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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 In the Department of Mathematics We accept t h i s t h e s i s as conforming t o the re q u i r e d 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 ex-tensive 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 finan-cial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver 8, Canada Date ^ £ k i i ABSTRACT In chapter I known asymptotic forms and expansions of the hypergeometric f u n c t i o n obtained by Erde'lyi [ 5 ] , Hapaev [ 1 0 , 1 1 ] , Knottnerus [ 1 5 L Sommerfeld [ 2 5 ] and Watson [ 2 8 ] are discussed. Also the asymptotic expansions of the hypergeometric f u n c t i o n o c c u r r i n g i n gas-flow theory w i l l be discussed. These expansions were obtained by Cherry [ 1 , 2 ] , L i g h t h i l l [ 1 7 ] and S e i f e r t [ 2 J ] . Moreover, u s i n g a paper by Thorne [ 2 8 ] asymptotic expansions of 2 F 1 ( p + l , -p; 1-m; ( l - t ) / 2 ) , - 1 < t < 1 , and 2 P 1 ( (p+m+2)/2, (p+m+l)/2; p+ 3/2-, t " ), t > 1 , are obtained as p-»» and m = -(p+ l / 2 ) a , where a i s f i x e d and 0 < a < 1 . The : expansions are i n terms of A i r y f u n c t i o n s of the f i r s t k ind. The hypergeometric equation i s normalized i n chapter I I . I t r e a d i l y y i e l d s the two t u r n i n g p o i n t s t ^ , i = 1 , 2 . I f we consider,the case the a=b i s a larg e r e a l parameter of the hypergeometric f u n c t i o n 2F- L(a,b; c; t ) , then the t u r n i n g p o i n t s coalesce w i t h the r e g u l a r s i n g u l a r i t i e s t = 0 and t = <*> of the hypergeometric equation as j a | ». In chapter I I I new asymptotic forms are found f o r t h i s p a r t i c u l a r case; t h a t i s , f o r 2 ^ (a, a] c ; t ) , 0 < T-^  _< t < 1 , and 2 F 1 ( a , a + l - c ; 1 ; t - 1 ) , 1 < t _< Tg < » , as -a^» . The asymptotic form i s i n terms of modified B e s s e l f u n c t i o n s of order 1 / 2 . Asymptotic expansions can be obtained i n a s i m i l a r manner. i i i Furthermore, a new asymptotic form i s derived for 2 F 1 ( c - a , c-a; c; t ) , 0 < <_ t < 1, as -a-»«, t h i s r e s u l t then leads to a sharper estimate on the modulus of n-th order derivatives of holomorphic functions as n becomes large. i v TABLE OP CONTENTS I 1. INTRODUCTION 1 2. RELEVANT PROPERTIES OF 2 F 1 ( a , b ? c ; t ) 5 5. KNOWN ASYMPTOTIC RESULTS 8 a. R e s u l t s obtained by Watson 8 b. R e s u l t s obtained by E r d e l y i 9 c. R e s u l t s obtained by Hapaev 10 d. R e s u l t s obtained by Khottnerus 11 e. R e s u l t s obtained by Cherry; and Sommerf e l d 12 f. R e s u l t s d e r i v e d from Thome's paper 15 I I THE NORMALIZED HYPERGEOMETRIC DIFFERENTIAL EQUATION 20 I I I THE ASYMPTOTIC BEHAVIOUR OF ^ ( a ^ a j c j t ) 22 and g F ^ a j a + l - c j l j t " 1 ) 22 IV REFERENCES Z>6 ACKNOWLEDGEMENT I wish t o acknowledge the I n v a l u a b l e guidance and as s i s t a n c e extended t o me by 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 the f i n a l manuscript. The generous f i n a n c i a l support of the Nat i o n a l Research Council and the 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 hypergeometric f u n c t i o n ^F^a,,}}', c; t ) has been i n v e s t i g a t e d by numerous authors, see f o r instance the references [ 5 , 14, 21, 22, 25, 5 1 ] . I t depends on the three parameters a,b and c and the v a r i a b l e t . The 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 Eul e r [ 5 ] have s u c c e s s f u l l y been em-ployed by Knottnerus [ 1 5 ] , S e i f e r t [ 2 3 ] and Sommerfeld [26] to d e r i v e asymptotic expansions as one or s e v e r a l of the para-meters tend t o OB. Both S e i f e r t and Sommerfeld used the method of steepest descents [ 6 ] on a s l i g h t l y modified form of the Euler i n t e g r a l r e p r e s e n t a t i o n . The expansions are v a l i d i n a t - i n t e r v a l which does not contain a t u r n i n g p o i n t [ 6 ] , Asymptotic expansions of the hypergeometric f u n c t i o n o c c u r i n g i n gas-flow theory on the other hand are v a l i d i n regions 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 which i n t h i s case corresponds t o the t r a n s i t i o n p o i n t of sub-sonic flow t o supersonic flow. The method used by the authors Cherry [ 1 , 2 ] , 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 