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Asymptotic expansions of the hypergeometric function for large values of the parameters Prinsenberg, Gerard Simon 1966

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ASYMPTOTIC EXPANSIONS OF THE HYPERGEOMETRIC FUNCTION FOR LARGE VALUES OF THE PARAMETERS  iy GERARD SIMON PRINSENBERG B.Sc., V i c t o r i a C o l l e g e , 1962  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS  I n t h e Department of Mathematics  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1966.  In presenting this thesis in partial fulfilment of. the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be.granted by the Head of my Department or by his representatives.  It is  understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of British Columbia Vancouver 8, Canada Date  ^  £k  ii  ABSTRACT  In c h a p t e r I known a s y m p t o t i c forms and expansions o f t h e hypergeometric  f u n c t i o n o b t a i n e d by Erde'lyi [ 5 ] , Hapaev  [10,11],  K n o t t n e r u s [ 1 5 L Sommerfeld [ 2 5 ] and Watson [ 2 8 ] a r e d i s c u s s e d . A l s o the a s y m p t o t i c expansions of t h e h y p e r g e o m e t r i c o c c u r r i n g i n g a s - f l o w t h e o r y w i l l be d i s c u s s e d . were o b t a i n e d by C h e r r y [ 1 , 2 ] ,  Lighthill [17]  function  These  expansions  and S e i f e r t [ 2 J ] .  Moreover, u s i n g a paper by Thorne [ 2 8 ] a s y m p t o t i c expansions o f -p; 1-m;  2  F (p+l,  2  P ( (p+m+2)/2,  1  (l-t)/2),  -1  < t < 1,  (p+m+l)/2; p+ 3/2-, t " ),  1  and t > 1,  are obtained  p-»» and m = -(p+ l / 2 ) a , where a i s f i x e d and 0 < a < 1 . e x p a n s i o n s a r e i n terms o f A i r y f u n c t i o n s o f t h e f i r s t  The h y p e r g e o m e t r i c  :  kind.  equation i s normalized i n chapter I I .  I t r e a d i l y y i e l d s t h e two t u r n i n g p o i n t s t ^ , i = 1 , 2 . c o n s i d e r , t h e case t h e a=b i s a l a r g e r e a l parameter hypergeometric  The  as  I f we  of the  f u n c t i o n F - ( a , b ; c; t ) , t h e n t h e t u r n i n g p o i n t s 2  L  c o a l e s c e w i t h t h e r e g u l a r s i n g u l a r i t i e s t = 0 and t = <*> o f t h e hypergeometric  e q u a t i o n as j a | ».  I n c h a p t e r I I I new a s y m p t o t i c forms a r e found f o r t h i s particular  case; t h a t i s , f o r 2 ^ 2  ( a , a] c ; t ) ,  F ( a , a + l - c ; 1; t 1  0 -  1  < T-^ _< t < 1 , and  ) , 1 < t _< Tg < » , as - a ^ » .  The a s y m p t o t i c form i s i n terms o f m o d i f i e d B e s s e l f u n c t i o n s o f order 1 / 2 . manner.  A s y m p t o t i c e x p a n s i o n s can be o b t a i n e d i n a s i m i l a r  iii Furthermore, a new asymptotic form i s derived f o r 2  F ( c - a , c-a; c; t ) , 0 < 1  <_ t < 1, as -a-»«, t h i s r e s u l t then  leads t o a sharper estimate on the modulus of n-th order d e r i v a t i v e s of holomorphic f u n c t i o n s as n becomes large.  iv  TABLE OP CONTENTS  1.  INTRODUCTION  2.  RELEVANT PROPERTIES OF F ( a , b ? c ; t )  5  5.  KNOWN ASYMPTOTIC RESULTS  8  a.  R e s u l t s o b t a i n e d by Watson  8  b.  R e s u l t s o b t a i n e d by E r d e l y i  9  c.  R e s u l t s o b t a i n e d by Hapaev  10  d.  R e s u l t s o b t a i n e d by K h o t t n e r u s  11  e.  R e s u l t s o b t a i n e d by Cherry; and Sommerf e l d  12  f.  R e s u l t s d e r i v e d f r o m Thome's paper  15  II  THE NORMALIZED HYPERGEOMETRIC DIFFERENTIAL EQUATION  20  III  THE ASYMPTOTIC BEHAVIOUR OF ^ ( a ^ a j c j t )  22  I  1 2  1  and g F ^ a j a + l - c j l j t " ) 1  IV  REFERENCES  22 Z>6  ACKNOWLEDGEMENT  I w i s h t o acknowledge t h e I n v a l u a b l e guidance and a s s i s t a n c e extended t o me b y Dr. C. A. Swanson and a l s o would l i k e t o thank him f o r h i s a s s i s t a n c e i n p r e p a r i n g t h e f i n a l manuscript. The generous f i n a n c i a l s u p p o r t o f t h e N a t i o n a l R e s e a r c h C o u n c i l and t h e U n i v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged.  1.  CHAPTER I  1.  INTRODUCTION The  h y p e r g e o m e t r i c f u n c t i o n ^F^a,,}}',  i n v e s t i g a t e d by numerous a u t h o r s , [5,  1 4 , 21,  a,b  and c and t h e v a r i a b l e t . The  22,  25,  51].  c;  t ) has been  see f o r i n s t a n c e t h e r e f e r e n c e s  I t depends on t h e t h r e e parameters  i n t e g r a l r e p r e s e n t a t i o n s of the hypergeometric  f u n c t i o n by Barnes' and E u l e r [ 5 ] have s u c c e s s f u l l y been emp l o y e d by K n o t t n e r u s [ 1 5 ] , S e i f e r t [ 2 3 ] and Sommerfeld [ 2 6 ] to d e r i v e asymptotic meters tend t o OB.  e x p a n s i o n s as one or s e v e r a l o f t h e p a r a -  B o t h S e i f e r t and Sommerfeld used t h e method  of s t e e p e s t d e s c e n t s [ 6 ] on a s l i g h t l y m o d i f i e d f o r m o f t h e Euler i n t e g r a l representation.  The e x p a n s i o n s a r e v a l i d i n a  t - i n t e r v a l w h i c h does n o t c o n t a i n a t u r n i n g p o i n t [ 6 ] , Asymptotic expansions of t h e hypergeometric f u n c t i o n o c c u r i n g i n gas-flow  t h e o r y on t h e o t h e r hand a r e v a l i d i n  r e g i o n s c o n t a i n i n g a r e g u l a r s i n g u l a r i t y and a t u r n i n g p o i n t w h i c h i n t h i s case c o r r e s p o n d s t o t h e t r a n s i t i o n p o i n t o f subsonic flow t o supersonic Cherry [ 1 , 2 ] , asymptotic [7,  flow.  The method used by t h e a u t h o r s  L i g h t h i l l [.17].,. and S e i f e r t . [ 2 J ] t o o b t a i n t h e  expansion i s described i n f o r instance, the references  », 19, 2 0 , 2 7 ] .  T h i s method concerns t h e a s y m p t o t i c  solu-  t i o n s o f d i f f e r e n t i a l e q u a t i o n s w i t h f i x e d t u r n i n g p o i n t s , or  2.  singularities. Some a u t h o r s , n o t a b l y E r d e l y i [ 5 ] Hapaev [ 9 , 10] and MacRobert [ l 8 ] used w e l l known e x p a n s i o n s o f t h e c o n f l u e n t h y p e r g e o m e t r i c f u n c t i o n ^ F ^ ( a ; b; t ) [ 2 4 , 31] t o d e r i v e asympt o t i c expansions of the hypergeometric f u n c t i o n . In a s i m i l a r f a s h i o n , asymptotic  e x p a n s i o n s as p -• « a r e  derived f o r g F ^ p + l , -p; 1-m;  (l-t)/2),  -1 < t < 1, and  F ((p+m+2)/2, (p+m+l)/2; p+3/2; t " ) , 2  2  1  where m = - ( p + 1/2)a, a f i x e d and 0 < a < 1.  t > 1 The e x p a n s i o n s  are d e r i v e d from a paper by Thorne [ 2 8 ] on t h e a s y m p t o t i c e x p a n s i o n s o f t h e a s s o c i a t e d Legendre f u n c t i o n f o r l a r g e degree and o r d e r .  