be t h e Wronskian o f the n f u n c t i o n s  ' ^Ct)  $ • (*-l,Z •••  (6)  (1.3.8)  \AI& =  (»->? Since  \AJ^)4=-0  equation ( 1 . 3 . 6 )  , the unique s o l u t i o n of the n a l g e b r a i c and ( 1 . 3 . 7 ) i s  (1.3.9) ( 1 . 3 * 4 ) , we  P u t t i n g these v a l u e s i n t o (1.3*10) where  " # = Z < ^ +  obtain  I.  a r e the n c o n s t a n t s o f i n t e g r a t i o n i n v o l v e d i n  computing the cjs  , and where  t  i s a f i x e d c o n s t a n t . Summing  the determinants under the i n t e g r a l s i g n i n ( 1 . 3 * 1 0 ) , we have  dx  (1*3.11)  V,<•*-) V)  Writing (1.3.12) and r e p l a c i n g (1.3.13)  £?  by i t s v a l u e from ( 1 . 3 . 7 ) ,  u « . i « i  r  ^ +  /  we get from  KUtfLi^V-Zr *)}  (1.3.11),  * *T*tix.  1  l  or upon i n t e r c h a n g i n g i n t e g r a t i o n and summation, (1.3.14)  «0 = Z*r*rti+LJ*NMl*&-Z&  ( WvJft  To t r a n s f o r m t h i s i n t o a V o l t e r r a I n t e g r a l E q u a t i o n , we need t h e Lemma. (1.3.15) Proof.  For r - / ^ . . [  3 W \^  , and K-/,Z - • - (.»•*•-() , the f o l l o w i n g { % ) f ^ - ^ ) -  From t h e d e f i n i t i o n o f  t>J]J * = 2r  (1.3.12),  =  0  ho  (7) ye]  WW  3*  ^ CO v, en  <n-l>  4? Chi,  V Or) K  Each determinant involved i n the partial derivatives  contains a non-derived f i r s t row.  Hence, for  =?  ,  this f i r s t row becomes identical with the last row, and the determinant vanishes;  therefore, 9  (1.3.16)  3x  From (1.3.12)  M^,*?  =  0  i t follows that each of the partial derivatives  contains only terms with  Cvi-i") ,  as factors, so by (1.3.16) (1.3.17) For  NC*.*) =  t-=./,z . . * . i  = , and  = z ..  w-k-/  , we have  Since each term i n the summation has as a factor one of the  (8) partial derivatives  l^NCzrf/a*  (j=o i-. (yt-2)), e q u a t i o n  shows t h a t the r i g h t s i d e vanishes when ?c=z . and  (1.3.17)  hence the Lemma i s proved. We now perform  (  *~)  p a r t i a l i n t e g r a t i o n s upon t h e  i n t e g r a l s on the r i g h t s i d e of (1.3.14), and use t h e Lemma at each stage t o s i m p l i f y t h e i n t e g r a t e d  part:  b  where  /3  e  =£  pending upon  K b  •  »  A  N  D  A  1  1  T  H  E  ^JI.K  a  e  c o n s t a n t s de-  P u t t i n g these v a l u e s i n t o t h e summation i n  (1.3.14), we g e t  + f*NC  where t h e  f  (1.3.19)  K (i,x) - f  r  r  ,  [R (x)o  .  f&Ju&Jx,  a r e c o n s t a n t s , and t h e k e r n e l i s g i v e n by ^ ^ " " £ c ^ { / V ^ ) [ ^ - ^ J ^ .  (9) The a r b i t r a r y constants o f i n t e g r a t i o n determined  from the i n i t i a l v a l u e s = Z  U®0>)  •  )V  are  .  From.(1.3.lg),  ,  y=.i v.  Now,  the expansion o f  3 i^Ct 7c)/^x ''' as a sum of determinants <u,J  i  0  )  c o n t a i n s e x a c t l y one term w i t h t h e f i r s t  row d e r i v e d once, as  w e l l as the other rows down to the (t«-^st d e r i v e d once, and c o n t a i n s a l l other terms w i t h a non-derived f i r s t for  * =2  row.  Hence,  , i t follows that  (1.3.20)  9  l  M  "  ;  ^x. '° - C->/*' W&)  1*1 (*, 9)/  iM  From t h i s and from the d e f i n i t i o n 3  N L  (1.3.12), ^  S u b s t i t u t i o n of t h i s and t h e v a l u e s  (1.3.21)  K(  = «  Z/  Hence,  a) -  .  i t i s seen t h a t  en  (1.3.17)  into  (1.3.;l9) <  gives  .  „  S i m i l a r l y , f o r t h e i-ru d e r i v a t i v e , we get an e x p r e s s i o n o f the  form  (1.3.22)  f a) =  where  •§-^(i>)  and  P  of  z  £  {%)  depends upon the i n i t i a l  (f = i i f  the l i n e a r e q u a t i o n  (1.3.23)  V  where  <w, •• •  values  m . j£=o, 1 z . - ••  iS*\ ^„.J-^;  • The  solution  (1.3.22) i s  = -7-  i s the Wronskian of the n f u n c t i o n s In  U  rffi)  .  c o n c l u s i o n , we have reduced the g e n e r a l Cauchy  (10) problem t o t h e problem o f s o l v i n g the i n t e g r a l  (1.3.20), where a l l the c o n s t a n t s (I.3.23) 1.4  tr  i n terms o f the Cauchy i n i t i a l  equation  are g i v e n by values.  The E q u a t i o n With The F i r s t D e r i v a t i v e Term M i s s i n g , . T h i s s e c t i o n a p p l i e s t o the case s e c t i o n 1.1.  in  kernel w i l l  w=A  discussed  From e q u a t i o n (1.1.16), we see t h a t t h e  simplify to  (1.4.1)  { S&?-Q$)}  if  Pfr?=0. We can arrange t h i s i n c e r t a i n problems by  c h o o s i n g t h e t r e q u a t i o n such t h a t  P=  • however, i n other  cases o f i n t e r e s t t h i s c h o i c e i s i n c o n v e n i e n t , and i n s t e a d we t r a n s f o r m t h e v a r i a b l e s so t h a t the new f u n c t i o n s P&9 and  are  zero i n (1.1.1) and (1.1.2).  We now show t h a t the  l a t t e r e n t a i l s no l o s s o f g e n e r a l i t y ; t h a t i s , s t a r t i n g the second  (1.4.2)  order d i f f e r e n t i a l  A"t?+  with  equation  Ler)ji'e;  + [nt) +  'hTei]JLe)=o  w i t h r e a l , continuous, d i f f e r e n t i a b l e c o e f f i c i e n t f u n c t i o n s L£)  t  Mt) , and Tf?  , and non-negative TP , i t i s p o s s i b l e t o  change t h e v a r i a b l e s so t h a t  (1.4.3)  (1.4.2) becomes  [ Qfr? -+ 7iJ up) ^ O .  F i r s t , introduction of the integrating  enables  (1.4.4) where If  (1.4.2) t o be w r i t t e n i n the s e l f - a d j o i n t form +  ItV+TVJ^ti^O,  <fi £-) -= p <& nfc) j  i n (1.4.4) we make the t r i a l  (1.4.5)  factor  ' JL& = tf£lug) ;  ff?-=f>&T&. substitution  ^ =  9#7£  x  (11) t h e n , i n order t h a t the r e s u l t i n g f i c i e n t s of U  11  U  e q u a t i o n have the c o e f -  and "}\U the same, and i n order t h a t the U '  term v a n i s h , the f u n c t i o n s  ^fr)  a n c  *  &fr)  must s a t i s f y the  equations  2p9^4-  [ pg'-h /&]  with solution  & =  P u t t i n g these i n t o  (g/p) Vz,  , and  >  p9  =  X  =  y  ?  (pf)~  ,//4  (1.4.5), we o b t a i n the change o f v a r i a b l e s  Jit) -  (1.4.6)  fi-=^0  (Pf)- +u(?)  ; ^=  l/  which changes (1.4.2) i n t o  }  (1.4*3)•  We now o b t a i n s p e c i a l p r o p e r t i e s o f the s o l u t i o n s of (1.4*3)*  Let. two l i n e a r l y independent  u/V; 4- [ Q#7 4  {  andu^fr]:  7v J <t &? =Oj t  -h [ Qfr) I f we s u b t r a c t U,  s o l u t i o n s be u  «J?) = O .  times t h e second  e q u a t i o n from  times  the f i r s t , we o b t a i n  I W>  -  < *) } "= ° •  u  /(  I n t e g r a t i n g and then d i v i d i n g both s i d e s by  ~~ ~^frf  V.fr) 2  Since  U(  and  , we o b t a i n  Uz. are l i n e a r l y independent,  the constant OZ i s  d i f f e r e n t from zero, and i t f o l l o w s t h a t (1.4.7)  C  u  &  f  *  ^  .  Of course, the Wronskian i s g i v e n by (1.4.8) 1.5  H/#;=  u ?)«/fr?~ v-vfr/u/fr) = t  C •  S o l u t i o n of The I n t e g r a l E q u a t i o n . The i n t e g r a l e q u a t i o n (1.3.18) i s t o be s o l v e d by the  (12) method of" s u c c e s s i v e approximations.  Up*)*  (1.5.1)  U6$r) +  By t h i s  method,  £*K(i x) ctpi&Uz t  i s the p - t h approximation t o (1.3.18), where  (1.5.2)  U &) * dAT^) + 6  .  The Liouville-Neumann Theorem (#) s t a t e s t h a t a s u f f i c i e n t c o n d i t i o n f o r the sequence  t o converge t o the unique  continuous s o l u t i o n of (1.3.18) i s t h a t U &)  be continuous  0  and K£t x') t  tinuities.  be bounded w i t h o n l y r e g u l a r l y d i s t r i b u t e d d i s c o n However, i n the problems t h a t we s h a l l c o n s i d e r ,  the theorem cannot be a p p l i e d because  of the unbounded  nature  of the k e r n e l , and s p e c i a l c o n s i d e r a t i o n i s r e q u i r e d . I n t h e type of problem t o be d i s c u s s e d i n Chapter  (1.4.3),  t h e d i f f e r e n t i a l e q u a t i o n t o be s o l v e d has t h e f o r m w i t h the f i r s t  d e r i v a t i v e term m i s s i n g , and f u r t h e r  M  In t h i s case,  where  i s a constant.  (1.5.3)  •< (  -  The i n t e g r a l e q u a t i o n  (1.5.4) where  4,  S-Q-  H  (1.1.16) g i v e s  y W^x). (1.1.15) t h e n has the form, (withft- o ),  c<*/\£-;-4- pi jf* /VO,x; u ^ d r ,  (1.1.14),(1.4*7), and (1.4.8) show t h a t = *r  (1.5.5)  f*fiJ5f- f \ % i f ] .  4  The e x i s t e n c e of the s o l u t i o n of the i n t e g r a l e q u a t i o n  (1.5.4)  under q u i t e g e n e r a l c o n d i t i o n s w i l l now be proved. Theorem finite where  1  Suppose t h a t  z-plane w i t h zeros of o r d e r  U  v*\j a t  i s an e n t i r e f u n c t i o n without  negative number, and { H &}  <v~,&)/{(?