the asymptotic expansion i s described i n f o r i n s t a n c e , the references [ 7 , » , 1 9 , 2 0 , 2 7 ] . This method concerns the asymptotic s o l u -t i o n s of d i f f e r e n t i a l equations w i t h f i x e d t u r n i n g points, or 2. s i n g u l a r i t i e s . Some authors, notably E r d e l y i [ 5 ] Hapaev [9, 10] and MacRobert [ l 8 ] used w e l l known expansions of the confluent hypergeometric f u n c t i o n ^F^(a; b; t ) [24, 31] t o derive asymp-t 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 expansions as p -• « are derived f o r g F ^ p + l , -p; 1-m; ( l - t ) / 2 ) , -1 < t < 1, and 2F 1((p+m+2)/2, (p+m+l)/2; p+3/2; t " 2 ) , t > 1 where m = -(p + 1/2)a, a f i x e d and 0 < a < 1. The expansions are derived from a paper by Thorne [28] on the asymptotic expansions of the ass 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 order. The expansions are i n terms of A i r y f u n c t i o n s of the f i r s t k i n d [ 5 ] . In t h i s chapter a d e s c r i p t i o n of the known asymptotic expansions f o r gF^(a,b; c; t ) w i l l be given. The expansions are v a l i d as one or s e v e r a l of the para-meters approaches », under v a r i o u s r e s t r i c t i o n s on the (other) parameters and t . The f o l l o w i n g t a b l e summarizes the r e s u l t s . 3. Author l a r g e Date parameters other r e s t r i c t i o n s G. N. Watson 1918 c a,b f i x e d , |arg c j_< 7r-e,e > 0 , | t | < 1. G.N.Watson 1918 a,b, c 1) (a+b)fixed, c f i x e d , l - 2 t + 2 ( t 2 - t ) 1 / 2 = e± p 2) (a-b),(a+b-c) f i x e d [ t - 2 + 2 ( ( l - t ) t " 1 ) 1 / 2 ] t " 1 = e± p T.M.MacRobert 1923 c 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=iv, 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 ) " 1 , a b = | c ( l - c ) ( Y - i r 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 arg 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 , | arg 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 < j t | < 1 R.C.Thorne 1956 a,b, c a,b l i n e a r In c,|t|<l and t > l . M. M. Hapaev 1958 a b,c f i x e d | t | ~ r ^ a | - 1 U. J. Khottnerus 19b0 c a,b f i x e d , t-plane cut from 1 t o 1 + ioe . 4. (continued) Author Date l a r g e parameters other r e s t r i c t i o n s U. J. Khottnerus 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 r e a l , r » . M.M. Hapaev l y o l a, c b, a c " 1 f i x e d , j t j < j c a " 1 j . 2. RELEVANT PROPERTIES OF ^ ( a ^ b ; c, t) The hypergeometric equation (1) t ( l - t ) ^ - | + (c - ( a + b + l ) t ) - ^ - abx = 0 dt has three 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 the s i n g u l a r i t y t = 0 i s developable i n a s e r i e s (2) 1 + (ab/c)t + ( a ( a + l ) ( b ( b + l ) ) / c ( c + l ) 2 j ) t 2 + ... 00 = I ( ( a J n ^ n / ( c ) n n - ' ) t n , c t 0, -1, -2, n=0 where ( a ) n i s the Pochhammer symbol and ( a ) n = r( a + n)/ P(a) = a(a+l)...(a+n-l), n = 1,2,3, [5]. The s e r i e s ( 2 ) , due t o Gauss, i s denoted by 2F 1( a' b» c* The s e r i e s converges f o r j t j < 1 and f o r t = 1 i t diverges 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 the p o i n t t = l i s excluded -{14]. I f c = -n, n = 0,1,2,..., then a s o l u t i o n of equation ( l ) which i s r e g u l a r at t=0 i s t n + 1 Y( (a+n+1 ) m(b+n+l ) m / ( n + 2 ) m ml ) t m m=0 (3) = t n + 1 ^ ( a + n + l , b+n+1 ; n+2 ; t ) . Also i f a = -n or b = -n, where n = 0,1,2,... and i f c = -m, where rn = n, n+1,... , then we define [5] 6. n (4) ^ ( - n . b ; -m S t ) ( ( - n ) r ( b ) r / ( - m ) p r ) t r ; r=0 s i m i l a r l y f o r 2F 1 ( a,-n; -mj t ) . Since (3) and (4) are s o l u t i o n s of e q u a t i o n (1) we see that the hypergeometric equation has a s o l u t i o n which i s a polynomial i n t whenever -a or -b i s a nonnegative i n t e g e r . The property (5 ) 2 F l ( a ' b > c ; X ) = f ( c ) r ( c - a - b ) /V(c-&)V(cr\>)) w i l l be used l a t e r . I t i s assumed th a t c, c-a-b, c-a and c-b are not negative i n t e g e r s [ 14J . There a re many r e s u l t s : r e l a t i n g the hypergeometric f u n c t i o n and the 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 of the 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 are i n terms o f contour 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 values o f m and p. For the t-plane cut from 1 t o -» and | l - t | < z. (6) Pp(t) = ( r ( l - m ) ) " 1 ( ( t - D / ( t + l ) ) - m / 2 ^ ( p + l r p j l - m j M / S ) and f o r | t | > 1 (7) e - m 7 r i ^ ( t ) = 2 P ( r ( p + l ) r ( P + m + l ) / r ( 2 p + 2 ) ) ( ( t 2 - l ) m / V P - m - 1 ) x 2F 1((p+m+2)/2,(p+m+l)/2; p+3/2; t " 2 ) For a l l t we have the c o n t i n u a t i o n formulae, which a l s o can be found i n Hobson [12], 7. («)• P j ( - t ) = e ^ F ^ t ) - f sih((p+m)ir)<£(t) and (9) Q£(-t) = - e ^ 1 (£(t), where the upper and lower signs are taken according as Im t > 0 or Im t < 0, 8. 