The e x p a n s i o n s a r e i n terms o f A i r y f u n c t i o n s o f  the f i r s t k i n d [ 5 ] . I n t h i s chapter  a d e s c r i p t i o n o f t h e known  asymptotic  e x p a n s i o n s f o r gF^(a,b; c; t ) w i l l be g i v e n . The  e x p a n s i o n s a r e v a l i d as one o r s e v e r a l o f t h e p a r a -  meters approaches », under v a r i o u s r e s t r i c t i o n s on t h e ( o t h e r ) parameters and t .  The f o l l o w i n g t a b l e summarizes t h e r e s u l t s .  3.  Author  Date  G. N. Watson  1918  large parameters c  other r e s t r i c t i o n s |arg c j_< 7r-e,e > 0 ,  a,b f i x e d ,  | t | < 1. G.N.Watson  1918  a,b, c  1) (a+b)fixed, c f i x e d , l-2t+2(t -t) 2  2)  1 / 2  = e±  p  (a-b),(a+b-c) f i x e d [t-2+2((l-t)t" ) / ]t" = 1  T.M.MacRobert  1923  c  1  2  1  e±  p  a,b f i x e d , | a r g c|<^+e, e > 0 , | a r g ( l - t ) | < V.i  H.  Sommerfeld  1939  a,b  a=ipv^ b = i v , c = l , c r e a l and f i x e d , v r e a l and v-^>, |t| < 1 . »  H. S e l f e r t  1947  a,b , c  a+b=c-l-(Y-l)" ,ab=|c(l-c)( -ir 1  Y  1<Y<», 0<t<l. M. J . L i g h t h i l l  1947  a>b, c  Same r e l a t i o n s between a,b and c complex,  j a r g 11 <TT,  l a r g ( l - t ) j <TT. T.M.Cherry  1950  a, b, c  Same r e l a t i o n s  between a,b  and c complex,  | a r g ( c - l ) |_<-^,  | a r g ( l - t ) |< 7T. A. E r d e l y i  1953  c  a,b f i x e d , | a r g cj_Cn--e, e > 0 , |t|>l,  j a r g ( l - t ) | < ir.  A. E r d e l y i  1953  b  a,c f i x e d , c ^ 0 , - l , - ^ , . . . , 0 < jt| < 1  R.C.Thorne  1956  a,b, c  a,b l i n e a r I n c , | t | < l and t > l .  M. M. Hapaev  1958  a  b,c f i x e d  U. J. K h o t t n e r u s  19b0  c  a,b f i x e d , t - p l a n e c u t f r o m 1 t o 1 + ioe .  |t| ~  r^a|  - 1  4.  (continued) Author  Date  large parameters  U. J. K h o t t n e r u s  lybO  a, b, c  ( a - b ) , ( a - c ) f i x e d , Ret > 1, Im t > 0, Re ( c - r ) >min$Re (a-r)., Re (b-r)J+l r real, r » .  M.M. Hapaev  lyol  a, c  b, a c "  other r e s t r i c t i o n s  1  fixed,  jtj<jca" j. 1  RELEVANT PROPERTIES OF ^ ( a ^ b ; c,  2.  t)  The h y p e r g e o m e t r i c e q u a t i o n (1)  t ( l - t ) ^ - | + ( c - ( + b + l ) t ) - ^ - abx = 0 a  dt has t h r e e r e g u l a r s i n g u l a r p o i n t s , namely, t = 0, t = 1, and t = One s o l u t i o n r e l a t i v e t o t h e s i n g u l a r i t y t = 0 i s d e v e l o p a b l e i n a series (2)  1 + (ab/c)t + ( a ( a + l ) ( b ( b + l ) ) / c ( c + l ) 2 j ) t  2  + ...  00  I  =  ((aJn^n/  ( ) c  n n  -')t , n  c t  0, - 1 , - 2 ,  n=0 where ( a ) i s t h e Pochhammer symbol and n  (a)  n  =  r( )/ a+n  P(a) =  a ( a + l ) . . . ( a + n - l ) , n = 1,2,3,  The s e r i e s ( 2 ) , due t o Gauss, i s denoted by F ( ' » a  2  b  1  c  [5].  *  The s e r i e s converges f o r j t j < 1 and f o r t = 1 i t d i v e r g e s when Re (c-a-b) _< - 1 , i t converges a b s o l u t e l y when Re (c-a-b) > 0 and i t converges c o n d i t i o n a l l y when -1 < Re(c-a-b) < 0 and t h e p o i n t t = l i s e x c l u d e d -{14]. I f c = - n , n = 0,1,2,..., t h e n a s o l u t i o n o f e q u a t i o n ( l ) w h i c h i s r e g u l a r a t t=0 i s t  n  +  1  Y ( (a+n+1 ) ( b + n + l ) / ( n + 2 ) m  m  m  ml ) t  m=0 (3)  = t  n  +  1  ^(a+n+l,  b+n+1 ;  n+2 ; t ) .  A l s o i f a = -n o r b = - n , where n = 0,1,2,... and i f c = -m, where rn = n, n+1,... , t h e n we d e f i n e [5]  m  6. n ^(-n.b;  (4)  -m  ((-n)  t)  S  r  (b)  r  /(-m)  p  r  ) t ; r  r=0 s i m i l a r l y f o r F ( a , - n ; -mj t ) . 1  2  S i n c e (3) and ( 4 ) a r e s o l u t i o n s o f e q u a t i o n (1) we see t h a t t h e h y p e r g e o m e t r i c e q u a t i o n has a s o l u t i o n w h i c h i s a p o l y n o m i a l i n t whenever - a o r -b i s a n o n n e g a t i v e i n t e g e r . The p r o p e r t y (5)  F 2  l( ' > a  b  w i l l be used l a t e r . are  c ;  X  )  =  f(c)r(c-a-b)  /V(c-&)V(cr\>))  I t i s assumed t h a t c, c-a-b, c-a and c-b  not negative integers [14J. There a r e many r e s u l t s : r e l a t i n g t h e h y p e r g e o m e t r i c  f u n c t i o n and t h e Legendre f u n c t i o n s PjJ(t) and QJj(t).  Hobson's  [12] d e f i n i t i o n s o f t h e Legendre f u n c t i o n w i l l be used.  These  d e f i n i t i o n s a r e i n terms o f c o n t o u r i n t e g r a l s and are v a l i d f o r u n r e s t r i c t e d v a l u e s o f m and p. 1 t o -» and  F o r t h e t - p l a n e c u t from  | l - t | < z.  Pp(t) = ( r ( l - m ) ) " ( ( t - D / ( t + l ) ) -  (6)  1  m / 2  ^(p+lrpjl-mjM/S)  and f o r | t | > 1 (7)  e-  m 7 r i  ^ ( t ) = 2P(r(p+l)r(P+m l)/r(2p+2))((t -l) /VP- - ) 2  m  m  1  +  x F ((p+m+2)/2,(p+m+l)/2; p+3/2; t " ) 2  2  1  F o r a l l t we have t h e c o n t i n u a t i o n f o r m u l a e , w h i c h a l s o c a n be found i n Hobson [ 1 2 ] ,  7.  («)•  Pj(-t) = e ^ F ^ t )  - f sih((p+m)ir)<£(t)  and (9)  Q£(-t) = - e ^  1  (£(t),  where t h e upper and lower s i g n s a r e t a k e n a c c o r d i n g as Im t > 0 or Im t < 0,  8.  3.  KNOWN ASYMPTOTIC RESULTS a.  R e s u l t s o b t a i n e d by Watson. Watson [ 2 9 ] i n v e s t i g a t e d t h e b e h a v i o u r o f F ( a , b ; c;. t ) 2  1  f o r l a r g e v a l u e s o f j a j , j b | and |c|. I f a, b and t a r e f i x e d and  | a r g c| _< w-e, e > 0 ,  |c| i s l a r g e w i t h t h e r e s t r i c t i o n  then f o r | t | < 1 pF-^a^b; c; t ) = l + ( a b / c ) t + . .. + ( ( a ) ( b ) / ( c ) n i ) t + 0 ( | c"  (10)  n  n  n  n  With a s l i g h t l y m o d i f i e d e x p r e s s i o n f o r t h e remainder term E r d e l y i [ 5 ] p r o v e d t h i s remains v a l i d even i f | t | > 1, MacRobert [ 1 8 ] p r o v e d  | a r g ( l - t ) | < 7r, p r o v i d e d t h a t Re c - ». (10)  f o r a range o f argument c w h i c h i s l a r g e r t h a n TT and v a l i d  for  a l l t provided that  |arg ( l - t ) |  < ir.  I n t h e case where more t h a n one o f t h e parameters approaches » Watson [29] d e r i v e d t h e f o l l o w i n g  I f we d e f i n e ? by t + ( t - l ) 2  1 - e  = (e - l)e  1 / / 2  = 2  a + b  and p u t  where t h e upper o r lower s i g n i s t a k e n  b  a c c o r d i n g as Im t >< 0 , t h e n f o r l a r g e (11)  =  results.  |\|  ( l / 2 t - l / 2 ) " " F ( a + X , a-c+l+X; a-b+l+2\; a  x  2  1  (r(a-b+l+2X)r(l/2)X-  x (1 + e - ? ) - ^ " / ^ ! 0  1  +  where | a r g x | _< v-b,  1 / 2  ^  (X- )L 1  0  6 > 0 and a l s o  2(l-t)  - 1  ]  1  9.  (12)  ^ ( a + X , b-X; ;  - jjt)  C  = 2  -  a + b  (r(i-b-rx)r(c)/r^)r( -b x))(i-e^)- 4 c  1  C  +  v ( l e ^ ) - - - \ x-'-2[e( - >* + e i c  a  b  X  b  i 7 r  +  ve-(  X + a  )*][l +  OdX" !)]  ( " i c  }  x  .x.  ,  1  where t h e L u p p e r or lower s i g n i s t a k e n a c c o r d i n g as Im t >< 0 and where | x | i s l a r g e , - ^TT - w  ?. = p + i n ,  + 6 < argX < -^TT + w  2  1  6 > 0  - 6  w  2  = t a n " ( h / p ) , -w-j^ = t a n " [ (h - TT)/ p ]  n > 0  w  2  = tan" [(h +  n j<  1  1  1  7r)/p],  -w •= t a n ( n / p ) _ 1  1  0  and t a n x denotes t h e p r i n c i p a l v a l u e o f t h e f u n c t i o n . _ 1  b >  R e s u l t s o b t a i n e d by E r d ' e l y i E r d e l y i d e r i v e d t h e f o l l o w i n g e x p a n s i o n f o r |b| l a r g e [ 5 ] .  I f a,c and t a r e f i x e d , c ^ if  |b| - * such t h a t -  (13)  0,-1,-2,... and 0 < | t | < 1, and  < argtb <  , then  g F ^ b ; c j ' t ) - g F - ^ b ; c; on  =[)_' ( ( a ) / ( c ) n  n.')(bt) ][l + n  n  Odbl" )] 1  n=0 and here t h e a s y m p t o t i c f o r m u l a s f o r t h e c o n f l u e n t hypergeometric f u n c t i o n o f a l a r g e argument [31] g i v e  10.  (14)  c; t)  pF-^b= e-  i 7 r a  = e-  [r(c)/r(c- )](bt)- [i-.-+  ocibtr )]  a  1  a  + [r(c)/na)]e and  i 7 r a  b t  (bt) - [i + a  c  1  ocibtr )] 1  •' "3  s i m i l a r l y , i f - ^ir < a r g b t < ^TT , we  (15)  2 i( ' ; > F  a  b  c  *)  e [r(c)/r(c-a)](bt)- [l  +  + [r(c)/r(a)]e (bt) - [i  + odbtr )].  i7ra  -  a  a  bt  c.  have  c  OClbtl" )] 1  1  R e s u l t s o b t a i n e d by Hapaev We  s h a l l now  r e f e r t o some p a p e r s by Hapaev [9>  He d e r i v e d an a s y m p t o t i c  10].  e x p a n s i o n i n terms of m o d i f i e d  Bessel  f u n c t i o n s of l a r g e o r d e r f o r t h e c o n f l u e n t h y p e r g e o m e t r i c f u n c t i o n F ( a ; c j t ) a s a -• », where | t | ~ 1  fixed.  1  Provided  laf  1  and  c is  t h e s e c o n d i t i o n s are s a t i s f i e d , he t h e n d e r i v e d  [10] es  (16)  ^ ( a ;  -c; t ) =  (c)(at^"  »  where B  m n  mn  Ix| < 1.  =  -1 , f  my  (m)  c ) / 2  JitaT )^ 1  2  ra=0  n=0  , (x,n) ^  and f ( x , n ) - (l+(|)x+(|)x +.'.. d  x=0  ) , n  11. He t h e n used t h i s r e s u l t t o d e r i v e (17)  2  F (a,b;c;t)= 1  es  (HcJ/Ra) y  B (t n,m=D  1  n  2 n + m  m n  h / 2 n nJ ) (r(a+in+2n-l)/r(c+nH-.2n-l))  F (a-l+m+2nj c+m+2n-l;2t), 1  where t h e e x p a n s i o n i s v a l i d f o r | t | ~ |2a|  _1  and b and c f i x e d .  In another one o f h i s p a p e r s [11] he o b t a i n e d a s y m p t o t i c expansions o f F ( a , b ; c ; t ) and F ( a ; c ; t ) where a=ae, C=Y£, a and 2  1  1  1  y a r e c o n s t a n t s and l -* eo, t h a t i s a/c i s f i x e d and b i s f i x e d . , .• Then f o r | t | < |c/a| he d e r i v e s (18)  ~  ^(aliynt)  e(  provided that | k t + Y l _ 1  Similarly, f o r Ikt^+Yl (19) ^ ( a ^ c j t )  a / Y ) t  (l+r ((aY" t) (a- -Y" 1  1  2  1  : L  )2" K^- (...)+..} 1  2  > |fcfcT 1, where k > 2n,n = 0,1,2, 1  > Ifct" ! 1  and | a Y " t | < 1 he d e r i v e s 1  ~ 1  ( l - a Y " t ) " { l + t " ( 2 - b ( b + l ) ( Y " - o " ) ( l - 2 t ) ( l - t ) " ) + *~ (...)+...}. 1  d.  b  1  1  1  2  2  R e s u l t s obtained by Khottnerus Next we s h a l l r e f e r , t o r e s u l t s d e r i v e d by K h o t t n e r u s [15]/  For l a r g e p o s i t i v e v a l u e s o f r and t h e t - p l a n e c u t from 1 t o 1+eoi he d e r i v e d t h e f o l l o w i n g r e l a t i o n (20)  gF^ajbjc+rjt) = 1 + (abt)r"  1  + 0(r ). 2  12.  The e x p a n s i o n i s v a l i d u n i f o r m l y i n t h e c l o s e d bounded r e g i o n G: fRe(t) 2  1  +  6  > 6 f i x e d and 6 > 0,  (21) < | t | < In. k f i x e d and k > 1 + 6; Im(t) _> G. [5]  U s i n g the w e l l known r e l a t i o n (22)  gF^ajbjcjt) = (l-t) " " c  a  b 2  F (c-a,c-b;c;t) 1  v a l i d f o r t i n the t-plane cut from t = l t o t=l+ooi  he r e a d i l y  derived the f o l l o w i n g r e s u l t . I f Re c > min{Re(a),Re(b)} + 1 and t s a t i s f i e s R e ( t ) > 1, Im(t) _> ° J t h e n f o r s u f f i c i e n t l y l a r g e p o s i t i v e r (23)  = ( 1 - t ) - " - { 1 + (c-*)  ^(a+r^+r^c+rjt)  c  a  b  r  The e x p a n s i o n i s v a l i d u n i f o r m l y p r o v i d e d  + (r 0  R e s u l t s o b t a i n e d by C h e r r y and Sommerfeld F i r s t we s h a l l c o n s i d e r t h e h y p e r g e o m e t r i c  occuring i n gas-flow  w  §  +  +  v  2  < w ^ 7 - t ?  I t i s s a t i s f i e d by X ( t ) = t ^ g F - ^ v - a ^ _ where 2 a 28  v  n—  = v + 6 + ^/v (1+26) + 6 = v + 6 - y v ( l + 2 6 ) .+ 6 2  v  equation-  theory  <? -  , 2  ,  )  y  )}  that t l i e s i n  a c l o s e d bounded r e g i o n o f t h e t - p l a n e c u t f r o m 1 t o l+»i.  e.  - 2  ' °'  v-b^; v+1;' t ) ,  13.  and p = - ( Y - 1 ) " ; Y > 1, i s t h e f i x e d a d i a b a t i c index. 1  The  (24) i s r e a d i l y g i v e n by t = ( l + 2 p ) .  t u r n i n g point of equation  _ 1  s The comparison  equation  2 d Z . l d Z . 2/ I N — 2 + - - j - + v (1 - - ) z dT T  / nr'\ (25)  '  n  •  = 0  2  i s s a t i s f i e d by t h e B e s s e l f u n c t i o n J , ( V T ) ,  ).  where T = (1-^—) (1-t  _1  s w i t h | a r g v| _< v - e and 0 _< t _< 1-e,  Then f o r v ~ «  e > 0, Cherry [ 1 , 2 ] d e r i v e d (26)  X (t)  ~ N(v/h ) /2 1  v  v  [ J v ( v T  .  ) ( 1  +  q 2 V  + Tj^(vT)(q v"  1  1  where t a n h "  1  ((1 - J ) 2  = tanh" N = h  v  (1-t) / 1  4  1  /  2  -  )  (1-T ) 2  -2  +  _  }  ...)],  1 / 2  t^tanh-^t^t^-tr ) / 1  1 / 2  2  1  /  v  2  the expansion  i s valid  u n i f o r m l y i n t and a r g v. To c a l c u l a t e t h e q , n = 1,2,..., W = (1 - T ) Z , and we s e t w((l-|-)VV(l-t) /*-l/2e)x,(t), n  2  1  u-tarff  ( ( 1 , - T  1  2  )  1  /  2  ) -  (  I  -  T  ^  then w = W(l + Q v~ 2  where  Q  X  2  + Q^v"' + . .. ) - - ^ ( Q T V ' " 4  = q-^(l - T ) /  Q2 = q  2  2  +  2 T  q  x  1  /  2  (2(1-T )), 2  2  and  4  T  v  - 4  5  = (r(a )r(l+v-b ))/(r(a -v)r(v)r(v+l)); v  v  + q^v" +  ((l- t-)(i-t) s - lp((l- )/(i- |-)) ' s 1  ^  +  2  + Q 5 v " ^ + . .. )  14.  %  = % + T q  /(2(1 - T ) ) ,  2  ...  2  3  and 2  d  Q  l  '  ,  2  = (<t>  2Q  = (<t> - §)Q2 + 2 $ ^ +  3  2  * = ((i-s ) / 4(i-t ) ){5?" 2  = (<j> - § ) Q  4  (1-T )  2  +  2Qg  2Q where|> = T ( 1 + ( 1 / 4 ) T ' ) /  x  2  3  +  3  Q  ,  x  Q  2  ,  3  ,  - (i+6t )r^(>4t )F.-  6  B  + Q  s  s  2 +  (i-2t xi-4t ); s  s  ;  - = T.1/2 S ' . I n t e g r a t i o n c o n s t a n t s a r e found f r o m t h e l i m i t i n g  form  of t h e X ( t ) e x p a n s i o n (26) f o r t = T = 0, w h i c h t h e n i s e q u a l v  to (27)  (ev" ) r(v+l)((2Trh 1  where  v  6 = a (l+a) a  _ 1  "  a = 1/2(1+2B)  a  1//2  6 ) / 2TTV) 2 v  v  1 / 2  ~  1 + Q ( 0 ) v " + Q ( 0 ) v " + . . ., 1  1  2  2  and - 1/2 .  We expand t h e l e f t hand member o f (27) by means o f (28)  and  l o g ( 2 7 r h ) ~ 2v l o g 6 + c-^v" + 1  v  . . .  