-t>fa&}is a n a l y t i c i n t h e  b  a  i s d e f i n e d as  zeros, p b  •  z.  i s a non-  Then the sequence  of s u c c e s s i v e approximations a s s o c i a t e d w i t h  converges t o t h e unique continuous s o l u t i o n of all  %),  2 =  (1.5.4)  (1.5.4) f o r  (13) Proof.  Let  (1.5.6)  (1.5.4)  U0  be r e w r i t t e n i n t h e form  --  L^;r)ur)J3C  ;  where  By h y p o t h e s i s , ^fr)  may be expanded about any o f the p o i n t s  i n a s e r i e s o f t h e form  (i.5.«)  *;« -  0(<*-yW],  w i t h i n f i n i t e r a d i i o f convergence, where we d e f i n e  S u c c e s s i v e approximations t o the s o l u t i o n of  (1.5.9)  «& =  -f  h  where  (1.5.10)  J  t  (1.5.6)  are  f^&j  LC^z) ^£c) ft^&Jx  and U. (?) =  ,  The proposed s o l u t i o n of  (1.5.11)  = 1  (1.5.6)  U.&) = *t  The nature o f t h e f u n c t i o n s  i s then  Z.  'frfe>.  ft^fc)  w i l l now be examined; from  (1.5.7) and (1.5.8),  (1.5.12) L( xl h  *0  J  - [^Ix-^+^te-^'XJfw^V.  (1.5.13) L ^ j ) .  }[^C -.p7 ]lH  2  Suppose now without l o s s o f g e n e r a l i t y t h a t the zeros a r e o r d e r e d a c c o r d i n g t o i n c r e a s i n g moduli  Since the s e r i e s on t h e r i g h t o f (I.5.8) has an i n f i n i t e r a d i u s of convergence, t h e f i r s t  series involving  %  in  (14) (1.5.13) i s u n i f o r m l y convergent can be i n t e g r a t e d termwise. be r e p r e s e n t e d  for  i n the c i r c l e .  C  , and ATJ$C)  (  -' 0 < I?-fyl< 6j  , by the product  can  , where  of the  S e r i e s g i v e n i n the second term on the r i g h t of e q u a t i o n  (1.5.12). tion i n  x  Now, the f u n c t i o n / v  an a r b i t r a r y s m a l l number r  Laurent  The product O  s e r i e s r e p r e s e n t s e i t h e r an a n a l y t i c  , or a f u n c t i o n w i t h an i s o l a t e d pole at  the expansion that  for a l l finite  ^'  func-  . Since  shows t h a t the l a t t e r i s not t h e case, i t f o l l o w s  ^C*) ^fr)  i s analytic  soning f o r i n c r e a s i n g ^  in C  until  exhausted shows t h a t ^P)^)  .  A p p l i c a t i o n of t h i s r e a -  a l l the p o i n t s  have been  i s analytic f o r a l l f i n i t e  and r e p r e s e n t a b l e by any of the u n i f o r m l y convergent g i v e n i n the second term on t h e r i g h t of  -x  ,  series  (1.5.12).  From (1.5.10) we then have  U.5.14)  l ^ l ^ ^ ^ ^ ^ ^ ^ ^ - t  By t h e same type  o f argument used above, the two s e r i e s i n square  b r a c k e t s on t h e r i g h t  side represent f u n c t i o n s  which are a n a l y t i c i n the f i n i t e p l a n e . ed from 2t^+ip + i  by p u t t i n g  £ ~o  %&)  and  F u r t h e r , t h e s e r i e s form-,  , dividing  , d i v i d i n g a l l the terms by  the f i r s t term by  (g-t? 2  , and dropping  superscripts i s  1=0  *  w i t h an i n f i n i t e r a d i u s o f convergence, where (1.5.15)  S'  =  ( 5 " ,  Since the a n a l y t i c i t y of  *=-°> and  n  a  s  D  e  e  n  e s t a b l i s h e d , we  (15) s h a l l h e r e a f t e r use the s e r i e s f o r j-o E q u a t i o n (1.5.14)  (1.5.16) 1^)1 where  , and drop s u p e r s c r i p t s .  then gives  <  x  ri±iP_'  i»- |*  A (  t  i s a bound on t h e s e r i e s  (1.5.17)  S<'  =  7  Z  S i "  with  Putting  where  (1.5.16) i n t o (li.5-.10). we get  A  i s a bound on t h e s e r i e s  r  with  By i n d u c t i o n , In where  A  i s a bound on t h e s e r i e s  K  S  (1.5.19)  . -  ^  with  .  (1.5.20) Since  '  b  I  r  ^ | / c  2  a comparison of (1.5.20)-with S  V  S  1 S  S  I  (1.5.15)  fora l l^  i t f o l l o w s t h a t the sequence o f nunbers by some number  (1.5.21)  f\  .  i f„&\  i p + ^ v - n - ^ ; ^ ^ ,^  ^ )  shows t h a t  and £ A^}  *  i s bounded above  Hence,  c <3-±*e=jr A  ,  ,,_ ," t  (16) so t h a t t h e s e r i e s on the r i g h t of•(1.5.11) i s dominated by  (1.5.22)  a*f { (  Zwf+2p  and t h e convergence  _,; A H Iz-fcJ*} ,  i s established.  By the same r e a s o n i n g as  used i n the Liouville-NeumanmTheorem satisfies  {&)  ^Too  U  ^C ? Z  (i.5.6) To show the uniqueness of the bounded s o l u t i o n ,  sup-  pose t h a t u)fej i s another bounded s o l u t i o n of (1.5.6),  (1.5.23)  toty =  Since  + V  &) / * L(i x) co<?)J?c . x  i s bounded, t h e r e e x i s t s a constant ) ufe)-u}&) I  f o r a l l z- .  £  such that  E I v,C^ I  <  (1.5.6) and (1.5-23),  From  (1.5.24) /  < f K ^ I (J \L(t,x)\\uCc)-vooo\\dx\  P u t t i n g t h i s back i n t o the i n t e g r a l on the r i g h t  s i d e of  (1.5.24), we o b t a i n C o n t i n u i n g t h e p r o c e s s , we get at the n-th stage, I U(2) by  LA) &) J <  1^",^) I I  I  O  as  -  (1.5.21), which proves the uniqueness. We can a l s o s t a t e the f o l l o w i n g  Corollary.  The same r e s u l t h o l d s i f t h e k e r n e l has the form  Hd xl t  where  for 1.6  Q ^)  -  ( S '=i,2 • • • i  j  DC*),  D^' (x.-bj)  *  S  ), and ( j=  o *, Z, f  • •' • }  The General S o l u t i o n Of A R e l a t e d I n t e g r a l E q u a t i o n . In t h e f o o t n o t e on page (2), i t i s observed that the  i n t e g r a l equation  (1.1.15) can be w r i t t e n  ).  (17) (1.6.1)  i n terms o f an i n d e f i n i t e i n t e g r a l .  I t i s t h i s form  (1.6.1)  which w i l l be used i n Chapter 3 , and hence i t i s of i n t e r e s t to  examine t h e convergence  approximations of the type  o f i t s s o l u t i o n by s u c c e s s i v e (1.5.1).  Since both the second  s o l u t i o n Wi0r) and t h e k e r n e l Kfa.x) a finite  may be unbounded a t  i> , t h e g e n e r a l s o l u t i o n o f  point  be expected t o e x i s t f o r a l l 2- .  cannot  (1.6.1)  However, i n the next theo-  rem we s h a l l show t h a t under c e r t a i n c o n d i t i o n s the s o l u t i o n does e x i s t s f o r a l l * Theorem 2 .  excluded from a s m a l l c i r c l e  Suppose t h a t the f u n c t i o n  a n a l y t i c i n the f i n i t e by C^^O /" %) where  Dftl - P (?-b)  S  {. Kn&l}  a  is  r  l  n  ^ w i t h one zero o f o r d e r f»o>/ z e r o s , and  1><> = L> p  is  F u r t h e r , l e t kfyx) = NOi,x) t>£t) , where  for positive integral  s . Then the sequence  o f s u c c e s s i v e approximations a s s o c i a t e d w i t h  converges f o r a l l  4- f  t o the unique s o l u t i o n o f  S u c c e s s i v e approximations  t o the s o l u t i o n of  (1.6.1) (1.6.1). (1.6.1)  are (1.6.2)  where  and (1.6.4)  and where we d e f i n e  1  The f i n i t e /*/  <  I zj  -» .  3  number.  s  Proof.  'V &)/{U-b)  i s an e n t i r e f u n c t i o n without  a non-negative  about  z-plane - w i t h zeros of order one a t  *  1  P  z-plane r e f e r s t o a l l v a l u e s of ? , where  3»  i s fixed.  f o r which  (18)  (1.6.$)  U &) = eL*/ &) -+J3^fr) •  The proposed  s o l u t i o n o f (1.6.1) i s then  (1.6.6)  u<t) -  0  % fr)=  K  I,  a  <*o &)j[_ ^«&)+  /3  t  Z  -&&^<^&).  h"-^-^) •  E s s e n t i a l l y t h e same procedure t h a t was used i n Theorem 1 g i v e s  for  finite  z  excluded from  f  , from which t h e convergence  f o l l o w s f o r these v a l u e s o f -? To show uniqueness o f t h e g e n e r a l s o l u t i o n , suppose that  i s a second s o l u t i o n o f (1.6.1) which i s f i n i t e f o r  all finite (1.6.7)  2-  o u t s i d e of  U)fr) =  7  ,  elA/fr) •+ /3*f fc) 4z  From„finiteness o f r  r  fe)  f  <C^X)  , uLz)  f  , i t follows that there exist  u)(x)Ax.  , and  constants  C  (  «-^<£9 o u t s i d e and  so  that  From (1.6.1)  and  (1.6.8") IU&-UJ&1  (1.6.7) s.  \4l*\K(%xn\«G)-»>M\lM  (1.6.9) 1 ^ - u>&\<. Vl^&iC, The  s u b s t i t u t i o n of  \% &\ {  + vC l£ &\ z  K  •  (1.6.9) i n t o the r i g h t s i d e of (1.6.8)  gives  R e p e t i t i o n o f the p r o c e s s g i v e s at the n-th stage  —>  O  •* —? —* .  The f o l l o w i n g g e n e r a l i z a t i o n can be proved by s i m i l a r reasoning :  (19) Theorem 3. analytic 2=b '  Suppose t h a t the f u n c t i o n  i n the f i n i t e , where  non-negative.  ^^/{(e-i/j^J-  z-plane w i t h zeros o f order  at  i s e n t i r e without z e r o s , and p  F u r t h e r , l e t KC% x)/'H(-^x) = 2- ^ )  is  C^-^O  is C ^f^ s  Then the sequence., of s u c c e s s i v e approximations of (1.6.1) converges f o r a l l v a l u e s of c l u d e d from s m a l l c i r c l e s unique  s o l u t i o n o f (1.6.1).  * ry  ( i n the f i n i t e plane) exabout  by  t o the  )  (20) CHAPTER  TWO  APPLICATION TO INITIAL VALUE PROBLEMS 2.1  Introduction. T h i s chapter c o n t a i n s a p p l i c a t i o n s of the r e s u l t s of  Chapter 1 to the type of i n i t i a l value problem i n which the d i f f e r e n t i a l equation t o be s o l v e d d i f f e r s from a known equat i o n by terms c o n t a i n i n g small parameters.  In p a r t i c u l a r , the  i n t e g r a l e q u a t i o n (1.1.5) i s used t o o b t a i n s o l u t i o n s as power s e r i e s expansions  i n one  or two  of these parameters.  A l s o , as  a g e n e r a l r e s u l t , the f o r m u l a t i o n of the g e n e r a l Cauchy problem as an i n t e g r a l e q u a t i o n , obtained i n s e c t i o n 1.3, to  g i v e m u l t i p l e power s e r i e s expansions  of h i g h e r order  f e r e n t i a l equations i n s e v e r a l parameters. lem c o n s i d e r e d , the m-th of  i s used  I n the  dif-  type of prob-  s u c c e s s i v e approximation t o the  solution  the i n t e g r a l e q u a t i o n y i e l d s a l l the terms of the m u l t i p l e  power s e r i e s having the sum  of t h e powers o f the v a r i o u s p a r a -  meters l e s s t h a n or equal t o m.  Since the d i f f e r e n t i a l equa-  t i o n s under c o n s i d e r a t i o n w i l l be assumed t o have no s i n g u lar  p o i n t s , so t h a t the k e r n e l  tfte,*)  are bounded, the Liouville-Neumann  and t h e second  Theorem guarantees  convergence of the g e n e r a l s o l u t i o n o f the i n t e g r a l by s u c c e s s i v e  2.2  solution the  equation  approximations.  A Problem With A P e r t u r b a t i o n In The F i r s t D e r i v a t i v e Term. Suppose t h a t the s o l u t i o n of the d i f f e r e n t i a l  (2.2.1)  u"(*l  h a v i n g the i n i t i a l  +  CLS-F&) uf(*)  v a l u e s tl(t>)  -f  and  equation  &&)ufr)=o u'd?)  i s r e q u i r e d , when  (21) ^k)  and  ^fr)  (2.2.2)  are known t o be s o l u t i o n s o f  v-'fr)-/-  QGr) «rfr) = O •  E q u a t i o n (2.2.1) d i f f e r s from (2.2.2) by a term c o n t a i n i n g a s m a l l parameter  s  where  Lfr,*)  (2.2.3) where  the  first  The m-th  From (1.1.14),(1.1.16), and  s  LC^x)  i s independent o f s .  u&)~  °L  .  ^  and /?  &)+-  From (1.1.15) and (1.2.7),  -f S J*  are l i n e a r i n s  (1.4.8),  .  L C^ ) x  u&^JXj  With  approximation t o (2.2.3) i s  approximation then has the form  g i v i n g the complete power s e r i e s up t o the term i n 5  •  As a simple i l l u s t r a t i o n , c o n s i d e r the problem, whose solution Q&)  i s e a s i l y o b t a i n e d by other methods, i n which  =  (for real ^  l<  ), are put i n t o  =—  as- cor  I f the i n i t i a l c o n d i t i o n s are yields  oi=-\  (2.2.1).  F^Sr)-/  In t h i s case,  .  u(o)=i  u^o) = o  t h e n (1.2.7)  , ^ — 3 £ , and the i n t e g r a l e q u a t i o n i s  U&) =  Cafvt-t- -4- %f St*** -  a  s  f*cosnC-x~*~)ufr)<}vC;  First  w i t h approximation A  (2.2.4)  f  U.&O  + Tc{/>* «*  ~ ^  "* '" s  *  rt  •  (22) T h i s checks w i t h the f i r s t  two terms i n the power s e r i e s  expansion of the known s o l u t i o n  (2.2.5)  u © =  where  2.3  ^  cos (Mi  =  c  4-  sin  J  ^__  Another S i n g l e Term P e r t u r b a t i o n Problem. Consider the problem of f i n d i n g t h e s o l u t i o n o f the  equation  eQclu &7+  (2.3.1)  [ Tfr)-(- sGCzll  /  with t h e p e r t u r b a t i o n $G(x) i n i t i a l v a l u e s u(j?)  (2.3.2)  i n the ufr?  and a %)  , where  trig) + P&) v'C*? +  ucxp^o  term, having the  ^ifr) and ^ & )  satisfy  7-fr?«rfr?=-o •  Again, use of ( 1 . 1 . 1 4 ) , ( 1 . 1 . 1 5 ) , and ( 1 . 1 . 1 6 ) shows t h a t the i n t e g r a l e q u a t i o n f o r the problem i s  (2.3.3)  Ufr) -  -h-ftf-ity-ir sj  where, from ( 1 . 2 . 7 ) ,  (2.3.3)  t  oi and f) as w e l l as Lfe x.') are indepeni  dent o f the parameter s l u t i o n of  L£* x) u&)Jx ,  The m-th approximation t o the so-  #  t h e n has the form  (2.3.4)  =  J l ~&&) £=0 *  s. X  For example, suppose t h a t the s o l u t i o n of  (2.3.5) with i n i t i a l  (2.3.6)  a" &) 4-  { s(x ^ax)4l  * } a&) = o  conditions  u(o) = /;  i s r e q u i r e d , where CL and n  u <?)=6 /  are r e a l c o n s t a n t s .  In t h i s  case, the k e r n e l i s whence the i n t e g r a l equation i s  (2.3.7)  U&J  •=* oi cos n-z 4 /3sm«* +-£f 5\*»(x~+) %x +ax} u&Ux . l  (23) (1.2.7), the  From  initial  *-'-kl.tl  -  and hence the f i r s t  (2.3.6)  conditions  /;  a-  i l l  give  II-°>  (2.3.7) i s  approximation t o o  o  where  -£,(2-} = COS**  2.4  Multiple Perturbation  Problems.  Consider the d i f f e r e n t i a l e q u a t i o n  (2.4.1)  u  H  &) •+ ZS.Ffr)  [ QCt) + t CCt)] u&) = 0 ,  c o n t a i n i n g two independent p e r t u r b i n g parameters  s  and 't ,  Suppose the s o l u t i o n of  (2.4.2) are  AT  +  *s (x) and  known t o be  (1.1.16), ufr)  QCz) yirfr?^ D •  t  Then, from  (1.1.15) and  s a t i s f i e s the V o l t e r r a I n t e g r a l E q u a t i o n  *^i&+/3* 0r)  (2.4.3)  7  +  sj\& )ufr)Jx+t(%h Ja&Ux x  tX  b  A> where the f u n c t i o n s  are  independent of 5 and  are  u(l>) and u ^ ? \ i t f o l l o w s from (1.2.7) t h a t t h e c o n s t a n t s  £~ .  I f the i n i t i a l values f o r  <K and ft are g i v e n by  l / ^  \ _  u  * - J-1 The f i r s t  u&)  approximation t o  (2.4.3)  ,  s  •l t h e n has t h e form  i  (24) which g i v e s a l l t h e l i n e a r terms i n t h e s o l u t i o n w i s e , at the m-th  giving  S  . Like -  stage of approximation,  a l l t h e c o n t r i b u t i o n s to terms i n  powers o f  u&j  and  u{t) h a v i n g the sum o f  tr l e s s t h a n or equal t o  *-n .  As an example, c o n s i d e r the d i f f e r e n t i a l e q u a t i o n  (2.4.5)  u"frl+  . s(  +• L^-ltvcJ  ( + *x)  ufr7=Oj  o f the type c o n s i d e r e d by Goveyou and M u l l i k g n (1).  In t h i s  case, the k e r n e l i s  ~  '  ^  {tx-asj  For t h e i n i t i a l c o n d i t i o n s  UQ>)  —  /  - Sc°s»(?c-?){ i-f-4?cj . ,  , equations  a^ol-O  (2.4.4)  give ^  /  ;  /»-  if. •  Hence, the i n t e g r a l e q u a t i o n f o r the problem i s  (2.4.6) With  U£7 = ra$>t 4-|-Jn-**- -f-/ *ni  ^ — t f eUx-sfcofiCx-yfl+arfutUx .  ;  S  Wo^ * f«»*-f|-/-i<i»i«3  s o l u t i o n of  (2.4.7)  , the f i r s t  approximation t o the  (2.4.6) i s +J^i o&) 5 +£oj&lt+2, &st  =A o& y  J  /l  +  A&s",  where -Ao <^  =  A . fr> = The f i r s t solution  c  u  n  ^  approximation  ^ / .  +  (2.4.7)  (£+-^V*  M ,  *>  g i v e s a l l t h e l i n e a r terms i n the  (2.4.6). Consider now  the s o l u t i o n  a g e n e r a l p e r t u r b a t i o n problem, i n which  of t h e n-th order l i n e a r d i f f e r e n t i a l e q u a t i o n  (25) (2.4.8)  cx%  + £  5 For)] u '"h U  "•**t  »-r —f  h a v i n g the i n i t i a l v a l u e s required.  U  ^  . ( £~o i i • - ••(-•>) ) i s J  J  /  E q u a t i o n (2.4.8) i s changed by terms c o n t a i n i n g  s m a l l parameters (2.4.9) ^frl  S^.y. 4- £  whose s o l u t i o n s  nr  known.  - O  h  from t h e e q u a t i o n frl «r *"}*)= o c  (x )  . (*"-/,? - - - "i ) are supposed t o be  From (1.3.18), the i n t e g r a l e q u a t i o n f o r t h i s problem i s U&) =  (2.4.10)  2.  2- " - J S  H  fCr?«/xr  L  where, from (1.3.19)  L^C*.*) Here,  N  = c-o"'~'^'>  { A/rw P  i s g i v e n i n terms of  (1.3.12).  