3. KNOWN ASYMPTOTIC RESULTS a. R e s u l t s obtained by Watson. Watson [ 2 9 ] i n v e s t i g a t e d the behaviour of 2 F 1 ( a , b ; c;. t ) f o r l a r g e values of j a j , jb| and |c|. I f a, b and t are f i x e d and |c| i s l a r g e w i t h the r e s t r i c t i o n |arg c| _< w-e, e > 0 , then f o r | t | < 1 (10) pF-^a^b; c; t ) = l+(ab/c)t + . .. + ( ( a ) n ( b ) n / ( c ) n n i ) t n + 0 ( | c"1 With a s l i g h t l y modified expression f o r the remainder term E r d e l y i [ 5 ] proved t h i s remains v a l i d even i f | t | > 1, | a r g ( l - t ) | < 7r, provided that Re c - ». MacRobert [ 1 8 ] proved (10) f o r a range of argument c which i s l a r g e r than TT and v a l i d f o r a l l t provided that |arg ( l - t ) | < ir. In the case where more than one of the parameters approaches » Watson [29] d e r i v e d the f o l l o w i n g r e s u l t s . I f we define ? by t + ( t 2 - l ) 1 / / 2 = and put 1 - e = ( e b - l ) e where the upper or lower s i g n i s taken according as Im t >< 0 , then f o r l a r g e |\| (11) ( l / 2 t - l / 2 ) " a " x 2 F 1 ( a + X , a-c+l+X; a-b+l+2\; 2 ( l - t ) - 1 ] = 2 a + b ( r ( a - b + l + 2 X ) r ( l / 2 ) X - 1 / 2 ^ x (1 + e - ? ) 0 - ^ " 1 / ^ ! + 0 ( X - 1 ) L where |arg x | _< v-b, 6 > 0 and a l s o 9. (12) ^ ( a + X , b-X; C; - j j t ) = 2 a + b - 1 ( r(i-b - r x ) r(c ) / r ^ ) r( C-b + x))(i-e^ ) - c 4 x v ( l + e ^ ) c - a - b - \ x-'-2[e( X- b>* + e i i 7 r ( c " i } .x. v e - ( X + a ) * ] [ l + O d X " 1 ! ) ] , where theLupper or lower s i g n i s taken according as Im t >< 0 and where | x | i s l a r g e , ?. = p + i n , - ^TT - w 2 + 6 < argX < -^ TT + w 1 - 6 6 > 0 w 2 = t a n " 1 ( h / p ) , -w-j^  = t a n " 1 [ (h - TT)/ p ] n > 0 w 2 = t a n " 1 [ ( h + 7r ) / p ] , -w1 •= t a n _ 1 ( n / p ) n j< 0 and t a n _ 1 x denotes the p r i n c i p a l value of the f u n c t i o n . b > R e s u l t s obtained by Erd'elyi E r d e l y i derived the f o l l o w i n g expansion f o r |b| l a r g e [ 5 ] . I f a,c and t are f i x e d , c ^  0,-1,-2,... and 0 < | t | < 1, and i f |b| - * such that - < arg tb < , then (13) g F ^ b ; c j ' t ) - g F - ^ b ; c; on =[)_' ( ( a ) n / ( c ) n n . ' ) ( b t ) n ] [ l + O d b l " 1 ) ] n=0 and here the asymptotic formulas f o r the confluent hyper-geometric f u n c t i o n of a l a r g e argument [31] give 10. (14) p F - ^ b - c; t ) = e - i 7 r a = e - i 7 r a [ r ( c ) / r ( c - a ) ] ( b t ) - a [ i - . - + o c i b t r 1 ) ] + [ r ( c ) / n a ) ] e b t ( b t ) a - c [ i + o c i b t r 1 ) ] 1 '• "3 and s i m i l a r l y , i f - ^ir < arg bt < ^TT , we have (15) 2 F i ( a ' b ; c> *) - e i 7 r a [ r(c ) / r(c-a)](bt)- a[l + O C l b t l " 1 ) ] + [ r(c ) / r(a)]e b t ( b t) a- c[i + o d b t r 1 ) ] . c. R e s u l t s obtained by Hapaev We s h a l l now r e f e r t o some papers by Hapaev [9> 10]. He derived an asymptotic expansion i n terms of modified B e s s e l f u n c t i o n s of l a r g e order f o r the confluent hypergeometric f u n c t i o n 1 F 1 ( a ; c j t ) a s a -• », where | t | ~ l a f 1 and c i s f i x e d . Provided these c o n d i t i o n s are s a t i s f i e d , he then derived [10] es (16) ^ ( a ; -c; t ) = ( c ) ( a t ^ " c ) / 2 JitaT1)^2 » ra=0 n=0 1 ( m ), ^ where B m n = - , f (x,n) mn my Ix| < 1. and f ( x , n ) - (l+(|)x+(|)x d+.'.. ) n , x=0 11. He then used t h i s r e s u l t t o de r i v e (17) 2 F 1 ( a , b ; c ; t ) = es (HcJ/Ra) y B m n ( t 2 n + m hn/ 2 n nJ ) (r(a+in+2n-l)/r(c+nH-.2n-l)) n,m=D 1F 1(a-l+m+2nj c+m+2n-l;2t), where the expansion 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 of h i s papers [11] he obtained asymptotic expansions of 2 F 1 ( a , b ; c ; t ) and 1 F 1 ( a ; c ; t ) where a=ae, C=Y£, a and y are constants and l -* eo, that 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 ( a / Y ) t ( l + r 1 ( ( a Y " 1 t ) 2 ( a - 1 - Y ": L) 2 " 1 K ^ - 2 ( . . . ) + . . } provided that | k t _ 1 + Y l > |fcfcT11, where k > 2n,n = 0,1,2, S i m i l a r l y , f o r I k t ^ + Y l > I fct " 1 ! and | a Y " 1 t | < 1 he d e r i v e s (19) ^ ( a ^ c j t ) ~ ( l - a Y " 1 t ) " b { l + t "1( 2 - 1 b ( b + l ) ( Y " 1 - o " 1 ) ( l - 2 t ) ( l - t ) " 2 ) + *~ 2(...)