S t i r l i n g s e r i e s and equate c o e f f i c i e n t s t o get ^ ( 0 ) ,  r = 1,2,.... and  Q . j \ T \  S i m i l a r r e s u l t s were o b t a i n e d by L i g h t h i l l  Seifert [23]. The  gF^-in,  [17]  '  asymptotic behaviour of the hypergeometric f u n c t i o n  -ipw;l;t) occuring  i n wave mechanics has been i n v e s t i g a t e d  15 by Sommerfeld [26]. he  F o r t = - 4 p ( 1 - p ) " s i n ( a / 2 ) , 0 < p < 1, 2  2  obtained  (29) -gF^-ivlpnajt) where u  ~ e-  7 r p n  (27ma)"  1 / 2  (iu; + 1  1  e *^)  = ( ( 1 - p ) / 2 ) ( l + i c o t ( 3 / 2 ) ) , c o t (a/2) £ 0,  Q  f ( u ) = i l o g ( u ( l - u ) " ( l - u t ) " ) and p  p  1  -a=f"(u ); 0  I f c o t (a/2) - 0, then f(u )=27r 0  u  = ( (1-p )/2) ( l + i ( 7 r - a ) / 2 + i ( 7 r - a ) / 2 4 5  Q  - i(l+p)log((l+p)/(l-p))+( /(l+p))i(7r-a /2 2  p  p  -p(l-p)(Tr-a) (l+p)- /6 + ... 2  3  -a = f " ( u ) = 6 p ( T r - a ) ( l - p ) " ( l + p ) - + ... 1  2  0  f.  R e s u l t s d e r i v e d from Thome's paper The f o l l o w i n g r e s u l t w i l l now be d e r i v e d from a paper [28]:  w r i t t e n by Thorne (30)  (r(l-m))" F (p+l,-p;l-m;(l-t)/2)  ~  1  2  1  {r(p+l+m)/np+l-m)} [(l-t)/(l+t)] 1/2  x (4 /(t 2  2  2 P  ))V^  (  M  (  Y  2;?  z  )  >l/3^  m/2  v  E  (  g  z  )  f  2A  „  s=0  as  s=0  p - »,  where m = - y a = p. = Jl-a 2  -(p+l/2)a, ,  a i s f i x e d and  0< a  < 1,  16.  |z(t)  = a cosh" (|(t- -l)"  3 / 2  1  2  1 / 2  )-cosh" tB" 1  1  and t h e f u n c t i o n s F ( z ) and E ( z ) a r e g i v e n b y t h e r e l a t i o n s (40) s s and (43) determined l a t e r . t-interval  The e x p a n s i o n i s v a l i d f o r t h e  | t | < 1.  Completely a n a l o g o u s l y t o t h e f o r e g o i n g , t h e f o l l o w i n g r e s u l t was found f o r t > 1:  (31)  F ((m+p+2)/2, (m+p+l)/2; p+ J ; t " ) ~ 2  2  1  , 7 7 " .  7T2-P1  1 2  m t  P  + m + 1  (i- 2y ^ t  »  }> (z) "  X  2s  s  "5) e " -5 2 1  (2p+2)/((r(p+l)(n:p+m+l)r(p+l-m))  Y  ( 4 z / ( t  2_ 2 7 p  ) )  U l ( Y  1  1  ±  -3 - 3 e  z )Y  " 3  2 _2 . _5 « + e'^A^Y e" z) " XF (z)Y"  X  _2 .  1  37ri  1  Y  s=0  s  2s  }  s=0  as p-*», where m,B,Y* and z a r e d e f i n e d b y (30);  the f u n c t i o n s  E (z) and F (z) a r e i d e n t i c a l t o t h o s e o f t h e former case as w e l l , s ' s ' v  v  We s h a l l now s e t out t o prove t h e f i r s t r e s u l t . of an a s y m p t o t i c e x p a n s i o n f o r P ^ ( t ) as p  Instead  » , an a s y m p t o t i c  e x p a n s i o n f o r t h e a s s o c i a t e d Legendre f u n c t i o n o f t h e second k i n d Q^(t) was used t o d e r i v e t h e second r e s u l t .  A s y m p t o t i c expan-  s i o n s f o r P ( t ) and < ^ ( t ) , m = -(p+-|)a, as p - > » can be o b t a i n e d m  by a method employed by Thorne [28]. The a s s o c i a t e d Legendre e q u a t i o n  (32)  (1-t ) i \ 2  dt  - 2 t dy $ dt  + (P(P+D - m ( l - t ) - ) y 2  2  1  =  0  has a fundamental system o f s o l u t i o n s c o n s i s t i n g o f P ^ ( t ) and  17.  Q^t).  I f we use Hobson's d e f i n i t i o n s  they are s i n g l e valued  of t h e s e f u n c t i o n s  a n a l y t i c i n the t-plane  then  cut along the  r e a l a x i s from 1 t o -a>, and a r e r e a l when t i s r e a l and t > 1. z = x, where - 1  For  < x < 1,  t h e fundamental s o l u t i o n s o f  ( 3 2 ) a r e t a k e n as P p ( ) and Qp^x) d e f i n e d by x  (33)  P p ( x ) = e~ *  (34)  2  P^(x + i . o )  where  e  m  7Ti  m  m 7 r i  and  Q*(x) = e ^ Q ^ x  tl  + i.o) + e ^  ^(x-i.o),  e>0.  f ( x + i . o ) =. l i m f ( x + i . e ) , 0  e  The f u n c t i o n s P p ( x ) and Qp(x) a r e g e n e r a l l y known as F e r r e r s ' f u n c t i o n s and a r e r e a l f o r x r e a l and 0 < x < 1. E q u a t i o n ( 3 2 ) i s n o r m a l i z e d by t h e 2  (t -1)  transformation  1/2  y = Y, t h e r e s u l t i n g  /  (35)  equation  = { p ( p + l ) ( t - l ) - + (m - l ) ( t 2  dt*  i s now s a t i s f i e d by ( t - l ) 2  1 / / 2  1  2  2  P ( t ) and ( t - l ) m  2  s e t rn = - ( p + 1 . 2 ) a , 0 < a < 1 and a f i x e d ,  1//2  I)' } Y 2  Q^( t)  then f o r p =  I f we J _ 2 1  a  equation (35) reduces t o (36)  = {(t - p )(p+l/2) (t -l)- -4- (t +3)(t -l)- } 2  2  2  2  2  1  2  2  2  dt w h i c h i s now s a t i s f i e d by ( t - l ) / 2 a ( p + l / 2 ) ^ j 2  1  p  (t -l) / ^(P+l/ ) 2  1  2  2  ( t )  & n d  .  I t i s p o s s i b l e now, a c c o r d i n g  to  Olver  [ 1 9 , 2 0 ] or  Thorne [ 2 7 ] , t o o b t a i n a s y m p t o t i c e x p a n s i o n s o f P p ( t ) and 0 ^ ( t ) , which are v a l i d uniformly w i t h respect  t o t , as p -» » , f o r t  18.  l y i n g i n a domain D^., s a y , i n w h i c h t h e p o i n t s t = l and t = 6 + i . o a r e i n t e r i o r p o i n t s and w h i c h extends t o i n f i n i t y . o f ( ( t - B ) ( p + l / 2 ) ( t - l ) " ) Y has double p o l e s  The c o e f f i c i e n t  2  2  2  2  t = +1 and t u r n i n g p o i n t s a t t h e  at the r e g u l a r s i n g u l a r i t i e s simple  2  z e r o s t = +B. To o b t a i n e x p a n s i o n s v a l i d a t t h e t u r n i n g p o i n t t=B+i.o [19?20]  we make t h e t - z t r a n s f o r m a t i o n  (37)  H = -(t - D ( t - P )- V/ , 2  X -|z^  2  = (^)  .t = - j ( s -B )(s - l) 8 2  2  2  2  _ 1 / 2  2  V  2  Y and  - -  1  _ 1  ds  = acosh~ a6" (t" -l) 1  where t h e l o w e r l i m i t o f t h e i n t e g r a l  1  " "-cosh" tB"" ,  2  i s p + i.o.  2  1  1  Then X ( z )  s a t i s f i e s the equation (38) where  £\ dz f^z') = ( | ) z  2  5  The comparison  {(p+l/2) z  =  2  + f , ( z ) }X, x  + 4  _ 1  z(t - l)(t 2  B )~ {t (4a -  2  2  3  2  2  l)+(l-a )}  equation 2  (39)  (p+l/2) zX  ^ - i» -dz^  2  i s s a t i s f i e d by t h e A i r y f u n c t i o n s A i ( z ) and B i ( z ) . Thorne [ 2 8 ] now showed t h a t i f (40)  E ( z ) = 1, Q  P (z) 8  W ? >  =  |z- jV {f (r) 1/2  1/2  1  - - i k^ F  +  f  f  «00  i<  E ( r ) - E»(r)} d r , and f l  r )  F  s< > r  d r  +  ° W  4  19.  where t h e sequence o f a  s _> 0, a r e i n t e g r a t i o n c o n s t a n t s ,  t h e n t h e f o l l o w i n g r e s u l t s w i l l be o b t a i n e d : (41)  e^  {r(p+m+l)/r(p-m+l)} ^ ( 4 z ( t - B ) " )  P (t) ~ m  2  2  _1 » {Ai(v z) " £ E (z) " s=0  x  1  3  Y  s  2  2  3  Y  Y  x  J  _5 »  + A i ( z ) " "3 Y F ( Z ) Y s=0  2 S  1  Y  ]  2 S  s  and . 2 r i + £i  1  1  n l  (42)  e" 2  2  - 7Ti  -  2  x{Ai( e" 1  * <g(t) ~ 7r{r(p+m+l)/r(p-m+l)3^(4z(t -B )- )  2  z)  1  Y  1  1 Y  - -TTi  0 8  ^E (Z)Y" + s=0 2 s  s  e  '  (43)  -  2  7Ti  A i ^ e  3  where t h e i n t e g r a t i o n c o n s t a n t s a  2  s  1  -  5  z) " Y  2  1  7r  x  °° 3  )\(Z)Y~ s=0  2 S  }  are s p e c i f i e d by the r e l a t i o n 1  0,  2  l ( % y '  8  +  PgY" " ) 2 3  s=0 where R = v ( l + v ) * " " , V  1  v  1  (V2) r(-m) m  ~J-m/tor  v = -5(a  -1  {rCp+nH-D/TCp-iw-l)}"  -1).  