The c o n s t a n t s  KV  »  C-^,-  ( "-^ , ' '  ) by  z  :  , g i v e n by  (I.3.23), w i l l i n  g e n e r a l have the form  where the exponent on each o f the parameters  5;  i s e i t h e r one  (y) or zero, and the c o e f f i c i e n t s the  i n i t i a l values  i4%  ,  • v ir^Vt/  „  , and  (2.4.10) i s X-/*/ ; Tis -f-  depend upon  F \) , (/^>/ •••(*>-(/ {  The m-th approximation of  (2.4.11) f u ^ - z z  dc  y 2  n <r*  T h i s g i v e s a l l the terms o f ^^/which have the sum o f the exponents of t h e v a r i o u s parameters to  •  -V  l e s s than o r equal  The second summation on the r i g h t o f  (2.4.11) g i v e s  incomplete c o n t r i b u t i o n s t o terms o f tt&) having t h e sum o f exponents of  5/  equal t o •*•*•/  •  c  (26) 2.5  Another Treatment Of The Problem I n S e c t i o n  2.2  C o n s i d e r t h e problem o f f i n d i n g t h e s o l u t i o n of the  second order d i f f e r e n t i a l  (2.5.1) of  +  <Ls F  (2.2.1)  the type  ®  Mfil -  ^  X^)=0 ;  (F(o)^cl  Q(£)~n , f o r a r e a l constant  with  which s a t i s f i e s t h e i n i t i a l  (2.5.2)  equation  /;  *  conditions  A'Co? =Q •  (2.5.2)  A c t u a l l y , the c o n d i t i o n s  can be r e p l a c e d by completely  a r b i t r a r y ones without e s s e n t i a l l y changing t h e r e s u l t . first  d e r i v a t i v e term i s removed from  formations  (2.5.1)  I f th  by the t r a n s -  £  (2.5.3)  AQrl  =  e~  S  '  u»l j  f  ^=x,  o b t a i n e d by f o l l o w i n g t h e procedure of s e c t i o n 1.4, t h e r e s u l t is  (2.5.4) where  u"0r)+ ~)  = /;  (2.5.4)  (2.5.6) with  i s compared  with  v"fr?4-  ^ ; =o  y  solutions l  /  v~-i(*') — st«i*  y  r e s u l t i n g i n t e g r a l equation i s  (2.5.7) With  conditions  = sa  ts gl — cos n<- > the  u&/=o,  1  must s a t i s f y t h e i n i t i a l  (2.5.5) When  z' Ffrf-ir iS}  Ufr)=  cos«* 4-  Uofc}- cos**-4-r<^  solution of  (2.5.7)  (2.5.8)  + **{^frtfiU  n  &^*-{*^x-tO[^F\^^{f &)]u97Jx /  ,  the f i r s t  approximation t o thi  is  -J L f ^ n ^ +  ^(f^^rrux  (27)  The sequence of s u c c e s s i v e approximations with  (2.5.7)  s o l u t i o n of  a c t u a l l y does converge t o t h e unique  (2.5.7)  with i t s f i r s t argument.  p r o v i d e d t h a t the f u n c t i o n  together  The m-th a p p r o x i m a t i o n s o f t h e form  by t h e second from l a t e r  S  higher than  s  f— C  i n &.5.I);  g i v e s f o r t h e approximation ,  U,&) =  In  contributions  approximations.  (2.5.8)  (2.5.9)  , represented  summation, w i l l i n g e n e r a l r e c e i v e  As an example, suppose t h a t  (2.5.9)  continuous  d e r i v a t i v e are bounded f o r a l l v a l u e s of the  where t h e powers o f  then  associated  A* f*tc_y#r -Ji« «*t-r  t h i s p a r t i c u l a r example, t h e f i r s t  approximation  g i v e s a l l terms up t o those c o n t a i n i n g &  , and  l i k e w i s e the m-th approximation g i v e s a l l terms up t o those containing at  s  , s i n c e t h e r e i s no o v e r l a p p i n g o f terms  t h e s u c c e s s i v e stages of approximation.  The r e s u l t  (2.5.9)  checks with t h e expansion of the known s o l u t i o n  (2.2.5)  up t o t h e  s  3  term.  (28)  CHAPTER THREE APPLICATION TO SPECIAL FUNCTIONS 3.1  Introduction. The object i n t h i s chapter i s to use the r e s u l t of  Section 1.1  to obtain expansions of special functions i n series  of better known functions. expand  X,@x)  and  pectively, where  Y„ 6*7 7<*7 K  Ikeda (4) f i r s t used t h i s method to i n terms of  and  V*.(x)  of f i r s t and second kinds o f order ving  J" c*?  and  M  Y*,fr/  are the Bessel Functions n  . In addition to r e d e r i -  Ikeda's formal r e s u l t s , we have examined the convergence of  the s e r i e s ; i n p a r t i c u l a r , we have found that the IT, &*) converges f o r a l l x be imposed upon (for  res-  all x  series  and a l l d. , but that a r e s t r i c t i o n must  i n order that the Y„'(_**)  series  converge,  excluded from a neighbourhood of the origin.) For  d e t a i l s , see M.A. Thesis of D.A. Trumpler (16). More recently, F. Tricomi (15) has obtained of the Confluent Hypergeometric Functions.  expansions  Function i n s e r i e s of Bessel  Using Laplace Transform methods, he arrived at an  expansion f o r the well-behaved  solution of the Confluent Hyper-  geometric Equation, and gave a four-term recurrence for the c o e f f i c i e n t s i n the series.  Also, by setting up an integral  equation s i m i l a r to that which we have derived i n Section 1.1, he obtained asymptotic formulae, but no general expansions.  In  t h i s chapter, we use the r e s u l t of Section 1.1 to obtain the general solution of the Confluent Hypergeometric series i n T^q well-behaved  and  Y^(x)  Equation as  , and as a special case, the  solution of t h i s equation as a series i n X (*') x  •  (29) F u r t h e r , we a r r i v e at a s i m i l a r s e r i e s o f B e s s e l F u n c t i o n s f o r the s o l u t i o n of a g e n e r a l i z e d Confluent Hypergeometric  Equation.  T h e o r e t i c a l l y , t h e procedure c o u l d be g e n e r a l i z e d t o o b t a i n expansions o f v a r i o u s other f u n c t i o n s i n terms o f known f u n c t i o n s except  f o r t h e computational  d i f f i c u l t i e s i n evaluating certain  i n t e g r a l s i n v o l v i n g the l a t t e r .  3.2  The Expansion Of The S o l u t i o n Of The Confluent  m e t r i c E q u a t i o n I n S e r i e s Of B e s s e l  Hypergeo-  Functions.  The o b j e c t i s t o express t h e s o l u t i o n W(a c t) o f )  the C o n f l u e n t Hypergeometric  (3.2.1)  irW ^  Equation  -h (c-6) W ' -  11  }  i n terms o f the s o l u t i o n s  «UJ<£) « O a n (  *  VI, <£)  of B e s s e l ' s  Equation  (3.2.2) We now proceed t i o n s of  ? -h ( 4- •£) Xit) = O -  trA"6r) +  t o set up an i n t e g r a l e q u a t i o n l i n k i n g the s o l u -  (3.2.1)  expression  and  (1.4.1)  (3.2.2).  In order t o o b t a i n t h e simple  f o r the k e r n e l , we use  (1.4.6)  t o get the  transformat i o n  (3.2.3)  t  which changes  (3.2.4)  (3.2.2)  ir&l;  17=*>  into  AT" Cf? +  [ 4 ^  •+ * ]ir#?  = • 0  L i k e w i s e , we can remove the f i r s t d e r i v a t i v e term f r o m by t h e change o f v a r i a b l e  (3.2.5)  <*lfc)=  which changes  (3.2.6)  (3.2.1)  tf* into  + J£ - 1L^> } t f ~ o^ U  (3.2.1)  (30) where 1^=  (3.2.7) The f u r t h e r  changes  f - /  \4=  ;  s=  .  transformation  (3.2.6) i n t o  (3.2.9)  aV;+ { % r - 7* + <}  o  }  where (3.2.10)  n= U!+\ = c-i  £ =  7V-^H -  We now use t h e r e s u l t of S e c t i o n 1.1 t o w r i t e the s o l u t i o n s of  (3*2.9) i n terms o f t h e known s o l u t i o n s Y^Cx)  From  of  cx^- X*fr?  and  (3.2.4) by a V o l t e r r a I n t e g r a l Equation.  (1.1.14, (1.1.16), and (3.2.3), we o b t a i n  upon t a k i n g identity  £&  , JL*,6r) =  x  ,  and u s i n g the  (See Watson (17) )  (3.2.11)  Yji<l  (3.2.12)  -  = £  = *  V  ;  f^GoY^)- x^y««;}  1  (1.1.15) i s t h e n jft«v,«-^r»j cVu^«vx.  The i n t e g r a l e q u a t i o n  (3.2.13)  '  =  s  To o b t a i n t h e s o l u t i o n of (3.2.13), we s h a l l need t h e f o l l o w i n g special results:  (3.2.14) (3.2.15)  t . - ^ ^ ^ - u t * ) Y  -<*) = £  Y^,<*; - Y^CO ;  (3.2.16) x =£f*[ZM&-i w\p>]{f)**J*)Jx H  1  See t h e f o o t n o t e  on page 2  ;  •t-fjk  M"!  (fj'x^ + 0 ^ ; + . ^ ; Hf» + I  (3D where  >  i s a positive  integer,  constants o fintegration. (3.2.15) are well-known  and  (3.2.17) w i l l  the  L  (16),  be e s t a b l i s h e d i/ @j = t  the solution of (3.2.I3)  (3.2.18)  and t h e r e s u l t s  (3.2.14) (3«2.l6)  i n S e c t i o n 3«6»  * * l^TQi) + /3 Y„&}  « » - .* f  will  and c a l c u l a t e  [ Xt) A  have t h e f o r m 3>>+  eM\*-)]  upon rearrangement o f t h e terms i n t h e s e r i e s . (3.2.18) into  tute  are  I z i  f e w a p p r o x i m a t i o n s o f ( 3 . 2 . 1 3 ) , i t becomes a p p a r e n t  first  that  cfy ( - , , ,^)  The r e c u r r e n c e r e l a t i o n s  and  I f we t a k e  and  (3.2.I3)  We t h e n  and d e t e r m i n e t h e n e c e s s a r y  rence formulae f o r the c o e f f i c i e n t s (3.2.13) i s s a t i s f i e d .  >  The r e s u l t  and  S  substirecurso t h a t  K  o f the s u b s t i t u t i o n i s  (3.2.14) and (3.2.15) t o t h e B e s s e l F u n c t i o n s i n t h e  Applying  summation under t h e i n t e g r a l s i g n ,  and t h e n u s i n g  ( 3 . 2 . 1 6 ) and  ( 3 . 2 . 1 7 ) , we o b t a i n i n t u r n  where (3  W, , and  by> +/ the  and  right  i n the  /?,  a r e c o n s t a n t s depending upon  ('='^,3, * first  ; * = '/V'. '  ). Replacing  d. r  and t h i r d terms under t h e summation o n  s i d e , we may r e w r i t e  t h i s i n t h e form  ,  «>  (32)  Equating  s e p a r a t e l y the c o e f f i c i e n t s of  (±y+ j  (*) and  3  V-' r e c u r r e n c e r e l a on both s i d e s , we o b t a i n the ~ 2  I  fc)  r+i J  Putting  (3.2.18) (3.2.20) where  4 ,  and  ^  t>r  | we can r e w r i t e  i n t h e form  u&-  ** f W, £  ^ ^ J ^ ) +  f  A  b (ff Y ^ J r  ?  