+...}. d. R e s u l t s obtained by Khottnerus Next we s h a l l refer, t o r e s u l t s d e r i v e d by Khottnerus [15]/ For l a r g e p o s i t i v e values of r and the t-plane cut from 1 t o 1+eoi he deriv e d the f o l l o w i n g r e l a t i o n (20) g F ^ a j b j c + r j t ) = 1 + ( a b t ) r " 1 + 0 ( r 2 ) . 12. The expansion i s v a l i d u n i f o r m l y i n the closed bounded reg i o n G: f R e ( 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. Using the w e l l known r e l a t i o n [ 5 ] (22) g F ^ a j b j c j t ) = ( l - t ) c " a " b 2 F 1 ( c - a , c - b ; c ; t ) v a l i d f o r t i n the t-plane cut from t = l to t = l + o o i 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 Re(t) > 1, Im(t) _> ° J then 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) ^ ( a + r ^ + r ^ c + r j t ) = ( 1 - t ) c - a " b - r { 1 + (c-*) + 0 ( r - 2 ) } The expansion i s v a l i d u niformly provided that t l i e s i n a closed bounded r e g i o n of the t-plane cut from 1 to l+»i. e. Results obtained by Cherry and Sommerfeld F i r s t we s h a l l consider the hypergeometric equation-occuring i n gas-flow theory w § + <? - + v 2 < w ^ 7 - t ? ) y ' °' I t i s s a t i s f i e d by X ( t ) = t ^ g F - ^ v - a ^ v-b^; v+1;' t ) , _ n— where 2 a v = v + 6 + /^v (1+26) + 6 , 28 v = v + 6 - y v 2 ( l + 2 6 ) .+ 6 2 , 13. and p = - ( Y - 1 ) " 1 ; Y > 1, i s the f i x e d a d i a b a t i c index. The t u r n i n g p o i n t of equation (24) i s r e a d i l y given by t = ( l + 2 p ) _ 1 . s The comparison equation 2 ' • / nr'\ d Z . l d Z . 2/n I N (25) — 2 + - - j - + v (1 - - 2 ) z = 0 dT T i s s a t i s f i e d by the Bessel f u n c t i o n J , ( V T ) , where T = (1-^—) (1-t )_1. s Then f o r v ~ « w i t h |arg v| _< v - e and 0 _< t _< 1-e, e > 0, Cherry [1,2] derived (26) Xv(t) ~ N ( v / h v ) 1 / 2 [ J v ( v T . ) ( 1 + q 2 V - 2 + ^ v - 4 + _ } + T j ^ ( v T ) ( q 1 v " 1 + q^v"5+ . . . ) ] , where t a n h " 1 ((1 - J 2 ) 1 / 2 ) - ( 1 - T 2 ) 1 / 2 = ta n h " 1 ( ( l - t - ) ( i - t ) 1 / 2 - t^tanh -^ t^t^ - t r 1 ) 2 / 2 s N = (1-t) 1/ 4 - l p ( ( l - T 2 ) / ( i - | - ) ) 1 / 4 and ' s h v = ( r ( a v ) r ( l + v - b v ) ) / ( r ( a v - v ) r ( v ) r ( v + l ) ) ; the expansion i s v a l i d u n iformly i n t and arg v. To c a l c u l a t e the q n , n = 1,2,..., we set w- ( ( l - | - ) V V ( l - t ) 1 / * - l / 2 e ) x , ( t ) , W = (1 - T 2 ) Z , and then u - t a r f f 1 ( ( 1 , - T 2 ) 1 / 2 ) - ( I - T ^ w = W(l + Q 2v~ 2 + Q^v"'4 + . .. ) - - ^ ( Q T V ' " 2 + Q 5v"^ + . .. ) where Q X = q-^(l - T 2 ) 1 / 2 Q2 = q 2 + T 2 q x / ( 2 ( 1 - T 2 ) ) , 14. and % = % + T 2 q 3 /(2(1 - T 2 ) ) , . . . 2 d Q l ' , x 2Qg = (<t> + Qx , 2 Q 3 = (<t> - §)Q2 + 2 $ ^ + + Q 2 , 2Q 4 = (<j> - §)Q 3 + Q 3 where|> = T 2 ( 1 + ( 1 / 4 ) T 2 ' ) / ( 1 - T 2 ) 3 , * = ( ( i - s 2 ) / 4 ( i - t B ) 2 ) { 5 ? " 6 - ( i + 6 t s ) r ^ ( > 4 t s ) F . - 2 + ( i - 2 t s x i - 4 t s ) ; ; - .1/2 S = T ' . I n t e g r a t i o n constants are found from the l i m i t i n g form of the X v ( t ) expansion (26) f o r t = T = 0, which then i s equal to (27) ( e v " 1 ) v r ( v + l ) ( ( 2 T r h v 6 2 v) / 2 T T V ) 1 / 2 ~ 1+Q 1 ( 0)v" 1+Q 2 ( 0)v" 2+. . ., where 6 = a a ( l + a ) _ 1 " a and a = 1/2 ( 1+2B) 1 / / 2 - 1/2 . We expand the l e f t hand member of (27) by means of ( 2 8 ) log(27rh v) ~ 2v l o g 6 + c-^v"1 + Q . j \ T \ . . . and 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,.... S i m i l a r r e s u l t s were obtained by L i g h t h i l l [ 1 7 ] and S e i f e r t [ 2 3 ] . ' The asymptotic behaviour of the hypergeometric f u n c t i o n g F ^ - i n , - i p w ; 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]. For t = -4p(1-p)" 2sin 2(a/2), 0 < p < 1, he obtained (29) -gF -^ivlpnajt) ~ e - 7 r p n ( 2 7 m a ) " 1 / 2 ( i u ; 1 + e 1 * ^ ) where uQ = ((1 - p)/2)(l+i c o t ( 3 / 2 ) ) , cot (a/2) £ 0, f(u)= i l o g ( u p ( l - u ) " p ( l - u t ) " 1 ) and - a = f " ( u 0 ) ; I f cot (a/2) - 0, then u Q = ( (1-p )/2) (l+i(7r-a)/2 + i(7r-a) 5 / 2 4 f(u 0)=27r p - i ( l + p ) l o g ( ( l + p ) / ( l - p ) ) + ( p / ( l + p ) ) i ( 7 r - a 2 / 2 -p(l-p)(Tr-a) 3(l+p ) - 2 /6 + ... -a = f " ( u 0 ) = 6p(Tr-a)(l-p)" 1(l+p)- 2+ ... f. R e s u l t s derived 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 derived from a paper w r i t t e n by Thorne [28]: (30) ( r ( l - m ) ) " 1 2 F 1 ( p + l , - p ; l - m ; ( l - t ) / 2 ) ~ {r(p+l+m)/np+l-m)} 1 / 2[(l-t)/(l+t)] m / 2 v x ( 4 2 / ( t 2 - P 2 ) ) V ^ ( M ( Y 2 ; ? z ) > l / 3 ^ E g ( z ) f 2 A „ s=0 s=0 as p - » , where m = -ya = -(p+l/2)a, a i s f i x e d and 0 < a < 1, p. = Jl-a 2 , 16. | z ( t ) 3 / 2 = a c o s h " 1 ( | ( t - 2 - l ) " 1 / 2 ) - c o s h " 1 t B " 1 and the f u n c t i o n s F (z) and E (z) are given by the r e l a t i o n s (40) s s and (43) determined l a t e r . The expansion i s v a l i d f o r the t - i n t e r v a l | t | < 1. Completely analogously to the forego i n g , the f o l l o w i n g r e s u l t was found f o r t > 1: (31) 2F1((m+p+2)/2, (m+p+l)/2; p+ J ; t " 2 ) ~ , 7 7 " . 1 " -51 7T2-P1 (2p+2)/((r(p+l)(n:p+m+l)r(p+l-m)) "5) e m 1 2 2 ± 1 t P + m + 1 ( i - t 2 y ^ ( 4 z / ( t2_ p2 ) )7 U l ( Y-3 e- 3 z ) Y " 3 X » _ 2 . 2 _ 2 . _ 5 « X }>s(z)Y"2s + e ' ^ A ^ Y 1 e" 3 7 r iz) Y" 1XF s(z)Y" 2 s } s=0 s=0 as p-*», where m,B,Y* and z are defined by (30); the f u n c t i o n s E (z) and F (z) are i d e n t i c a l t o those of the former case as w e l l , s v ' s v ' We s h a l l now set out t o prove the f i r s t r e s u l t . Instead of an asymptotic expansion f o r P^(t) as p » , an asymptotic expansion f o r the as s o c i a t e d Legendre f u n c t i o n of the second kind Q^(t) was used t o derive the second r e s u l t . Asymptotic expan-sions f o r P m ( t ) and <^(t), m = -(p+-|)a, as p->» can be obtained by a method employed by Thorne [28]. The a s s o c i a t e d Legendre equation dy dt (32) (1-t2) i \ -2t $ + (P(P+D - m 2 ( l - t 2 ) - 1 ) y = 0 dt has a fundamental system of s o l u t i o n s c o n s i s t i n g of P^(t) and 17. Q ^ t ) . I f we use Hobson's d e f i n i t i o n s of these f u n c t i o n s then they are s i n g l e valued a n a l y t i c i n the t-plane cut along the r e a l a x i s from 1 to -a>, and are r e a l when t i s r e a l and t > 1. For z = x, where - 1 < x < 1, the fundamental s o l u t i o n s of (32) are taken as P p ( x ) and Qp^x) defined by (33) Pp(x) = e~ * P^(x + i . o ) and m 7 T i  mt l (34) 2 e m 7 r iQ*(x) = e ^ Q ^ x + i . o ) + e ^  ^ ( x - i . o ) , where f ( x + i.o) =. l i m f ( x + i . e ) , e > 0 . e 0 The f u n c t i o n s Pp(x) and Qp(x) are 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 are r e a l f o r x r e a l and 0 < x < 1. Equation (32) i s normalized by the tr a n s f o r m a t i o n 2 1/2 (t - 1 ) / y = Y, the r e s u l t i n g equation ( 3 5 ) = { p ( p + l ) ( t 2 - l ) - 1 + (m 2- l ) ( t 2 - I ) ' 2 } Y d t * i s now s a t i s f i e d by ( t 2 - l ) 1 / / 2 P m ( t ) and ( t 2 - l ) 1 / / 2Q^( t ) I f we set rn = -(p+ 1 . 2)a, 0 < a < 1 and a f i x e d , then f o r p = J 1 _ a 2 equation (35) reduces t o ( 3 6 ) = { ( t 2 - p 2 ) ( p + l / 2 ) 2 ( t 2 - l ) - 2 - 4 - 1 ( t 2 + 3 ) ( t 2 - l ) - 2 } dt which i s now s a t i s f i e d by ( t 2 - l ) 1 / 2 p a ( p + l / 2 ) ^ j & n d ( t 2 - l ) 1 / 2 ^ ( P + l / 2 ) ( t ) . I t i s p o s s i b l e now, according t o Olver [ 1 9 , 2 0 ] or Thorne [ 2 7 ] , t o ob t a i n asymptotic expansions of Pp(t) and 0 ^ ( t ) , which are v a l i d u niformly w i t h respect to t , as p -» » , f o r t 18. l y i n g i n a domain D^., say, i n which the p o i n t s t = l and t = 6 + i . o are i n t e r i o r p o i n t s and which extends t o i n f i n i t y . The c o e f f i c i e n t of ( ( t 2 - B 2 ) ( p + l / 2 ) 2 ( t 2 - l ) " 2 ) Y has double poles at the r e g u l a r s i n g u l a r i t i e s t = +1 and t u r n i n g p o i n t s at the simple zeros t = +B. To ob t a i n expansions v a l i d at the t u r n i n g p o i n t t=B+i.o we make the t - z tra n s f o r m a t i o n [ 1 9 ? 