The e x p a n s i o n s ( 4 l ) and (42) a r e v a l i d throughout t h e t - p l a n e c u t from +1 t o -» except f o r a pear-shaped"domain s u r r o u n d i n g t h e s i n g u l a r i t y t = - l and a s t r i p l y i n g i m m e d i a t e l y below t h e r e a l t - a x i s f o r w h i c h |Ret| < B+6, 0 > Im t > -6, 8 > 0.  I n b o t h t h e s e r e g i o n s a s y m p t o t i c e x p a n s i o n s can be o b t a i n e d  by use o f t h e c o n t i n u a t i o n expansion o f  f o r m u l a e (8) and (9). The a s y m p t o t i c  (p+l-m, -p-m; 1-m; ( l - t ) / 2 ) i s now an immediate  consequence o f ( 4 l ) , (6) and r e l a t i o n (33), where we have t o take the + sign.  The e x p a n s i o n o f o  2  F ((m+p+2)/2, (m+p+l)/2; p+3/2; t 1  ) f o l l o w s i m m e d i a t e l y from  the a s y m p t o t i c e x p a n s i o n (42) and t h e r e l a t i o n ( 7 ) -  2  20.  CHAPTER I I  THE NORMALIZED HYPERGEOMETRIC DIFFERENTIAL EQUATION  The h y p e r g e o m e t r i c (1)  t ( l - t ) A_| dt  i s normalized x(t)  + [  equation - (a+b+ljtjg*- - abx = 0  c  by s e t t i n g  = y(t) f  |(i-t)( - " " c  a  b  x(t) = y ( t ) t " l ( t - l ) ( - " c  Then e q u a t i o n  b  )  /  l ) / 2  ,  2  ,  0 < t < 1, and t > 1.  ( 1 ) becomes  UL dt  (2)  a  l  + {(At I  where  2  + B t + C ) / 4 t ( l - t ) \ y = 0, ' 2  2  A = 1 - (a-b) , B = 2c(a+b-l)-4ab,  and  C = c(2-c) . L e t us c o n s i d e r t h e case t h a t a=b i s a l a r g e r e a l parameter. Then t h e t u r n i n g p o i n t s t quadratic  equation  (3)  t  2  , i = 1,2, a r e t h e r o o t s o f t h e  i  + [2c(2a-l) - 4 a ] t + c(2-c) 2  =  0  For a l a r g e (4)  t _~ ]  c(2-c)/4a  2  and  t  ~ 4a . 2  2  I f we now make t h e s u b s t i t u t i o n  t = c(2-c)z/4a , then  21.  equation (2) for a=b is transformed into 2  (5)  dz  + 4 a f(c(2-c)(l-z) + f(z))/g(z) 4  I  )  where  f ( z ) = (c (l-c)(2a-l)z/2a ) +  and  g ( z ) = z (4a - c ( 2 - c ) z ) ~ .  2  2  2  2  y  = o,  (c (2-c) z /l6a ) 2  2  2  4  ?  For z bounded f ( z ) = ©(a" ) as a - <x>. 1  The t u r n i n g p o i n t now  o c c u r s a t z = l and t h e s i n g u l a r i t i e s occur a t z=0 and z = 4 a / c(2-c). 2  E q u a t i o n (5) can be w r i t t e n i n Thome's form (6)  1-| + ( ( 2 a ) z - ( l - z ) p (z) + z dz 1 2  2  2  2  2 q i  ( )| y = 0 , ) Z  2  where p-^z) = c ( 2 - c ) / ( 4 a - c ( 2 - c ) z ) , p-j_(z) does n o t v a n i s h and i s r e g u l a r f o r z < 4a / c ( 2 - c ) ; f u r t h e r m o r e q (z) x  = 4a f(z) / (4a - c(z-c)'z) , 2  2  2  _2  f o r z bounded q-j_(z) = 0 ( a ) as a -* » . I f we s e t y = ( 2 - c ) and u = 2a, t h e n c  2 (7)  ^-g + (( /4)(l-z)/z dz ( Y  2  + [(Y/2) (1-Z)/U Z][(2- Z/U )/(1-YZ/U ) 2  2  2  2  Y  2")  + q _(z)/z j y = 0. ]  Now Y ( 1 - Z ) / 4 Z  2  i s n o t t h e dominant term, t h e t u r n i n g  p o i n t t-^ and t h e s i n g u l a r i t y t, = 0 c o a l e s c e bounded.  f o r _a l a r g e and z  N u m e r i c a l a n a l y t i c methods can determine t h e v a l u e s  f o r 2 ] _ ( , j > t ) i n t h e r e g i o n i n w h i c h t h e t u r n i n g p o i n t and F  a  a  c  the s i n g u l a r i t y t=0 c o a l e s c e .  F u r t h e r m o r e , ^F-^a, a; c; t )  probably i s the simplest f u n c t i o n demonstrating t h i s behaviour.  22.  ' CHAPTER I I I  THE ASYMPTOTIC BEHAVIOUR OF F - ( a , a ; c ; t ) AND 2  L  F (a,a+l-c;l;t ) _ 1  2  1  A s o l u t i o n of the normalized hypergeometric equation 2^ ^ - g + {(At ' r "  (1)  2  + B t + C) / 4 t ( l - t ) ) y ' ) 2  2  =  0,  p  where  A  =  l-(a-b) ,  B  =  2 c ( a + b - l ) - 4ab, and  C  =. c ( 2 - c ) ,  is  |- ^ ( a ^ b j c j t J t ^ C l - t ) ^ ^ - ^ ,  (2)  1  \  0  0 < t < 1, and  2  ^ ( a ^ a + l - c r a - b + l ^ - ^ t ^ ^ ^ ^ C t - l ) ^ ^ - ^ 1  0  t > 1. Let us c o n s i d e r  t h e case t h a t a = b i s a l a r g e  parameter.  Then e q u a t i o n ( l ) has t h e form  (3)  ^ dt  =  real  [up(t) + q ( t ) l y ,  p(t) = l/(4t(1-t) ),  where  2  q ( t ) = ( c ( c - 2 ) - t ) / ( 4 t ( l - t ) ) and 2  2  u  = 4a  2  - 2c(2a-l).  2  New v a r i a b l e s Z and Y a r e i n t r o d u c e d f *' 1  2 0  !  ts. Z = [(1/2)" [p(s)] t , d t j-1/2 \ dz * J  1  ;  y  1 / 2  ds]  2  by t h e r e l a t i o n s  23.  T h e r e f o r e , we r e a d i l y o b t a i n 1/2  t  CO  = 1  =  (1A)|" "o  (5)  z  Equation  1  s  x /  *(l-s)  (-l/4)ln((l-  1/2 =  (-lA)f  -TTp^ - (l/2)coth" (/t) s ' (1-s)  =  (-i/4)m((./ u-i)/(./t+i)), t > i .  1  t  t  (3) i s then t r a n s f o r m e d =  juz"  1  + YZ"  where Y = -3/l6 and f o r z = z h(z). =  t < 1, and  ft)/(l-hrt)),  »  2  (6)  = (l^tanh" ^)  2  into  + h(z)z" | Y , 1  2  (-3/4)-(l/4)cosech (2z 2  1 / 2  )+(c(c-2)+3/4)tanh (2z 2  ).  1 / 2  For t > 1 r e l a t i o n (5) i m p l i e s t h a t t = c o t h ( 2 z  ).  2  T h i s l e a d s t o t h e f o l l o w i n g r e s u l t s f o r t > 1 and z s z : 2  t = (||) =  2z"  1 / 2  coth(-2z  )cosech (2z  1 / 2  2  q[t(z)] = (l/4)[c(c-2)sinh (2z 4  1 / 2  1 / 2  ),  )tanh (2z 4  1 / 2  ) - sinh (2z 4  l / 2  and t h e r e f o r e (t) q[t(z)] = [c(c-2)z" ][tanh (2z 2  yz' 2  1  + h(z)z  _ 1  = t i ,  2  (t"  1 / 2  )+(t)  1 / 2  1 / 2  )]-[z~ coth (2z 1  2  1 / 2  )],  q[t(z)]  dz = (-3/l6)z" + z ' H 2  + (c(c-2) +  (-3/4)-(l/4)cosech (2z 2  (3/4))tanh (2z 2  1 / 2  )].  1 / 2  )  )]  24.  T h e r e f o r e , y = - 3 / l 6 and i f we s e t 2 u  = 1 + 4  = 1/4,  Y  then w i t h o u t l o s s o f g e n e r a l i t y we can t a k e p. = 1/2. I f we now p u t  = u + c ( c - 2 ) = (2a-c)  t h e n the b a s i c  e q u a t i o n (6) becomes (7)  ^ | dz  where h-^z) =  = ( u ^ " + ( ( p -l ) / 4 ) z " 1  2  2  +  h (z) " )Y, 1  Z  1  (-l/4)cosech (2z / )-(c(c-2)+3/4)sech (2z ' ). 2  1  2  2  1  / 2  A p a i r o f l i n e a r independent s o l u t i o n s o f t h e comparison e q u a t i o n , t h a t i s e q u a t i o n (7) w i t h h-^z) = 0, are Y-L = z  I  and K  1 / 2  K  (2(u  l Z  )  1 / 2  )and  a r e m o d i f i e d B e s s e l f u n c t i o n s of o r d e r (jt . 2  For convenience,  i f we now r e p l a c e z, ^  2  by z  and u.j/4  r e s p e c t i v e l y , t h e n e q u a t i o n (7) becomes (8)  where ^  ^ |  =  z"  1  ||  +  iju  = - 2 ( 2 a - c ) and f o r z s z  f(z) = 4h (z ) 2  1  = -cosech (2z) 2  2  +  Hz"  +  2  f ( z ) | Y,  = (l/2)coth~ (,/t) 1  2  - 4 ( ( c ( c - 2 ) + 3/4) s e c h ( 2 z ) ) . 2  The f u n c t i o n f ( z ) i s an even f u n c t i o n o f z and i s r e g u l a r i n an unbounded s i m p l y - c o n n e c t e d  open domain D,  f(z) = 0(|z|" where a i s c o n s t a n t and a > 0.  1 _ a  )  actually  as |z| - • ,  25.  Let  D' be any s i m p l y - c o n n e c t e d domain l y i n g w h o l l y i n D,  the  b o u n d a r i e s of w h i c h do not i n t e r s e c t t h e b o u n d a r i e s of D.  