now  (3-2.21)  j<?„  =  ^  ^ « g)*i!±L  Since the c o n d i t i o n s of Theorem 2 o f S e c t i o n 1.6.  ;  are s a t i s f i e d ,  the sequence o f s u c c e s s i v e approximations to t h e s o l u t i o n o f  (3.2.13)  converges f o r a l l f i n i t e  neighbourhood of the o r i g i n . been obtained  Z-  excluded from a small  However, the s e r i e s  t r e a t e d i n more d e t a i l i n the M.A.  in  appropriate  (3.2.20).  The constant  normalization.  second s o l u t i o n of  (3.2.9)  T h i s question i s  T h e s i s of D.A.  To "obtain the s o l u t i o n which i s f i n i t e  the  has  by rearrangement of terms, so t h a t the convergence  of t h i s s e r i e s does not f o l l o w immediately.  /3, = O  (3.2.20)  Trumpler  at the o r i g i n ,  oi  ]  we set  i s determined f r o m  I t can be shown (16)  i s o b t a i n e d by t a k i n g  (16).  t h a t the opooial  (33) s p e c i a l values f o r  ^  and  The s o l u t i o n from  (3.2.20) (3.2.5)  using  (3.2.22)  ?  of  t  (3.2.1)  i s obtained  by changing back t o t h e o r i g i n a l v a r i a b l e s ,  (3-2.8) :  and  WC^C;*)  where the constant  (3.2.23) 3.3  \A/(^ c-  = 'A  ^ -  * -t«~  e  C((2-[  €  (3.2.21)  i n the recurrence He-4-4-  ^ i s g i v e n by  •  A G e n e r a l i z a t i o n Of The Problem. Consider a g e n e r a l i z a t i o n of the Confluent Hypergeo-  metric Equation  (3.3.D  V'&)  +  ( f> =l  }  (3.2.6) ^ ~t  i s r e p l a c e d by —4 ' "'  l  f  f  which i s d i f f e r e n t from  (3.2.6)  the problem.  )•  -  i~±x t -'+J±.  As i n  p  ^ } ^ ; = o ,  i n t h a t the term  — ij_  ' , where  i s a constant,  (3.2.6), y\  ^  and  The case d i s c u s s e d e a r l i e r  J?  in  are c o n s t a n t s of  (f=/) i s r e l a t e d to  the quantum mechanical problem f o r an harmonic o s c i l l a t o r i n space.  The more g e n e r a l form here (and t h e g e n e r a l i z a t i o n  c o n s i d e r e d i n S e c t i o n 3«4)  c o u l d t h e r e f o r e be i n t e r p r e t e d as  an anharmonic o s c i l l a t o r i n space.  (3.2.8)  transforms  (3.3.2) where  u > ) + { - ^ !  into  +  _  {  y u^r) -  Oj  now  (3.3.3)  *  -  As i n S e c t i o n 3*2, pare  (3.3.1)  The change of v a r i a b l e  ItkfJ?  we use the r e s u l t s o f S e c t i o n 1.1  (3.3.2) w i t h (3.2.4),  (3.3.4)  .  Ufr) = ^TS^i  + /3  t o com-  and o b t a i n t h e i n t e g r a l e q u a t i o n  X® + ^ ^ f / f t ^ X ^ ) - W t o J ^ « W c / > r  I n o r d e r t o o b t a i n the s o l u t i o n o f  (3.3.2) which  i s f i n i t e at  (34) the o r i g i n , we take a-o the g e n e r a l  ; a simple  m o d i f i c a t i o n would give  s o l u t i o n , (as i n S e c t i o n 3 . 2 ) ' .  Following the  method o f S e c t i o n 3 . 2 , we l o o k f o r the s o l u t i o n o f ( 3 . 3 . 4 ) i n the form (3.3.5) and  u&>=*>*  2  <v ( f f ^ c * - ;  determine the necessary  i n order t h a t ( 3 . 3 . 4 )  ficients (3.3.5)  r e c u r r e n c e formulae f o r the be s a t i s f i e d .  coef-  Putting  i n t o ( 3 . 3 . 4 ) , we get  (3.3.6)  1  a ^jjr-)  -  r  zc*i+i-Z/fr^a-w^^  In t h e e v a l u a t i o n of the i n t e g r a l , we need the r e s u l t (3-3-7) where  q ^ # f  Section 3 . 6 .  -  i s a positive integer. Substituting ( 3 . 3 . 7 )  change o f dummy  s=r-p-f  >  T h i s w i l l be proved i n into (3-3.6)  ( 3 . 2 . 1 5 ) , we get, upon i n t e r c h a n g i n g  The  C-^)}7^J)  and u s i n g  summation and i n t e g r a t i o n ,  leads t o  (3.3.8)/  i-<o  4  VI/ I  A f t e r the dummy o f summation been r e p l a c e d by  s  i n the r i g h t member has  V  r  , equating the c o e f f i c i e n t s of ff/ J " (*?  (35) g i v e s the r e c u r r e n c e  This gives  formula  <3-r-+zp+\  t  e  r  m  s  of  . - ••  <lr+p  . In  order t o a p p l y the r e c u r s i o n , however, we need t h e e x p l i c i t forms of d  a  0i  (3.3.10)  •• -  t>  4  - ';  0  Now, i n ( 3 . 3 . 8 ) , t o J~  (t)  4*? 4,  . =  (3.3.8)  From •• •  -  <zp - o -  the o n l y term on t h e r i g h t which c o n t r i b u t e s  i s t h e f-o  term i n the f i r s t  single  summation.  Hence,  S i m i l a r l y , we o b t a i n  (3.3.11)  *  p  H  r  =c-o*-'(lJ^[(^(^+o-~  for  t=l l  (3.3.11)  g i v e the c o e f f i c i e n t s  :>  (3.3.9)  2.4  c*+r)] . i n summary, ( 3 . 3 . 1 0 ) and  -•••{>  )  4  a  6  t h e n g i v e s a l l subsequent  <Z  • •-  dtp  , and  = f*'j V"*^  ) *  z  r  A Further G e n e r a l i z a t i o n . Consider a f u r t h e r g e n e r a l i z a t i o n of the Confluent  Hypergeometric  \/"e)+{-i2 Ct ~+¥ -  where the |H  and  X ^ ^  are c o n s t a n t s  ( £ =',l  where  •• • p ;  t  (3.4.1) C  t  z  ,* '  ),  As i n S e c t i o n 3 . 3 , the  are again constants.  change of v a r i a b l e ( 3 . 2 . 8 ) transforms (3.4.2)  ^)=0;  $  (3.4.D  and  Equation  x s  %  into  * + * 1 « & =  0  ,  (36) A= ^ M ;  (3.4-3)  c  f r - fr-'O*"' / KJ. •  Using t h e r e s u l t s of S e c t i o n 1.1 t o compare (3.2.4), we g e t , f o r /i = o (3.4.4)  (3.4.2) w i t h  ,  um-^^r^-f-f,^?f  Lw&}-X-*flji)h^%*  •  Mi  S u b s t i t u t i o n o f the proposed solution,(3*3*5) i n t o  (3.4*4),  and use of (3.3*7) and (3.2.15) g i v e s  + 221 ^  %«m  ft  The s u c c e s s i v e changes of dummy t=  Q^pz [(>*»«<)•••  S-2.p-f2.g_  (3*4*6)^ ~°  s = r-f+^  , and  t r a n s f o r m (3.4*5) i n t o  .  where  and'  ^  < j c V T i T ^ ^  * * f ^ " -  ^LL%z-^  & ^[(^- c^^j x+t+s®  +c  •+  E q u a t i n g t h e c o e f f i c i e n t s of  y ^ "4- t^> —1 r  the r e c u r r e n c e r e l a t i o n (3.^.7) ^ where  ^  ' ^  ---  g i v e n example.  4^  i  l  ^  M  i n (3.4.6) g i v e s  ^  ^  ^  ^  are o b t a i n e d from (3.4.6) i n any  However, t h e complicated nature o f the f u n c t i o n s  ,  (37) and  l-Jv,Cip  makes i t i n c o n v e n i e n t t o g e t g e n e r a l  expressions f o r these For f i n i t e  coefficients.  numbers  P  , T h e o r e m 3 o f S e c t i o n 1.6  shows t h a t t h e s e q u e n c e o f s u c c e s s i v e a p p r o x i m a t i o n s siated with the integral equation to t h e unique continuous  (3.4.4) a c t u a l l y  s o l u t i o n o f (3.4.4).  assoconverges  As i n S e c t i o n  3.2, h o w e v e r , we h a v e r e a r r a n g e d t e r m s i n o b t a i n i n g  (3.3.5),  so t h a t f u r t h e r a t t e n t i o n i s r e q u i r e d i n o r d e r t o e s t a b l i s h t h e convergence.  3.5  Solutions of Related D i f f e r e n t i a l Equations  I n Terms Of B e s s e l  Expanded  Functions.  I n t h i s s e c t i o n , i t w i l l b e shown t h a t t h e s o l u t i o n s o f a number o f i m p o r t a n t to t h e Confluent  are related  Hypergeometric Function through  changes o f v a r i a b l e . be u s e d t o e x p r e s s  various  H e n c e , t h e r e s u l t s o f S e c t i o n 3.2 c a n  these  Functions. Numerical be  d i f f e r e n t i a l equations  solutions as series of Bessel  values for these  solutions could  computed a c c u r a t e l y b y making u s e o f t h e e x t e n s i v e  l a t i o n of t h e B e s s e l F u n c t i o n  series  w o u l d be needed t o g u a r a n t e e a c c u r a t e r e s u l t s The W h i t t a k e r  Function  P u t t i n g i*i=vt?-A£  (3.5.1) which,  v"Gr)+  i-i_ +  (3.2.19)  (9).  (18).  into  (3.2.6) g i v e s  %+'^irl ^= ' vC  6  b y (3.2.5), h a s a s i t s s o l u t i o n t h e W h i t t a k e r  tion  (3.5.2)  » „ . . ft -  tabu-  (17), and i n f a c t , f o r A « 1  o n l y a f e w terms o f t h e r a p i d l y convergent  (a)  then  H / < V ; t)  Func-  ,  (3d) Since  (3.2.7) g i v e s  equation (3.5.3)  f  ,  H =  €  -  c  ,  t  (3.5.2) can be w r i t t e n Al^^,  e  ^ £  * U/(*»*~i-hs  ^  which i s now i n a form t o which (3.2.21) can be a p p l i e d . (b) The Laguerre  Function  Density;  By the s u b s t i t u t i o n s (3.5.4) equation  ±  ^  ;  (3.2.6) becomes  (3.5.5) with  -  L."tf+  i -4 - UQ>  --fr]  L&?^6;  solution  0.5.6)  L j ,)= n  (c)  n,,^&  (  The " A s s o c i a t e d Hermite The  (3.5.7)  = n „  r ' W f  i  t  .  H  Equation".  s o - c a l l e d A s s o c i a t e d Hermite L  -(2*HKtj+i?