2 0 ] (37) H = -(t 2- D ( t 2 - P2)-VV/2 , X = ( ^ ) _ 1 / 2 Y and - -1 - | z ^ 2 = - j ( s 2 - B 2 ) ( s 2 - l ) _ 1 d s = a c o s h ~ 1 a 6 " 1 ( t " 2 - l ) " 2"-cosh" 1tB"" 1, .t 8 where the lower l i m i t of the i n t e g r a l i s p + i . o . Then X(z) s a t i s f i e s the equation ( 3 8 ) £\ = { ( p + l / 2 ) 2 z + f , ( z ) }X, dz x where f^z') = ( | 5 ) z 2 + 4 _ 1 z ( t 2 - l ) ( t 2 - B 2 ) ~ 3 { t 2 ( 4 a 2 - l ) + ( l - a 4 ) } The comparison equation 2 (39) ^ - i » ( p + l / 2 ) 2 z X -dz^ i s s a t i s f i e d by the 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 th a t i f (40) E Q ( z ) = 1, P 8 ( z ) = | z - 1 / 2 j V 1 / 2 { f 1 ( r ) E f l ( r ) - E»(r)} dr, and W ? > - - iFk^ + f f i < r ) F s < r > d r + ° W «00 19. where the sequence of a s _> 0, are i n t e g r a t i o n constants, then the f o l l o w i n g r e s u l t s w i l l be obtained: (41) e ^  P m ( t ) ~ {r(p+m+l)/r(p-m+l)} ^ ( 4 z ( t 2 - B 2 ) " 1 ) J x 2 _ 1 » 2 _ 5 » x { A i ( v 1 z ) Y " 3 £ E s ( z ) Y " 2 S + A i ( Y 3 z ) Y" "3 Y F s ( Z ) Y - 2 S ] s=0 s=0 and . 2 n l r i + £i 1 1 (42) e" 2 * <g(t) ~ 7r {r(p+m+l)/r(p-m+l)3^(4z(t 2-B 2)- 1) 7 r x 2 - 2 7 T i - 1 0 8 - - T T i 2 - 2 7 T i - 5 °° x { A i ( Y 1 e " 1 z ) Y 1 ^ E s ( Z ) Y " 2 s + e 3 A i ^ e 1 z ) Y " 3 ) \ ( Z ) Y ~ 2 S } s=0 s=0 where the i n t e g r a t i o n constants a are s p e c i f i e d by the r e l a t i o n s ' 0 , 1 (43) l ( % y ' 2 8 + PgY" 2 3" 1) ~J-m/tor (V2) mr(-m) {rCp+nH-D/TCp-iw-l)}"2 s=0 where R = v V ( l + v ) * " 1 " v , v = -5(a - 1 -1). The expansions ( 4 l ) and (42) are v a l i d throughout the t-plane cut from +1 to -» except f o r a pear-shaped"domain surrounding the 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 immediately below the r e a l t - a x i s f o r which |Ret| < B+6, 0 > Im t > -6, 8 > 0. In both these regions asymptotic expansions can be obtained by use of the c o n t i n u a t i o n formulae (8) and (9). The asymptotic expansion of (p+l-m, -p-m; 1-m; ( l - t ) / 2 ) i s now an immediate consequence of ( 4 l ) , (6) and r e l a t i o n (33), where we have to take the + s i g n . The expansion of o 2F 1((m+p+2)/2, (m+p+l)/2; p+3/2; t ) f o l l o w s immediately from the asymptotic expansion (42) and the r e l a t i o n (7)-20. CHAPTER I I THE NORMALIZED HYPERGEOMETRIC DIFFERENTIAL EQUATION The hypergeometric equation (1) t ( l - t ) A_| + [ c - (a+b+ljtjg*- - abx = 0 dt i s normalized by s e t t i n g x ( t ) = y ( t ) f | ( i - t ) ( c - a " b " l ) / 2 , 0 < t < 1, and x ( t ) = y ( t ) t " l ( t - l ) ( c - a " b - l ) / 2 , t > 1. Then equation (1) becomes (2) UL + {(At2 + Bt + C ) / 4 t 2 ( l - t ) 2 \ y = 0, dt I ' where A = 1 - (a-b) , B = 2c(a+b-l)-4ab, and C = c(2-c) . Let us consider the case that a=b i s a l a r g e r e a l parameter. Then the t u r n i n g p o i n t s t i , i = 1,2, are the roo t s of the quadratic equation (3) t 2 + [ 2 c ( 2 a - l ) - 4 a 2 ] t + c(2-c) = 0 For a l a r g e (4) t ]_ ~ c ( 2 - c ) / 4 a 2 and t 2 ~ 4a 2. I f we now make the 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) + 4a 4 f(c(2-c)(l-z) + f(z))/g(z) dz I ) y = o, where f ( z ) = ( c 2 ( l - c ) ( 2 a - l ) z / 2 a 2 ) + ( c 2 ( 2 - c ) 2 z 2 / l 6 a 4 ) 2 2 ? and g(z) = z (4a - c ( 2 - c ) z ) ~ . For z bounded f ( z ) = ©(a" 1) as a - <x>. The t u r n i n g p o i n t now occurs at z=l and the s i n g u l a r i t i e s occur at z=0 and z = 4a 2 / c(2-c). Equation (5) can be w r i t t e n i n Thome's form (6) 1-| + ( ( 2 a 2 ) 2 z - 2 ( l - z ) p (z) + z 2 q i ( Z ) | y = 0 , dz 1 ) 2 2 where p-^z) = c(2-c)/(4a - c(2-c)z) , p-j_(z) does not van i s h and i s r e g u l a r f o r z < 4a / c ( 2 - c ) ; furthermore q x ( z ) = 4 a 2 f ( z ) / (4a 2 - c ( z - c ) ' z ) 2 , _2 f o r z bounded q-j_(z) = 0(a ) as a -* » . I f we set y = c(2-c) and u = 2a, then 2 (7) ^-g + ( ( Y / 4 ) ( l - z ) / z 2 + [ ( Y / 2 ) 2 ( 1 - Z ) / U 2 Z ] [ ( 2 - Y Z / U 2 ) / ( 1 - Y Z / U 2 ) dz ( 2") + q ]_(z)/z j y = 0. Now Y ( 1 - Z ) / 4 Z 2 i s not the dominant term, the t u r n i n g p o i n t t-^ and the s i n g u l a r i t y t, = 0 coalesce f o r _a l a r g e and z bounded. Numerical a n a l y t i c methods can determine the values f o r 2 F]_( a, a j c> t ) i n the r e g i o n i n which the t u r n i n g p o i n t and the s i n g u l a r i t y t=0 coalesce. Furthermore, ^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 2F- L(a,a;c;t) AND 2 F 1 ( a , a + l - c ; l ; t _ 1 ) A s o l u t i o n of the normalized hypergeometric equation 2^  ' r " ' ) (1) ^-g + {(At2 + Bt + C) / 4 t 2 ( l - t ) 2 ) y = 0, p where A = l- ( a - b ) , and B = 2c(a+b-l) - 4ab, and C =. c ( 2 - c ) , i s (2) |- ^ ( a ^ b j c j t J t ^ C l - t ) ^ ^ 1 - 0 ^ 2 , 0 < t < 1, \ ^ ( a ^ a + l - c r a - b + l ^ - ^ t ^ ^ ^ ^ C t - l ) ^ ^ 1 - 0 ^ t > 1. Let us consider the case that a = b i s a l a r g e r e a l parameter. Then equation ( l ) has the form (3) ^ = [up(t) + q ( t ) l y , dt where p ( t ) = l / ( 4 t ( 1 - t ) 2 ) , q ( t ) = ( c ( c - 2 ) - t 2 ) / ( 4 t 2 ( l - t ) 2 ) and u = 4 a 2 - 2 c ( 2 a - l ) . New v a r i a b l e s Z and Y are introduced by the r e l a t i o n s f 1 * ' 2 0 ! t s . Z = [ ( 1 / 2 ) " [ p ( s ) ] 1 / 2 d s ] 2  J t , dt j-1/2 1 \ dz ; y * 2 3 . Therefore, we r e a d i l y o b t a i n 1/2 t CO = (1A)|" = ( l ^ t a n h " 1 ^ ) 1 "o s x / * ( l - s ) = ( - l / 4 ) l n ( ( l - ft)/(l-hrt)), t < 1, and 1/2 » (5) z 2 = ( - l A ) f -TTp^ - ( l / 2 ) c o t h " 1 ( / t ) t s ' (1-s) = (-i/4)m((./ tu-i)/(./t+i)), t > i . Equation (3) i s then transformed i n t o (6) = j u z " 1 + Y Z " 2 + h ( z ) z " 1 | Y , where Y = -3/l6 and f o r z = z 2 h(z). = ( - 3 / 4 ) - ( l / 4 ) c o s e c h 2 ( 2 z 1 / 2 ) + ( c ( c - 2 ) + 3 / 4 ) t a n h 2 ( 2 z 1 / 2 ) . For t > 1 r e l a t i o n (5) i m p l i e s that t = coth ( 2 z 2 ). This leads t o the 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 = (||) = 2 z " 1 / 2 c o t h ( - 2 z 1 / 2 ) c o s e c h 2 ( 2 z 1 / 2 ) , q [ t ( z ) ] = ( l / 4 ) [ c ( c - 2 ) s i n h 4 ( 2 z 1 / 2 ) t a n h 4 ( 2 z 1 / 2 ) - s i n h 4 ( 2 z l / 2 ) ] and t h e r e f o r e ( t ) 2 q [ t ( z ) ] = [ c ( c - 2 ) z " 1 ] [ t a n h 2 ( 2 z 1 / 2 ) ] - [ z ~ 1 c o t h 2 ( 2 z 1 / 2 ) ] , yz' 2 + h ( z ) z _ 1 = t i , ( t " 1 / 2 ) + ( t ) 1 / 2 q [ t ( z ) ] dz = (-3/l6)z"2+ z ' H ( - 3 / 4 ) - ( l / 4 ) c o s e c h 2 ( 2 z 1 / 2 ) + (c(c-2) + ( 3 / 4 ) ) t a n h 2 ( 2 z 1 / 2 ) ] . 24. Therefore, y = -3/l6 and i f we set u2 = 1 + 4 Y = 1/4, then without l o s s of g e n e r a l i t y we can take p. = 1/2. I f we now put = u + c(c-2) = (2a-c) then the b a s i c equation (6) becomes (7) ^ | = ( u ^ " 1 + ( ( p 2 - l ) / 4 ) z " 2 + h 1 ( z ) Z " 1 ) Y , dz where h-^z) = ( - l / 4 ) c o s e c h 2 ( 2 z 1 / 2 ) - ( c ( c - 2 ) + 3 / 4 ) s e c h 2 ( 2 z 1 ' / 2 ) . A p a i r of l i n e a r independent s o l u t i o n s of the comparison equation, that i s equation (7) w i t h h-^z) = 0, are Y-L = z 1 / 2 K ( 2 ( u l Z ) 1 / 2 ) a n d I and K are modified B e s s e l f u n c t i o n s of order (jt . 2 2 For convenience, i f we now repla c e z, ^ by z and u.j/4 r e s p e c t i v e l y , then equation (7) becomes (8) ^ | = z" 1 | | + i j u 2 + H z " 2 + f ( z ) | Y , where ^ = -2(2a-c) and f o r z s z 2 = ( l / 2 ) c o t h ~ 1 ( , / t ) f ( z ) = 4 h 1 ( z 2 ) = -cosech 2(2z) - 4 ( ( c ( c - 2 ) + 3/4) s e c h 2 ( 2 z ) ) . The f u n c t i o n f ( z ) i s an even f u n c t i o n of z and i s r e g u l a r i n an unbounded simply-connected open domain D, a c t u a l l y f ( z ) = 0 ( | z | " 1 _ a ) as |z| - • , where a i s constant and a > 0. 25. Let D' be any simply-connected domain l y i n g wholly i n D, the boundaries of which do not i n t e r s e c t the boundaries 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 sector |arg z| < TT/2, then the domain comprises those p o i n t s z of D' which can be j o i n e d to 6 by a contour which l i e s i n D1 and does not cross e i t h e r the imaginary a x i s or the l i n e through z p a r a l l e l to the 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 of the sector |arg z| < TT/2, which may be at » , and e to be an a r b i t r a r y p o s i t i v e number. Then D 2 c o n s i s t s of those p o i n t s z of D' f o r which |arg z| _< J>ir/2, Re z < Re d and a contour can be found j o i n i n g z and d which s a t i s f i e s the 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'; ( i i ) i t l i e s w holly to 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 ; ( i i i ) i t does not cross the negative imaginary a x i s i f A p a i r of l i n e a r independent s o l u t i o n s of the comparison equation The b a s i c equation (8) has s o l u t i o n s Y-^(z) and Y 2 ( z ) such that f o r Re \x > 0 ( i v ) TT/2 _< arg z _< J>TT/2. and does not cross the p o s i t i v e imaginary a x i s i f -37r/2 _< arg z _< -TT/2; i t l i e s outside the c i r c l e | r | =, e|z|. (9) are now Y - L = z l (u-jz) and Y 2 = z K M ( u l z ) * 26. ( i ) i f z l i e s i n Dx, M-l Y x ( z ) = z l ^ ^ z ) V A s ( z ) u " 2 s + 0 ( u " 2 M ) U=0 1-1 £ B s ( z ) u ' 2 s + ( z / ( l + | z | ) 0 ( u - 2 M ) fM-lls=0 as u^ -* <», uniform w i