Let  6 > 0 be an a r b i t r a r y r e a l p o i n t i n the s e c t o r  t h e n t h e domain  |arg z| < TT/2,  comprises t h o s e p o i n t s z o f D' w h i c h can be  j o i n e d t o 6 by a c o n t o u r w h i c h l i e s i n D  1  and does n o t c r o s s  e i t h e r t h e i m a g i n a r y a x i s or t h e l i n e t h r o u g h z p a r a l l e l t o t h e imaginary a x i s . Next suppose d t o be an a r b i t r a r y p o i n t o f the s e c t o r z| < TT/2, w h i c h may be a t » , and e t o be an a r b i t r a r y  |arg  p o s i t i v e number. which  Then D  c o n s i s t s of those p o i n t s z of D' f o r  2  |arg z| _< J>ir/2, Re z < Re d and a c o n t o u r can be found  j o i n i n g z and d w h i c h s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s [19,20]5 (i)  i t l i e s i n D';  (ii)  i t l i e s w h o l l y t o the r i g h t of the l i n e through z p a r a l l e l t o the imaginary a x i s ;  (iii)  i t does n o t c r o s s t h e n e g a t i v e i m a g i n a r y a x i s i f TT/2 _< a r g z _< J>TT/2. and does n o t c r o s s t h e p o s i t i v e i m a g i n a r y a x i s i f -37r/2 _< a r g z _< -TT/2;  (iv)  i t l i e s o u t s i d e the c i r c l e  | r | =, e|z|.  A p a i r of l i n e a r independent s o l u t i o n s o f t h e comparison equation  (9) are  now  Y-L = z l (u-jz) Y  2  =  z  K  ( l )* u  M  and  z  The b a s i c e q u a t i o n (8) has s o l u t i o n s Y-^(z) and Y ( z ) such 2  t h a t f o r Re \x > 0  26.  (i)  i f z lies in  D, x  M-l Y (z) = z l ^ ^ z ) V A ( z ) u " + 0(u" ) U=0 M 1-1 -l £ B (z)u' 2 s  x  2 M  s  f  +  2 s  s  (z/(l+|z|)0(u- ) 2 M  ls=0 as u^ -* <», u n i f o r m w i t h r e s p e c t t o z, (ii)  i f z l i e s i n B>>, M-l  Y (z) = z y u 2  l  )  Z  \  I  A (z)u"  2 s  s  + 0(u" ) 2 M  ^s=0 - ( z ^  M-l ) ^ ! ^ ) ! £ B (z)u" (s=0 i  1  + (z/(l+|z|) 0(u" )  2 s  2 M  s  as u^ - <*>, u n i f o r m w i t h r e s p e c t t c . z . Since f ( z ) = 0 ( | z | ~ expansion  )  1 _ a  as  |z| - oo and a > 0, the  i s v a l i d f o r z tending to i n f i n i t y .  asymptotic  The  sequences of  f u n c t i o n s A ( z ) and B ( z ) are g i v e n "by the r e l a t i o n s A (z) = 1 G  2B (z) = - A i ( z ) + J ^ f ( t ) A ( t ) - ( 2 p + l ) t - A ( t ) j dt, 1  s  2A  s  s  ( z ) = ( 2 n + l ) - B ( z ) - B l ( z ) + J * |f ( t ) B ( t )jdt + C 1  s + 1  where C  Z  g  fi  ,  i s a c o n s t a n t and 6 > 0 i s f i x e d .  6  Asymptotic (i)  s  forms, t h a t i s M = 1, of t h e s e s o l u t i o n s are  i f z l i e s i n L^,  Y ( z ) = z l ( u z ) ^1 + 0 ( u " ) ^ 2  1  ( j  +  (  1  a  u  i ) n i 1  I  +  (  u  i  z  )  ^ o B  ( z )  +  ( /( Z  1 +  I D) Z  ° K ^ 2  27.  as  -  », u n i f o r m w i t h r e s p e c t  (ii)  t o z i n D^,  i f z l i e s i n Dg,  Y (z) = zK^(u z) | 1 + 0(u" ) 2  2  1  - ( z u - ^ K ^ ^ z ) | B ( Z ) + (z/(l+|z|)) 0(u~ ) 2  0  as u-j^ - <*>, u n i f o r m w i t h r e s p e c t  t o z i n Dg. /  The f u n c t i o n B  q 2  (Z)  B ( z ) i s now  S  B (z) =  Q  J  (1/2)  o2  g i v e n by t h e i n t e g r a l  Z  = (-1/2) J  6 > 0  f(s)ds,  6 z  cosech (2s) 2  6  + 4(c(c-2)+3/4)sech (2s)  ds  2  = where C  2 &  (1/4)coth(2z)+(c(c-2)+3/4)tanh(-2z)+C , 2 6  i s a c o n s t a n t and 6 > 0 i s f i x e d .  S i n c e any s o l u t i o n i s l i n e a r l y e x p r e s s i b l e i n terms of two l i n e a r independent s o l u t i o n s and since' Y(t[  Z l  ]) =  x  0 < t < 1, (10)  (^)- /2p dz£ 1  ( a  ,  c/2  a ; C ; t ) t  1  and s i m i l a r l y , ?  .  t )  (2a l.c)/2 +  (8),  i s a s o l u t i o n of equation Y(t[z ]) = c (u )Y (z )  Y(t[z ]) -  ( 1  1  1  1  1  1  +  i t follows that  c (u )Y (z ), 2  1  2  1  since  ( ^g)" / dzj 1  i s a s o l u t i o n of ( 8 ) ,  P (a,a l-c;l;t- )t( • 1  2  2  T 1  +  i t implies that  c  2 a  )/ (t-l)( 2  t >  2 a + 1  1,  - )/ , c  2  28.  (11)  Y(t[z ]) = d (u )Y (z ) 2  Now,  1  2  + d (u )Y (z ). 2  1  2  2  approach  2  I t i s w e l l known [30] t h a t  I (z) ~ ( 2 7 r z ) as  1  and l e t t -» 1, t h a t i s z-^ and z  fix  infinity.  1  1 / 2  (e + z  e  ±^  + 1  / ) 2  e" ),  i r i  |arg z| < 3^/2 - e, e > 0,  z  |z| -» » u n i f o r m l y w i t h r e s p e c t t o a r g z; the upper s i g n  a p p l i e s t o the range -TT/2 + e _< a r g z _< 3T/2 - € and the t o the range -3t/2 + e _< a r g z _< TT/2 - €. K f j  (z) ~  (V2z)  1 / 2  lower  Also,  e" , z  |arg z| _< 3ff/2 - e u n i f o r m l y w i t h r e s p e c t t o a r g z as |z| -» ». Near the r e g u l a r s i n g u l a r i t y t = l [ p ( t ) ] / 1  and t h e r e f o r e a c c o r d i n g t o e q u a t i o n s  ~ 1-t  ,  e ^ 2 ~ t-1  ,  e  -4z  x  z  ~ l/(2(l-t))  2  (4) and (5)  0 < t <1 t > 1.  A p p l y i n g t h e s e e s t i m a t e s t o the s o l u t i o n s (2) f o r a=h we g e t  (12)  -^(a.ajc^Xl-e^D^te-Sjt ^)/  y(t[Zl])  2  ~  (r(e)r<oa.)/<r<c.) ) 2  )  e  -  2  (  2  a  +  i  -  c  )  z  2  i  as t -* 1" and - a i s l a r g e ; y(t[z ]) ~ ^ ( a ^ + L c j U f ^ l + e ^ ^ l i ^  2  2  ~ as t  1  +  (r(c-2a)/(r(l-a)r(c-a)))e- ( 2  2 a + 1  ^ / ^  4  ^ ) ^  2  - ) 2 c  z  and - a i s l a r g e .  L e t t i n g t -» 1  +  i n equation  (11) and u s i n g the  approxi-  m a t i o n s f o r the m o d i f i e d B e s s e l f u n c t i o n s as | z | - », we r e a d i l y P  )  see  that d ( i) = u  0  2  s i n c e u^ i s l a r g e and B  and  d  l ( l ) ( - | dz u  )  d  / 2 Y 1  q 2  I(  ( Z ) remains bounded as  z  j -» •  ) ~ e( l- ) 2(7ru )- / d (u ) , u  2  |  2  2  z  1  2  1  1  1  2  as  | z | -» OB ,  | arg z |  2  <_ 3T/2  2  Prom r e l a t i o n (12) we (13) The  d (u [a]) 1  =  1  constant  - e, € > 0, u n i f o r m l y i n a r g  then f i n d  that  (r(c>2a)/(r(l-a)r(c-a)))(2(c-2a)7r) / . 1  2  d ^ u - J a ] ) i s w e l l d e f i n e d f o r Re(c-2a) > 0 since  t h e i n v e r s e of the gamma f u n c t i o n i s e n t i r e . Because p M a . a + l - C i l i t - ) = (-SS-)l/2 (2a-c)/2 _ d | 1  t  ( t  l ) (  c-2a-l)/2  x  x  Z  x Y (z )d (u [a]), 1  2  1  1  we t h e n have d e r i v e d the f o l l o w i n g r e s u l t . For a and 1 < t _< T  c r e a l , c-2a  1  Z  / 2  I  ( i 2) u  l / 2  z  [ l +  °( l ) U  2  B(a)t(  ]  "  2 a  ( 2 2  -  2  u n f o r m l y w i t h r e s p e c t t o t as  c + 1  / 2 u  + z /(l+|z | )0(u^)] 2  t-interval  < «  2  (14) g F ^ a + l - c j l j t - ) -  x| 2  > 0 and the  *•• oo,  / )/ (t-l)( " 2  2  l ) 3/2 1  I  c  ( u  l 2^ Z  [ B  2 a  )/ x 2  o2( 2) z  z  2  where  = -2(2a-c), c f i x e d , z  = ( l ^ c o t h " ^ ) , 1  2  B(a) = B  o 2  (r(c-2a)/r(l-a)r(c-a))(2u Tr)  1 / 2  1  (z ) = ( l A ) t  1  /  - (c(c-2) + 3 A ) t -  2  2  , 1 / 2  + Gm , 2 x  T<  » and f i x e d .  