_  _ t f  +  Equation  j r 6 ; = o ,  M  obtained from (3.2.6) by t h e s u b s t i t u t i o n s (3.5.8)  t=  -f  ;  v&=  gfrbi  has the s o l u t i o n (3.5.9)  n>)  k r ^ r ^ H  -  ( £ i  or  (d)  Hermite s Equation T  (18)  Hermite s Equation T  (3.5.10)  t>£(*)-r E( +i) 3  (or Weber's Equation) i s  -  P#  .  which i s r e l a t e d t o (3.2.9) by the t r a n s f o r m a t i o n s (3.5.11)  * =  x  «(*)=p &>  =  d  ;  «=i  Hence, t h e s o l u t i o n of (3.5.10) may be w r i t t e n  where i n t h e expansion  (3.2.19) f o r  U  ' n d i n the fa  (39) r e c u r r e n c e r e l a t i o n (3.2.20),  ; (e)  ij + i  •  The E q u a t i o n F o r The Harmonic O s c i l l a t o r I n Space This equation i s  (3.5.12)  K/'fc?*  ^C-A  i  f+vJ  ^b)=o  J  which i s related, t o (3.2.9) by t h e t r a n s f o r m a t i o n (3.5.13) The  "= CJ  solution  (3'.5.14)  3.6  K^f;  U*.to).  of '(3.5.12) i s then K, <b?~  «(  :  .  Appendix To Chapter Three. (a) Proof Of (3.2.16) and (3.2.17). Consider t h e i n t e g r a l  (3.6.1)  / * Z.  ft>  ?•  ** - f*[*~* Z<iO ll**" ?*}} H  **  4  Sine e (3.6.2)  *~'X.g)--&L»- l^};  &[*"* 7  Mt  a p a r t i a l integration  0T-£*rv  M,  o f (3.6.1) g i v e s  /**.*> X, „ «x^Jx  - E-7  h  M  7^  **»] * + / * ^  T _ x** «/x l  u s i n g t h e r e l a t i o n (17) (3.6.3)  =  A p a r t i a l integration  * X ^ ^ J T ' W  -  of the l a s t i n t e g r a l on the r i g h t  (3.6.4)  ,*  /  fT j  . x - J  AH  . /r*  Upon use o f t h e i d e n t i t y (17) (3.6.4) becomes T ^ T T  x  +  v-  z  A + 2  =  j  gives  ^ %  * /  (40) Upon t r a n s p o s i n g terms i n t h i s equation, we get f i n a l l y  (3.6.5)  f'lnlfU™*-^[XuZfoF-JjoZj*)?*]*  Since the f u n c t i o n s Y fr) s a t i s f y the same r e c u r r e n c e r e l a n  tions  as  T*(z)  , the f o l l o w i n g r e s u l t  i s obtained i n the  same way as (3*6.5) (3.6.6)  S\*>T„e)  -^  lWT^c*>^\  T ^ ? n * .  0  Upon use o f (3.6.5) and (3.6.6), we get  Now, from (3.6.3) and (3.2.11), (3.6.7) and  Y  (tF) Z&) -  H H  the r e s u l t  &  Y..G-) =  a/rr* ,  (3.2.16) f o l l o w s .  Again, f o l l o w i n g the same procedure with  r e p l a c e d byY (3.6.8)  ^ [ l ^ l ^ - r c ^ l ^ x ^ ]  (3.6.9) f\ nl *)x'* J*n  f r o m which the r e s u l t  £ Y A W ^ - Y « U V  ^  ,  (b)  (x)  , we get, i n s t e a d of (3.6.5) and (3.6.6)  J^XVX+M  l  X.+.  (3.2.17) f o l l o w s .  Proof Of (3.3.7) We need the f o l l o w i n g  Lemma.  F o r J?-!^, --- f> , t h e f o l l o w i n g r e l a t i o n holds  (3.6.10)  = C? F  We now prove the r e s u l t  For  p-/  (3.6.11)  (3.3.7) by f i n i t e  i n d u c t i o n upon P  , (3.3.7) gives £  &) - &G}*Z+ X? - ( i ) l ^ l lC  •  (41) where f o r convenience, we put which i s c o r r e c t t r u e f o r P=fr (3.6.12)  n-tr  by (3.2.14).  ^obtain,  ( y--o,/ z - - - - ) j  )  Assuming the r e s u l t  (3»3»7) i s  w i t h the h e l p of (3.2.14),  Z & -  {T'f-H-Vj-L  Tfi,  where  ( X-  0,1,1,  • •--  Hence,  upon a p p l i c a t i o n of the Lemma. we get  which completes the proof by  Putting  f  induction.  t h i s into  (3.6.12),  (42) CHAPTER PHYSICAL 4.1  FOUR  APPLICATIONS  Introduction. Although the method of S e c t i o n 1.1 was: o r i g i n a l l y  designed f o r i n i t i a l value>• problems, i t can be adapted t o s o l v e boundary value problems.  In t h i s chapter, we  shall  d i s c u s s a type of boundary value problem which a r i s e s i n Quantum Mechanics.  Now,  i n the u s u a l problems t r e a t e d i n  Quantum Mechanics, i t i s r e q u i r e d to f i n d the s o l u t i o n s of the Schrodinger  Wave E q u a t i o n which s a t i s f i e s a set of  " n a t u r a l boundary c o n d i t i o n s " , f o r which the p o s i t i o n o f the mass p a r t i c l e i s u n r e s t r i c t e d .  The  probability inter-  p r e t a t i o n of the wave f u n c t i o n t h e n l e a d s to the boundary c o n d i t i o n s o f f i n i t e n e s s at the s i n g u l a r p o i n t s o f the wave equation.  I f , however, the  system under c o n s i d e r a t i o n i s en-  c l o s e d , t h e n these c o n d i t i o n s are r e p l a c e d by the  "artificial  boundary c o n d i t i o n s " t h a t the wave f u n c t i o n v a n i s h at c e r t a i n o r d i n a r y p o i n t s of the d i f f e r e n t i a l e q u a t i o n .  In f a c t , f o r  these s o - c a l l e d bounded Quantum Mechanical problems, the boundary c o n d i t i o n s r e q u i r e t h a t the wave f u n c t i o n v a n i s h on some s u r f a c e i n f i n i t e three-space, cone.  The  corresponding  an i n f i n i t e l y high and this  surface.  4.2  The  such as a sphere or a  p h y s i c a l c o n d i t i o n i s t h a t there  i n f i n i t e l y steep p o t e n t i a l w a l l  I n t e g r a l E q u a t i o n For The  be  on  Bounded Quantum Mechanical  Problem. Let g e n e r a l i z e d c u r v i l i n e a r c o o r d i n a t e s  x, ,  , and  x  3  (43) i n t h r e e dimensional E u c l i d e a n space be chosen so t h a t the s u r f a c e on which t h e wave f u n c t i o n where  C  i s a constant.  vanishes i s ^ -  ,  c  We assume t h a t t h e surface i s o f  s u f f i c i e n t l y simple nature t h a t the Schrodinger Wave E q u a t i o n i s separable  (12) i n t h e chosen c o o r d i n a t e s x  x  , "*z_ , ^  .  equation The  space dependent w a v e i s , f o r a p a r t i c l e o f A  mass  M  ,  (4.2.1) where  ~ ^ V >  =  Ce-Vlifs,  i s Planck's constant  d i v i d e d by  , £T  i s the  i. energy c o n s t a n t ,  V  i s t h e p o t e n t i a l energy, and  L a p l a c i a n o p e r a t o r i n t h e c o o r d i n a t e system The  V  i s the  , ^  , ^c . 3  substitution  permits t h e s e p a r a t i o n o f (4.2.1) i n t o t h r e e o r d i n a r y d i f f e r e n tial  equations f o r the f u n c t i o n s  Y^Cxv) and y faj)  Xfa) •  The equations f o r  have t h e same s o l u t i o n s as i n the unbounded  3  problem, ahd t h e l a t t e r a r e supposed known.  The  X (^)  equa-  t  t i o n has the form  <^ - ' 2  2  4  [  &  )  a ; ]  -t-  i•%(•*,)+  YA)'0  where JP  i s t h e quantum number a r i s i n g f r o m the  equation.  From S e c t i o n 1.4, (4.2.2)  (4.2.3)  u'<fc]  -f  problem, ufc) must be continuous  .  into  C ?}(xl+ n 3 «Cr?^0;  w i t h no o t h e r s i n g u l a r p o i n t s between them.  a£  X  can be transformed  which we suppose has the two s i n g u l a r p o i n t s  and must v a n i s h at  X^l J  for a l l  b  and ^  I n t h e bounded ^  satisfying  Hence, t h e boundary c o n d i t i o n s t o  ,  (44) be s a t i s f i e d a r e (4.2.4)  Fv'rte ;  Following  u C^ ?-=0 • 0  S e c t i o n 1.1, we compare (4.2.3) with the  equation (4.2.5) where  +  ^Ofl  E  F &1+  1 l<r0c>Oj  £  o  s a t i s f i e s the boundary c o n d i t i o n s  (4.2.6)  srQ,] F,«,+e  ;  =  o  We suppose t h a t a s o l u t i o n o f (4.2.5) which i s a n a l y t i c i n the f i n i t e plane i s known t o be (4.2.7)  t/7  =  Mt<&,  and t h a t t h e eigenvalue the  i s known.  From S e c t i o n 1.4,  second s o l u t i o n of (4.2.5) and t h e Wronskian o f t h e  two s o l u t i o n s are g i v e n by (4.2.8)  ^ 7  =  C  j  ffgfof,  and  (4.2.9)  - C -  Hence, from (1.1.14), (1.1.15), and (1.1.16), t h e i n t e g r a l equation  connecting  the s o l u t i o n s o f (4.2.3) and (4.2.5) i s  (4.2.10) The  convergence o f t h e s o l u t i o n o f (4.2.10) by s u c c e s s i v e  approximations i s e s t a b l i s h e d by Theorem 1 o f S e c t i o n 1.5, s i n c e i t has been assumed t h a t f i n i t e plane.  The f i r s t  that  I f we take u &)  /3~o  .  -#\ fr) i s a n a l y t i c i n the h  of conditions 0  (4.2.4) r e q u i r e s  A*&,  the f i r s t  approxi-  mations t o t h e s o l u t i o n of (4.2.10) i s (4.2.11)  u, & -  A* m [<-  A) ( * f  i«?T<hc $  x  * f^fg  *]  By a p p l y i n g the second o f (4.2.4) t o (4.2.11), we g e t f o r t h e  (45) first  approximations  t o the eigenvalue  ^ [ i ^ ^ ^ ^ ^ y  (4.2.12,  1  The problem i s then reduced t o e v a l u a t i n g i n t e g r a l s o f t h e type appearing on the r i g h t  of (4.2.12).  However, we are now  prevented from c o n t i n u i n g the g e n e r a l d i s c u s s i o n because of our i n a b i l i t y  4.3  t o o b t a i n e x p r e s s i o n s f o r these  integrals.  