2  F i n a l l y , i f We now c o n s i d e r the i n t e r v a l 0 < 1^ _< t < 1, 1/2 -1 = ( l / 2 ) t a n h ~ (./t). Analogous t o r e s u l t s o b t a i n e d  then  t s ^|  b e f o r e , we now have  and  Y  z~  2  + h^z^zl 1  = t 1/2  = 8z^  ^ dz 1  t"  - -(Vl6)z"  2  tanh(2z^  2  1  /  2  +  / / 2  )sec h(2 J 2  / / 2  z  t^q(t[ ]) Z l  + z ^ i h ^ )  +  c(2-c)),  where h ^ z - ^ = ( l / 4 ) s e c h ( 2 J / ) + ( c ( c - 2 ) + 3 / 4 ) c o s e c h ( 2 ^ 2  )  2  2  z  / / 2  z  T h e r e f o r e , as b e f o r e y = -(3/l6) such t h a t n = 1/2  ).  and  2  = u + c ( 2 - c ) = (2a-c) ; f u r t h e r m o r e , i f a g a i n we r e p l a c e z^ 2 2 and by z^ and u.j/4 r e s p e c t i v e l y , t h e n  z± = ( l ^ J t a n h " ^ / ^ ) , 1  u^. = - 2 ( 2 a - c ) and f(  Z l  ) = 4h ( 1  2 z  ) =  sech (2z )+(4c(p^2)+3)cosech (2z ). 2  2  1  I t i s seen t h a t f (  Z ; L  1  ) = 0(| | Z l  ) as | | - », a > 0. Z l  As b e f o r e , the r e a s o n f o r m o d i f y i n g the l a r g e parameter make, f (  Z l  solutions  ) = 0(|  Z l  |~  1 _ a  ) as | | - », a > 0, such t h a t t h e Z l  Y ( z ) and Y ( 1  1  i s to  2  Z l  ) o f equations- (8) c o r r e s p o n d i n g t o  z = z-j^ have a s y m p t o t i c e x p a n s i o n s v a l i d as z^ tends t o i n f i n i t y .  3T.  T h i s a l s o makes i t t h e r e f o r e p o s s i b l e t o compare r e l a t i o n s ( 1 1 ) and  (12) near t = l and u s i n g w e l l known e s t i m a t e s f o r the modi-  f i e d B e s s e l f u n c t i o n s as | z ^ | t e n d s t o i n f i n i t y , t h e c o n s t a n t s c ^ ( u ) and c ( u ) can be o b t a i n e d . 1  2  1  To f i n d these c o n s t a n t s , we f i r s t have t o o b t a i n t h e function B ^ ^ ^ . B  ol/ l) = (V2)J z  B  Z l  i( i) z  0  l  s  n  o  g i v e n by the  w  integral  f(s)ds =  6 ( l / 4 ) t a n h ( 2 ) - (c(c-2) + 3/4) c o t h ( 2 ) + C Z ; L  6 > 0 and f i x e d .  Z l  1 6  ,  L e t t i n g t - 1~ i n r e l a t i o n (10) and k e e p i n g  u-^ f i x e d we see t h a t c ( u ) = 0, 2  1  1/2  V l> z  ^("IKTJ! )  ~ e( l- ) l(7ru )- / c (u ) u  2  z  1  2  1  1  1  as \ \ - . , Z l  I _< 37r/2 - e, e > 0, u n i f o r m l y i n a r g z^; s i n c e  | arg  is  a l a r g e parameter and B ^ ( z ) remains bounded as \z-^\ - « . Q  Prom the e s t i m a t e s  1  (12) we now r e a d i l y g e t  c ( u [ a ] ) = (r(c)r(c-2a)/(r(c-a)) )(2(c-2a)7r) / . 2  1  1  2  1  The  constant  c ( u [ a ] ) i s w e l l d e f i n e d f o r Re c > 0 and 2  1  R e ( c - 2 a ) > 0, s i n c e t h e i n v e r s e o f t h e gamma f u n c t i o n i s e n t i r e . Because 2  F  l (  a,a c t) ;  ;  = #2)V2t-=/2 dz-^  ( 1  .  t )  (c-2 -l)/2Y 8  we t h e n have d e r i v e d t h e f o l l o w i n g - r e s u l t .  ( !  )C  (^)  32.  For a and c r e a l , c > 0, c-2a > 0 and t h e t - l n t e r v a l 0 < T (15)  < t < 1  1  ^ ( a ^ c j t ) [Y  x  z  V 2  2  - ( l z  / 2 u  (  = A(a)t(-  u  i  z  i  )  [  i  l ) 3/2( l l 1  I  U  Z  c  +  ^/ (l-t)( 2  +  °( i  ) [ B  ol( l)  u  C  2  a  )  /  2  x  2 ) ]  z  z /(l+iz l)0(u- )]|  +  2  1  1  u n i f o r m l y w i t h r e s p e c t t o t as u^ -» where  u.^ = ( l / 2 ) t a n h ~ , / i , 1  A(a) = B  o l  (  Z l  -  (r(c)r(c-2a)/(r(c-a)) )(2u 7r) / , 2  1  2  1  ) = (l/4')t  1 / 2  -(c(c-2)+3A)t-  1 / 2  +C  T  ,  T-j^ > 0 and f i x e d . C o m p l e t e l y a n a l o g o u s l y t o t h e f o r e g o i n g complete  asymp-  t o t i c e x p a n s i o n s can be o b t a i n e d f o r 2F^(a,a+l-cjl;t )  and  _ 1  2F^(a,a;c;t) u s i n g t h e complete instead  as - a -• » ,  a s y m p t o t i c e x p a n s i o n s o f Y-^(z) and Y2(z)  of t h e i r asymptotic  forms.  • By e m p l o y i n g t h e a l t e r n a t i v e forms o f t h e h y p e r g e o m e t r i c function  and t h e e x p r e s s i o n s f o r t h e a n a l y t i c c o n t i n u a t i o n o f  the hypergeometric f u n c t i o n ,  i t i s possible  t o deduce v a r i o u s  other asymptotic expansions f o r the hypergeometric For i n s t a n c e , f o r 0 < t < 1 F (a,a;c;t) =( l - t ) ( " c  2  1  so we have t h e f o l l o w i n g  2 a  ^ F (c-a,c-ajc;t);  result.  2  a  function.  For 0 < T (16)  2  F (c-a,c-a;c;t)-A(a)t(-  -  1  /  (z  uniformly Q l  c + 1  1  x |z  B  c-2a  > 0 and  the t - i n t e r v a l  1  < t <  x  c r e a l , c > 0,  a and  I  2  1 / 2  1 / 2  (u  u- )l  l Z l  1  V 2  2  1  2 a  - )/ c  x  2  2  (u z )[B 1  o l  (z )  + z /(l+|z |) 0(u~ )]j 2  1  w i t h r e s p e c t t o t as  ( z ) are g i v e n by  2  0(uj )]  )[l +  1  / )/ (l-t)(  1  1  •* » , where u^,z^,A(a) and  (15).  As an a p p l i c a t i o n of the a s y m p t o t i c f o r m (16) c o n s i d e r the  complex z-plane and  i s h o l o m o r p h i c i n |z| < R and  l e t us  suppose t h a t F ( z ) = U ( z ) + i V ( z )  c o n t i n u o u s i n |z| _< R.  F u r t h e r m o r e , f o r z = r e " ^ l e t us denote Max  F(re  1  1 - e  ) by  0<Q<2ir  M ( r ; F ) , r _< R.  Then Cauchy's f o r m u l a f o r the n t h  order  d e r i v a t i v e of F ( z ) s t a t e s  that  P r o v i d e d F ( z ) ^ M(R;F) we  t h e n have from the Maximum modulus  p r i n c i p l e the (18) But  inequality  | F ( z ) | < %Ln  for  M(R;P) J  -M} | t - z n+1  |z|=R  I z j = r < R the i n t e g r a l  I 2T?  worked out We For  [11]  therefore  and  i s found t o be  J  jz|=R  o < T  x  _< ( £ )  2  I  dt  I ..,  |t-z|  can  be  n + 1  F ( ^ ( n + l ) , ^ ( n + l ) ; l ; (-yr) ). 2  2  ] L  have the f o l l o w i n g r e s u l t .  j z | = r < R and  , r < R.  < 1 ,  34,  (19)  | F ( z ) | < n<  M(R;F) F (^(n+l)^(n+l);l;(|) ),  n  here  2  2  1  F ( ^ ( n + l ) ^ ( n + l ) ; 1 ; (-|) ) f o r l a r g e n i s g i v e n by (16), 2  2  1  where u  = 2w,  x  c z  =  1  = (l/2)tanh (|)  ,  _ 1  x  A[a(n)] B  0 l  (  =r(n)(r(-|(n+l)))- ( 4 7 m ) 2  ) = (l/4)(rR" + Rr" )+ C 1  Z l  T  1  T  1 / 2  and  ,  > 0 and f i x e d .  1  A s i m p l i f i c a t i o n o f n.'A(n) i s o b t a i n e d  by u s i n g t h e d u p l i c a t i o n  f o r m u l a o f t h e gamma f u n c t i o n  (20) T T  1 / 2  (2n) = 2  T(n) T(n+l/2).  2 n _ 1  I t t h e n i s found t h a t (21)  n.»A[a(n)] = 2  2 n + 1  ( F ( ^ + 1) ) ( n 7 r ) " / . 2  1  2  I f we now expand t h e gamma f u n c t i o n i n a S t i r l i n g s e r i e s , we get f o r n l a r g e (22)  (23)  n»A[a(n)] ~ 2 ( 2 n ) ' e " ( m r ) n  A[a(n)]  ^2(  n+1  n  or  1 / 2  / ). 2  T h e r e f o r e , i n view o f t h e a p p r o x i m a t i o n f o r m u l a e o f m o d i f i e d B e s s e l f u n c t i o n s - f o r l a r g e v a l u e s o f t h e v a r i a b l e we can r e s t a t e r e l a t i o n (19) as f o l l o w s : for  | z | = r < R and o < T _< ( r R x  - 1  )  2  < 1 ,  35.  (24)  | F ( z ) | < nJ M(RjP) p F ^ n + l ) y | ( n + l ) ; l ^ ) ) , n  2  here f o r n l a r g e n J ^ ^ n + D ^ C n + l ; ! ; ^ ) ) ~ 2(2n) e- (n7r) / 2  n  x (rR- )- / (l-(rR- ) )- / 1  1  2  1  = ^(2n) e" (rR' )" n  since z z  and  1 / / 2  I-  =  1  L /  n  1  2  1 / 2  n  n  zJ/ I  2  2  (l-(rR" ) ) 1  2  1 / 2  2  1  _ n / 2  1  sinh(n 1  1  x  2  (2nz )  , (2nz ) = (sinh(2nz )) (2nir)" /  tanh" (rR- )), 1  1  2  (l/2)tanh" (rR ). 