The Bounded Hydrogen Atom Problem. The Dutch P h y s i c i s t s M i c h e l s , de Boer, and B i j l  (10) were i n t e r e s t e d i n the behaviour o f gaseous  matter  under p r e s s u r e , and i n p a r t i c u l a r they wished t o determine the e f f e c t of pressure upon the s p e c t r a l l i n e s o f Hydrogen gas.  I n order t h a t the mathematical  assumed t h a t t h e e f f e c t infinitely  problem be s o l v e d , i t i s  of pressure can be r e p l a c e d by an  h i g h and i n f i n i t e l y  s u r f a c e o f a sphere o f f i n i t e  steep p o t e n t i a l w a l l on the radius  o b j e c t i o n s t o such an assumption  2  0  .  Although  physical  have been p o i n t e d out, (de  Groot and t e n Seldam (2),) i t i s n e v e r t h e l e s s u s e f u l t o s o l v e the quantum mechanical  problem of f i n d i n g the e i g e n f u n c t i o n s  and the e i g e n v a l u e s f o r the Hydrogen atom wave equation, under the c o n d i t i o n t h a t the atom be enclosed i n a sphere of r a d i u s  ^  •  The  Q(&>? and  p a r t s o f the wave equation  c l e a r l y have s o l u t i o n s which are i d e n t i c a l w i t h the s o l u t i o n s corresponding t o the n a t u r a l boundary c o n d i t i o n I t remains t o solve the r a d i a l part o f t h e Hydrogen atom wave e q u a t i o n (4.3.1)  W&+  [ f ~  + 1]  « & ~  O,  under the a r t i f i c i a l  (4.3.2)  boundary c o n d i t i o n s  F^rte  ;  i4(*J=o,  i n s t e a d of the n a t u r a l boundary c o n d i t i o n s "(0?  ( 4 - 3 . 3 )  The  F.Vrfe  u(o°7=o  ;  .  c o n d i t i o n s (4.3.3.) g i v e s o l u t i o n s of ( 4 * 3 « 1 )  by the Frobenius method; the e i g e n v a l u e s are for  positive integers  easily  7i,(w? =  =L  ,  , and the e i g e n f u n c t i o n s a r e t h e  Laiguerre F u n c t i o n D e n s i t i e s . However, (4.3.2) r e q u i r e t h a t equation Laguerre  u C  ;  ) = o  'A  s a t i s f y the  , where c<£>j ^  denotes the  F u n c t i o n D e n s i t y (corresponding t o t h e eigenvalue " A , )  which i s r e l a t e d t o the Confluent Hypergeometric  U/(a,c> *) ( c f . approximations  equation (3.5.6)).  Function  M i c h e l s et a l (10) have found  f o r t h e eigenvalues of the ground l e v e l ,  and  de Groot and t e n Seldam (2) have extended t h e i r method to the 2s and the 2p l e v e l s , g i v i n g graphs and t a b l e s f o r the in  X  .  shift  Soon afterward, Sommerfeld and Welker ( 1 4 ) a p p l i e d  the formulae  of Michels et a l f o r v a l u e s of  and f o u r times the Bohr r a d i u s . s t r e s s e d the importance  2»  equal t o t h r e e  A l s o , Sommerfeld and Welker  of a g e n e r a l i n v e s t i g a t i o n of the  haviour of the Confluent Hypergeometric  F u n c t i o n near  be-  =2- = <=*>  Sommerfeld and Welker ( 1 4 ) have a l s o d i s c u s s e d a g r a p h i c a l method f o r o b t a i n i n g the e i g e n v a l u e s , which g i v e s a c c u r a t e l y the curve  " X - A ^ 7  f©  r  s m a l l v a l u e s of ^  t h i s method, the known standard s o l u t i o n s u C ? ; for of  various positive integral  n  these s o l u t i o n s are l o c a t e d .  against  *  By  are p l o t t e d  , and the f i r s t  zeros  These f u n c t i o n s are then s o l u -  t i o n s of the problem f o r the p a r t i c u l a r values A graph of  .  i s drawn, and by  2^  of  ^  8  interpolation,  .  (47) the value of n(i ) corresponding t o a g i v e n value  Z  t  taken from the graph. ^ L%) ~-i/n  •  is  6  Then the ground l e v e l eigenvalue i s  The e i g e n v a l u e s f o r h i g h e r l e v e l s a r e  o b t a i n e d by a s i m i l a r  procedure.  The preceeding g i v e s an h i s t o r i c on the problem up t o the p r e s e n t .  sketch of work done  We now proceed t o g i v e our  own treatment, u s i n g the method of S e c t i o n 4.2.  The mathema-  t i c a l problem amounts t o s o l v i n g e q u a t i o n (4*3.1) under the boundary c o n d i t i o n s (4.3*2), when we know that the s o l u t i o n s of t h e e q u a t i o n (4*3*4)  i r " & + [ f - £Ug)  + * J  = o,  s a t i s f y i n g the boundary c o n d i t i o n s (4*3*5)  •vfc)  Fi'«i-fe  v<**>l — o  •  are v @)  (4*3*6)  x  =  A\&)  , with  Since the problem thus presented  i s of the same type  consi-  dered i n S e c t i o n 4*2, t h e V o l t e r r a I n t e g r a l E q u a t i o n c o r responding  t o e q u a t i o n (4*3*1) i s (4*2.10), w i t h f i r s t  mation (4.2.11).  For the ground l e v e l ,  Ai& and  -  -  (4*2.11) g i v e s  u ®=  (4*3*7) With  approxi-  t  U -  a partial  *ec~*[ i-(* + 0 f**e-**  /*  integration gives  , and  r^-^r]  J* d  tf=  x  l  - </r w  c  (48) or -i  f + f  =  *t)[  -  f [ ± + £ + i b } * < } .  Now,  Hence,  _ *  4 s c- IL± I  _,  1  M  *  1 1  ^  z  2 -*wy  - _ x S" _&*2— and (4.3.7) g i v e s  (4.3.8)  2"-  A p p l i c a t i o n o f the f i r s t  (4.3.9)  * f f (^ , -I  ^  Z  ^ f k ~] -  o f the c o n d i t i o n s (4*3«2) g i v e s  +~±%i,  r  The second approximation t o t h e i n t e g r a l equation i s obt a i n e d by p u t t i n g  (4.3.8) back i n t o (4.2.11): e—JtcU  Again i n t e g r a t i n g by p a r t s and l e t t i n g ' 6  (4.3.10) u «, -*«.-'«/• l - f i & L Upon making t h e approximation  A  0J  fa+l)  0  , we get  +< ^ Z A Z M = C V •+ I,)  7  J  , where  i s g i v e n by (4.3*9), and a p p l y i n g (4-3«2), we get f o r  the second approximation t o the e i g e n v a l u e ,  (4.3.11)  The r e s u l t  \ , 0.) - X^o) +  ,~  W l *  (4.3.9) i s i n e s s e n t i a l agreement w i t h t h a t ob-  t a i n e d by de Groot and t e n Seldam (2), and i n f a c t the c a l -  (49) c u l a t e d e i g e n v a l u e s f i t the curve of Sommerfeld and Welker ( c f . the bottom o f page 46) b e t t e r t h a n the r e s u l t s of de Groot  and t e n Seldam.  The v a l u e s o f  percent of the c o r r e c t v a l u e when  z  are w i t h i n one i s at l e a s t f i v e  g  times  the Bohr r a d i u s . Sums of t h e type appearing are most e a s i l y handled  i n (4.3*9) and (4.3*11)  by u s i n g a method of de Groot and  t e n Seldam, which depends upon the p r o p e r t i e s of t h e exponential  integral /  •x  — **»  which i s t a b u l a t e d ( 6 ) .  4.4  The Bounded R i g i d  Rotator.  F o r the g e n e r a l r o t a t o r problem i n t h r e e - s p a c e , a mass p a r t i c l e distance p  f  i s r e s t r i c t e d t o r o t a t e at a constant  <x. from t h e o r i g i n , and f o r the bounded problem,  i s not allowed t o e n t e r a cone d e f i n e d by an azimu-  t h a l angle.  In o t h e r words, t h e r e i s an i n f i n i t e l y high and  i n f i n i t e l y steep p o t e n t i a l w a l l on t h e s u r f a c e of t h e cone, and  i n s o l v i n g the quantum mechanical  t h a t t h e wave f u n c t i o n v a n i s h t h e r e .  problem, i t i s r e q u i r e d T h i s r o t a t o r problem  has been c o n s i d e r e d g r a p h i c a l l y by Sommerfeld and Hartmann (13), who used the "one-sided boundary c o n d i t i o n s " t h a t M be r e s t r i c t e d from e n t e r i n g o n l y the lower nappe of the cone; t h a t i s , they a p p l i e d the boundary c o n d i t i o n t h a t the wave f u n c t i o n v a n i s h o n l y f o r 0= 0  O  near  TT  .  , where  &  0  i s an angle  They o b t a i n e d the eigenvalues g r a p h i c a l l y by  c o n s t u c t i n g nodal curves analogous t o t h e curves used by  (50) Sommerfeld and Welker i n t h e hydrogen atom problem, ( c f . s e c t i o n 4*3)  and a l s o a r r i v e d at an a n a l y t i c r e s u l t f o r the  ground l e v e l i n the l i m i t i n g case t h a t They gave r e f e r e n c e s  (11) i n  ~  —  been used.  &  A  ^  1= E  , and rotator  V  has t h e constant value  r  The u s u a l s e p a r a t i o n  0 (&)  with  4-  /V-P  where the f a c t t h a t  where  ,  r  the Schrodinger wave e q u a t i o n f o r the r i g i d  ^  of the v a r i a b l e s i s  s a t i s f i e s t h e equation  _  ^^a -  £  2  £  By (1.4.6), the change o f v a r i a b l e  (4.4.2)  ug-) _  ©(©7  -  <9  t r a n s f o r m s (4.4.1) i n t o  (4.4.3)  r ^ - ^ +  The boundary c o n d i t i o n s  ^ -+  -ii^=o.  t o be s a t i s f i e d by the s o l u t i o n s  of (4.4.3) are (4.4.4)  u(oy  where Now,  Fr^-te <-  i t i s known t h a t  (4.4.5)  *r &? 4  s a t i s f y i n g the  tt  1T .  1926.  In s p h e r i c a l p o l a r c o o r d i n a t e s ,  i s near  t o the o r i g i n o f the problem l e a d i n g  back t o a paper by P a u l i n g  <P  &„  ar„  ;  u(Xj = o  /  TT .  s o l u t i o n s o f the e q u a t i o n  [  conditions  4 *  4 ±]  v6r?=  O  has  (5D (4.4.6)  ir(o) F<Wife •  <TS(IT? = o  are  (4.4.7)  ^('0 = - f l J Y * ) -  where  sin  p^tox),  1  are t h e a s s o c i a t e d Legendre F u n c t i o n s , and t h a t  the corresponding  e i g e n v a l u e s are  $($+0 f o r ^ - ° ^ ' " • /  U&)  Hence, r e f e r e n c e t o (4.2.10) shows us t h a t the i n t e g r a l  J  satisfies  equation  i n which 0 =• O  by the f i r s t  of conditions  (4.4-.4).  The  e x i s t e n c e o f the s o l u t i o n of t h i s equation i s guaranteed by Theorem 1 of S e c t i o n 1.$. 4.5  The Ground L e v e l Of The Bounded R i g i d R o t a t o r . For the ground l e v e l ,  so t h a t (4.4.8) becomes  (4.5.1) The f i r s t  = oifm  - /W.wV J si» x U(x)) J-=£ • x  approximation t o the s o l u t i o n of (4.5.1) i s  (4.5.2)  ^ ^ . A ^ - A ^ v i ^ x ^ i ^ ^ - j ^ ^ ^ j ^ J  or, upon making the s u b s i t u t i o r i s  (4.5.3)  y - *>* c  ,  or  c  ^ J V t ^ [/+ j\ A  i  j  o  ,  €  J  ~l-  (4.5.4)  t  ±  ^  T  ] .  A p p l i c a t i o n of the second o f c o n d i t i o n s (4«5*4) g i v e s f o r t h e first  approximation t o t h e ground l e v e l eigenvalue,  (52) (4-5.5)  y «  l  %  . = ±  )  >  where (4.5«6)  -»)  14- °s  =  •  c  The second approximation t o the s o l u t i o n o f ( 4 « 5 « D i s obtained  by p u t t i n g  (4.5.4) back i n t o the r i g h t s i d e of  (4.5-1) :  Again making the s u b s t i t u t i o n (4.5*3), we  obtain  or X (4*5*7)  ^  In order t o evaluate  the  expression  (4.5.8) - T - ^ ^ l  /!*(,+5)^  on the r i g h t s i d e o f (4.5*7), we (4.5.9)  (\(  i+>))<ly = 0+W)-CkCI-*«) -C< + ui) -  (4.5.10) J ^ V H - ^  =  4  w  P u t t i n g these i n t o  Since of  i*J  1=  (i+*)J^(i+»)Mi-u,)+Ci-ut-2J^z)J^(i-**)  *  (4.5.8) and s i m p l i f y i n g , we  -20»Ct + ^)  i s c l o s e t o -/  (4.5.12) i s c l o s e t o - 2  Riemann Z e t a - F u n c t i o n . i n g the v a l u e  -fr 2 j  64-^)^Vl^-^6+wJ^/W^'+^-2i; 2-l-4in2-4j  (4.5.11) ,.jf A,CMry? ^ ( i - y ?  (4.5.12)  need the f o l l o w i n g r e s u l t s :  2^2.-1  + u) 4  , the summation on the r i g h t s i d e i  - I }  Putting  = ^/^>  obtain  , where $ Q>)  i s th  (4.5.12) i n t o (4-5.7), and us-  , we get  (53) or (4.5.13 )  « ^si'n^g / )+ A i ,  l  ±  ^  J  By a p p l y i n g the second of c o n d i t i o n s  ~ A /U ' ~ +/.l4S-^Jfj  t  ±  £  (4.4.4) t o equations  (4.5.13) and using the approximation 7*= f/\ ' f'. where >jOJ L 0.0 6; i s g i v e n by (4.5.5), we o b t a i n f o r the second a p p r o x i 0,0 (  t  mation t o t h e eigenvalue  (4.5.14) where  ^  i s g i v e n by (4.5.6).  Again, t h e t h i r d approximation to t h e eigenvalue i s found t o be ^3)  (4.5.15) where  III the l i m i t ^ -» o  (4.5.16)  , e q u a t i o n (4.5.14) reduces t o  -  which i s t h e r e s u l t  o b t a i n e d by Sommerfeld and Hartmann (13)  by a d i f f e r e n t method. values o f  ^  +fiTll'  The f o l l o w i n g t a b l e g i v e s , f o r small  , a comparison of t h e e i g e n v a l u e s o b t a i n e d  from (4.5.16) and those from the f i r s t  t h r e e approximations  (4.5.5), (4.5.14) and (4.5.15). TABLE I CSH)  rr-z  A 0,0  *)  e  o°4S'  I0'  2  il'zf  zw~  18° ll'  5 W0"  l  y  O. \0^\  0. IU2  01450  OA443  OJ4SS  o.iSU  O.ZIOZ  0.1.20-3  0.9.111  0.XH-3Q  O.I3I5  10"  Z5°5\  OAOHS  \0"* O. tow  10"  z  o.illi  0-3/22  0.333S 0.3100  0.30*2 0.344C 0.3UO  0.44+1  (54)  4.6  Higher Energy L e v e l s Of The Bounded R i g i d R o t a t o r . For  t h e 1-1 l e v e l ,  (4.4.8), and f o l l o w i n g t h e same  P u t t i n g these v a l u e s i n t o  procedure t h a t was used i n S e c t i o n 4.5, we get f o r the first  and second  approximations t o t h e eigenvalue 7.-  (4.6.1)  ^  (^=i+c z ) oi  * or  f o r s m a l l v a l u e s of ->\  (4-6-2) The r a p i d for  s m a l l v a l u e s of  table:  (  A  W  ^  £ £ i i > * *  +  o f t h e s u c c e s s i v e approximations ->)  \f'  i sillustrated  =  ',1  ^  i n the following  ~ 2)  ' hi  TABLE 2  A *?,  ^7  %°34  J  r  G'  \oID"  3  2  1.0011  2 0021  own  1.01S5  10211  OOZ<\\  2.0S&S  O-0S(>5  If" 2 7 '  ix lo" '  20541  18° ll'  5XlcT  2.12(>(p 2-1343 0.i3>42>  2S°s\  y  +  ,  * [ W convergence  o  -*t 2.  a.  1  t  10''  2- 230 2.-40{  2-253  o.xss 0.4C5  (55) The  f o l l o w i n g graph compares t h e r e s u l t s o f TABLES 1 and 2:  GRAPH OF EIGENVALUES FOR THE (0,0) and (1,1)  *  The  *o C<*c r«e*) *0  IS  S  graphs i l l u s t r a t e t h a t ,  tO  as ->j-^o  ENERGY LEVELS  S  o  , the increments  i n t h e eigenvalues o f t h e bounded r o t a t o r approach those of t h e f r e e r o t a t o r f o r t h e (0,0) and (1,1) energy l e v e l s . Further,  there are v e r t i c a l and h o r i z o n t a l tangents r e s -  p e c t i v e l y t o t h e (0,0) and (1,1) curves at the o r i g i n . F o r h i g h e r energy l e v e l s , we have c a l c u l a t e d only approximation t o t h e e i g e n v a l u e s .  the f i r s t  The r e s u l t s a r e t a b u l a t e d  below. TABLE 3  (a*)  Pj (2,2)  (3,3)  sin** sin * 3  • H  S i n at  Cr r)  Si'n'ac  (o 0  cos  }  }  x  /2 20  2  (0,1)  OA)  0-,3)  -5/  SlU X cos*  C  cos*  l-Z  S i ' h * x  its  -l/C^f-  C-in  2  ~  -7 / (JU 2 "  V")  Jos  V-)  (56*) I n TABLE 3, t h e extreme r i g h t hand column i n v o l v e s o n l y t h e h i g h e s t power o f ^ll^,'  lae for  °\  which appears  i n t h e exact formu-  • A l s o , f o r the ( r , r ) l e v e l ,  KV  is a  > o  constant which can be determined of  ; f o r example,  r  The  f o r each p a r t i c u l a r  - /j  <  3  = 2;  <  H  eigenvalues o b t a i n e d from the formulae  value  = G• i n the  extreme r i g h t hand column a g a i n approach those o f t h e f r e e r o t a t o r as  -* °  . The tangents t o the curves are v e r t i c a l  f o r the (r,0) c u r v e s , (r=.<v_,z- - '' ) and h o r i z o n t a l f o r t h e ;  other curves.  The q u a l i t a t i v e r e s u l t s are i n agreement w i t h  those o b t a i n e d g r a p h i c a l l y by Sommerfeld and Hartmann (13). I n c o n c l u s i o n , the l a s t procedure  chapter g i v e s a systematic  f o r r e f o r m u l a t i n g a g i v e n quantum mechanical  lem as a V o l t e r r a I n t e g r a l E q u a t i o n . has  Reference  prob-  t o Chapter 1  shown t h a t , under q u i t e g e n e r a l c o n d i t i o n s , t h e sequence  o f s u c c e s s i v e approximations converges Although  t o t h e unique computational  a s s o c i a t e d w i t h such an e q u a t i o n  continuous s o l u t i o n of the e q u a t i o n . difficulty  i n evaluating certain  n i t e i n t e g r a l s has prevented us from o b t a i n i n g g e n e r a l  defiresults,  we have n e v e r t h e l e s s demonstrated the use of t h e method i n e v a l u a t i n g e i g e n v a l u e s for- the Hydrogen atom problem and the bounded r i g i d r o t a t o r  problem.  (57) BIBLIOGRAPHY 1.  R.R. Coveyou and T.W. M u l l i k e n , U.S. Atom. En. Comm. Rep. AECD-2407, (1948)  2.  S.R. de Groot and C.A. t e n Seldam, P h y s i c a , 12, 669, (1946)  3.  F. F u b i n i , Rend. L i n c e i , 6, 26, 253, (1937)  4.  Y. Ikeda, Math. Z e i t . , 22 , 16, (1925)  5.  E.L. Ince, O r d i n a r y D i f f e r e n t i a l E q u a t i o n s , Longmans and Green, (1927)  6.  E. Jahnke and F. Emde,  T a b l e s of F u n c t i o n s with Formulae  and Curves, L e i p z i g - B e r l i n , B.G. Teubner, (1938) 7.  J. l i o u v i l l e ,  J . de Math., 2, 24, (1837)  8.  W.V. L o v i t t ,  Linear I n t e g r a l Equations, 1st e d i t i o n ,  New York, McGraw-Hill (192*) 9.  M. K a r l i n , J . Math. Phys., 23, 43,- (1949)  10. A. M i c h e l s , J . de Boer, and A. B i j l ,  P h y s i c a , /j-, 981  (1937) 11. E P a u l i n g , Phys. Rev., 2 J , 568, (1926) 12. H.P. Robinson, Math. Ann.,  749, (1927)'  13. A. Sommerfeld and H. Hartmann, Ann. Phys., 5, 2 2 , 333',(1940) 14. A. Sommerfeld and H. Welker, Ann. Phys. 5, 32,56, (1938) 15. F. T r i c o m i , Ann. Mat. Pura. 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