1  - 1  A f u r t h e r s i m p l i f i c a t i o n i s obtained tanh" (rR" ) = \ l n ( ) , l-rR" 1  1  1  1  +  r  R  1  rR  - 1  by n o t i n g  that  < 1, and t h e r e f o r e f o r r R < _ 1  x  s i n h ( n t a n h - ^ r R " ) ) = ±+I*±)*/ 1-rR 1  2  i n the estimate 1 0  <  T  (25)  i S. (  r R  )  . ( l ^ n/2 l+rR-1  2  {  (24) f o r n l a r g e ,  g  o  t  h  a  t  | z | = r < R and  2  < 1 we now have  nJ F (-^(n+l)^(n+l);l;(|) ) ~ 2  2  1  (l/2)./2 ( 2 n ) e - ( r " R ) n  n  1  1 / 2  ((l-rR" )" 1  n  - (1 + r R " ) ' ) .  asymptotic form ( 2 5 ) y i e l d s a b e t t e r estimate  The  ^  1  | F ( z ) | _< 2nJ R(R-r ) " n  n _ 1  M(R;F),  1  then  v a l i d f o r |z|=r, 0 < r < R.  T h i s r e l a t i o n f o l l o w s r e a d i l y from ( l 8 ) upon n o t i n g (R-r)"" "" n  1  1 1  i s an upper bound f o r t h e i n t e g r a n d .  that  36.  REFERENCES  C h e r r y , T.M.,  Uniform asymptotic formulae f o r f u n c t i o n s  w i t h t r a n s i t i o n p o i n t s . Trans. Amer. Math. Soc. 6 8 , 1950,  224-257.  C h e r r y , T.M.,  Asymptotic expansion f o r the hypergeometric  f u n c t i o n s o c c u r i n g i n g a s - f l o w t h e o r y . P r o c . Roy. Soc. London, Ser. A 2 0 2 , 1 9 5 0 ,  507-522.  Coddington, E.A. and L e v i n s o n N., Theory o f 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 . M c G r a w - H i l l Book Co., New York,  1955-  Dettman, J . , The s o l u t i o n o f a second o r d e r l i n e a r  dif-  f e r e n t i a l e q u a t i o n near a r e g u l a r s i n g u l a r p o i n t . The Amer. Math. M o n t h l y , 71 No. 4 , A p r i l 1 9 6 4 , 378-385.  E r d e l y i , A. e t a l , H i g h e r t r a n s c e n d e n t a l f u n c t i o n s , v o l . 1,  M c G r a w - H i l l Book Co., New York,  1953-  E r d e ' l y i , A., A s y m p t o t i c e x p a n s i o n s , Dover P u b l i c a t i o n s , New Y o r k ,  1956.  E r d e l y i , A., A s y m p t o t i c s o l u t i o n s o f d i f f e r e n t i a l e q u a t i o n s w i t h t r a n s i t i o n p o i n t s or s i n g u l a r i t i e s . Journal of Mathematical P h y s i c s , 1960a,  16-26.  E r d e l y i , A., A s y m p t o t i c s o l u t i o n s o f o r d i n a r y 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 s . C a l i f o r n i a I n s t i t u t e of Technology, Hapaev, M.M.,  1961.  E x p a n s i o n s o f h y p e r g e o m e t r i c and degenerate  hypergeometric f u n c t i o n s i n s e r i e s of B e s s e l f u n c t i o n s . V e s t n i k Moskov. Univ. Ser. Mat. Meh. A s t r . F i z . Him., 1958,  No. 5 , 1 7 - 2 2 .  (Russian)  I  [10]  Hapaev, M.M.,  37-  A s y m p t o t i c e x p a n s i o n s o f h y p e r g e o m e t r i c and  confluent hypergeometric  functions.  I z v . Vyss. Ucebn. Zaved. M a t i m a t i k a , 1961,  No.5 ( 2 4 ) ,  98-101. ( R u s s i a n ) [11]  H i l l e , E., A n a l y t i c f u n c t i o n t h e o r y . V o l . I I . Ginn and Co., 1962.  [12]  Hobson, E.W.,  S p h e r i c a l and e l l i p s o i d a l  harmonics.  Cambridge U n i v e r s i t y P r e s s , 1931. [13]  H o c h s t a d t , H., D i f f e r e n t i a l e q u a t i o n s , a modern approach. H o l t , R h i n e h o r s t and W i s t o n , New York, 1963.  [14]  I n c e , E.L., 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 . P u b l i c a t i o n s , New York,  [15]  Dover  1956.  K h o t t n e r u s , U. J . , A p p r o x i m a t i o n f o r m u l a e f o r g e n e r a l i z e d hypergeometric f u n c t i o n s f o r l a r g e values of the parameters. J.B. W a l t e r s , Groningen, t h e N e t h e r l a n d s ,  i960. [16]  Kummer, E.E., Uber d i e h y p e r g e o m e t r i s c h e  Reihe.  J. R e i n e Angew, Math. 15, 39-83, 1836, 127 -172. [17]  L i g h t h i l l , M.J., The hodograph t r a n s f o r m a t i o n i n trans-sonic flow I I .  A u x i l i a r y theorems on t h e  h y p e r g e o m e t r i c f u n c t i o n s Y (Y)» n  London, Ser. A, 191, [l8]  [19]  1947,  P r o c . Roy. Soc.  341-351.  MacRobert, T.M., F u n c t i o n s of a complex v a r i a b l e . 4th ed., M a c M i l l a n and Co. L t d . , London, England, O l v e r , F.W.J., The a s y m p t o t i c s o l u t i o n o f l i n e a r  differen-  t i a l e q u a t i o n s o f t h e second o r d e r i n a domain c o n t a i n i n g one t r a n s i t i o n p o i n t . P h i l . Trans. Roy.  S o c , A 249, 1956/57, 65-97.  1954.  38 • [20]  O l v e r , F.W.J., U n i f o r m a s y m p t o t i c e x p a n s i o n s o f s o l u t i o n s of second o r d e r d i f f e r e n t i a l e q u a t i o n s f o r l a r g e v a l u e s o f a parameter.  P h i l . Trans. Roy. Soc.,  A 250, 1958, 479-517. [21]  P o o l e , E. G. C. , I n t r o d u c t i o n t o t h e h i s t o r y o f l i n e a r d i f f e r e n t i a l equations. C l a r i d o n P r e s s , Oxford, England,  [22]  1936.  R a i n v i l l e , E.D., S p e c i a l f u n c t i o n s . The M a c M i l l a n Co.,  New York, i960. [23]  S e i f e r t , H., D i e h y p e r g e o m e t r i s c h e n g l e i c h u n g e n d e r Gasdynamik.  Differential-  Math. Ann. 120,  1947,  75-126. [24]  S l a t e r , L . J . , Confluent hypergeometric f u n c t i o n s , Cambridge U n i v e r s i t y P r e s s , i960.  [25]  Snow, C., The h y p e r g e o m e t r i c and Legendre  functions  w i t h a p p l i c a t i o n s t o i n t e g r a l equations of p o t e n t i a l  [26]  t h e o r y . A p p l i e d S e r i e s 19,  U. S. Government P r i n t i n g  O f f i c e , Washington,  1952.  D.C.,  Sommerfeld, A., Atombau und S p e k t r a l l i n i e n , 2nd V o l . , F r i e d e r i c h Vieweg and Sohn, B r u n s w i c k ,  1939,  800-806. [27]  Thorne, R.C., The a s y m p t o t i c s o l u t i o n o f l i n e a r second o r d e r d i f f e r e n t i a l e q u a t i o n s i r a domain c o n t a i n i n g a t u r n i n g p o i n t and r e g u l a r s i n g u l a r i t y . Techn. R e p o r t 12, Dept. Math.,Cal. I n s t , o f Techn., 1956.  [28]  Thorne, R.C., The a s y m p t o t i c e x p a n s i o n o f Legendre f u n c t i o n s o f l a r g e degree and o r d e r . 13, Dept. Math., C a l . I n s t , o f Techn.,  [29]  Watson, G.N., Trans. Cambridge P h i l .  277-308.  Techn. R e p o r t 1956.  Soc. 22,  1918,  39.  [30]  Watson, G.N.,  Theory o f B e s s e l f u n c t i o n s . Cambridge  University Press, [31]  1944.  W h i t t a k e r , E.T. and Watson, G.N., A course o f modern a n a l y s i s , 4 t h ed. Cambridge U n i v e r s i t y P r e s s , 1952.  

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