t h respect to z, ( i i ) i f z l i e s i n B>>, M-l \ Y 2 ( z ) = z y u l Z ) I A s ( z ) u " 2 s + 0 ( u " 2 M ) ^s=0 M-l - ( z ^ 1 ) ^ ! ^ ) ! £ B s ( z ) u " 2 s + ( z / ( l + | z | ) 0 ( u " 2 M ) (s=0 i as u^ - <*>, uniform w i t h respect t c . z . Since f ( z ) = 0 ( | z | ~ 1 _ a ) as |z| - oo and a > 0, the asymptotic expansion i s v a l i d f o r z t e n d i n g to i n f i n i t y . The sequences of f u n c t i o n s A (z) and B (z) are given "by the r e l a t i o n s A G ( z ) = 1 2B s(z) = - A i ( z ) + J ^ f ( t ) A s ( t ) - ( 2 p + l ) t - 1 A s ( t ) j d t , 2 A s + 1 ( z ) = ( 2 n + l ) Z - 1 B s ( z ) - B l ( z ) + J* |f ( t ) B g ( t )jdt + Cfi , where C6 i s a constant and 6 > 0 i s f i x e d . Asymptotic forms, that i s M = 1, of these s o l u t i o n s are ( i ) i f z l i e s i n L^, Y 1 ( z ) = z l ( j ( u 1 z ) ^1 + 0 ( u " 2 ) ^ + ( a u i 1 ) I n + i ( u i z ) ^ B o ( z ) + ( Z / ( 1 + I Z D ) ° K 2 ^ 27. as - », uniform w i t h respect t o z i n D^ , ( i i ) i f z l i e s i n Dg, Y 2 ( z ) = zK^(u 1z) | 1 + 0(u" 2) - ( z u - ^ K ^ ^ z ) | B 0 ( Z ) + ( z / ( l + | z | ) ) 0(u~ 2) as u-j^  - <*>, uniform w i t h respect t o z i n Dg. The f u n c t i o n B q 2 ( Z ) S B Q ( z ) i s now given by the i n t e g r a l Bo2(z) = ( 1/2) JZ f ( s ) d s , 6 > 0 6 z = ( -1/2) J cosech 2 ( 2 s ) 6 + 4 ( c ( c - 2 ) + 3 / 4 ) s e c h 2 ( 2 s ) ds = ( 1 / 4 ) c o t h ( 2 z ) + ( c ( c - 2 ) + 3 / 4 ) t a n h ( - 2 z ) + C 2 6 , where C 2 & i s a constant and 6 > 0 i s f i x e d . Since 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 ] ) = ( ^ ) - 1 / 2 p ( a , a ; C ; t ) t c / 2 ( 1 . t ) ( 2 a + l . c ) / 2 x dz£ 1 0 < t < 1, i s a s o l u t i o n of equation ( 8 ) , i t f o l l o w s t h a t (10) Y ( t [ z 1 ] ) = c 1 ( u 1 ) Y 1 ( z 1 ) + c 2 ( u 1 ) Y 2 ( z 1 ) , and s i m i l a r l y , since Y ( t [ z ? ] ) - ( ^ g ) " 1 / 2 2 P T ( a , a + l - c ; l ; t - 1 ) t ( c - 2 a ) / 2 ( t - l ) ( 2 a + 1 - c ) / 2 , d z j • 1 t > 1, i s a s o l u t i o n of ( 8 ) , i t i m p l i e s that 28. (11) Y ( t [ z 2 ] ) = d 1 ( u 1 ) Y 1 ( z 2 ) + d 2 ( u 1 ) Y 2 ( z 2 ) . Now, f i x and l e t t -» 1, that i s z-^ and z 2 approach i n f i n i t y . I t i s w e l l known [30] that I (z) ~ ( 2 7 r z ) - 1 / 2 ( e z + e ± ^ + 1 / 2 ) i r i e " z ) , |arg z| < 3^ /2 - e, e > 0, as |z| -» » uniformly w i t h respect t o arg z; the upper s i g n a p p l i e s t o the range -TT/2 + e _< arg z _< 3T/2 - € and the lower to the range -3t/2 + e _< arg z _< TT/2 - €. A l s o , K f j ( z ) ~ ( V 2 z ) 1 / 2 e " z , |arg z| _< 3ff/2 - e unifo r m l y w i t h respect t o arg 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 / 2 ~ l / ( 2 ( l - t ) ) and t h e r e f o r e according t o equations (4) and (5) e - 4 z x ~ 1-t , 0 < t < 1 e ^ z2 ~ t-1 , t > 1. Applying these estimates t o the s o l u t i o n s (2) f o r a=h we get (12) y ( t [Z l] ) - ^ ( a . a j c ^ X l - e ^ D ^ t e - S j t 2 ^ ) / 2 ~ ( r ( e ) r < o a . ) / < r < c . ) ) 2 ) e - 2 ( 2 a + i - c ) z i as t -* 1" and -a i s l a r g e ; y ( t [ z 2 ] ) ~ ^ ( a ^ + L c j U f ^ l + e ^ ^ l i ^ 2 ^ / ^ 4 ^ ) ^ 2 ) ~ ( r ( c - 2 a ) / ( r ( l - a ) r ( c - a ) ) ) e - 2 ( 2 a + 1 - c ) z 2 as t 1 + and -a i s l a r g e . L e t t i n g t -» 1 + i n equation (11) and usin g the approxi-mations f o r the modified B e s s e l f u n c t i o n s as | z P | - », we r e a d i l y see t h a t d 2 ( u i ) = 0 and since u^ i s l a r g e and B q 2 ( Z 2 ) remains bounded as | j -» • d l ( u l ) ( - d | - ) 1/ 2 YI ( z 2 ) ~ e ( u l - 2 ) z 2 ( 7 r u 1 ) - 1 / 2 d 1 ( u 1 ) , d z 2 as | z 2 | -» OB , | arg z 2 | <_ 3T/2 - e, € > 0, u n i f o r m l y i n arg z 2 Prom r e l a t i o n (12) we then f i n d that (13) d 1 ( u 1 [ a ] ) = ( r ( c > 2 a ) / ( r ( l - a ) r ( c - a ) ) ) ( 2 ( c - 2 a ) 7 r ) 1 / 2 . The constant d ^ u - J a ] ) i s w e l l defined f o r Re(c-2a) > 0 since the 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 - 1 ) = ( - S S - ) l / 2 t ( 2 a - c ) / 2 ( t _ l ) ( c - 2 a - l ) / 2 x x d Z | x Y 1 ( z 2 ) d 1 ( u 1 [ a ] ) , we then have deri v e d the f o l l o w i n g r e s u l t . For a and c r e a l , c-2a > 0 and the t - i n t e r v a l 1 < t _< T 2 < « (14) g F ^ a + l - c j l j t - 1 ) - B ( a ) t ( 2 a - c + 1 / 2 ) / 2 ( t - l ) ( c " 2 a ) / 2 x x | Z 2 / 2 I l / 2 ( u i z 2 ) [ l + ° ( U l 2 ) ] " ( 2 2 / 2 u l 1 ) I 3 / 2 ( u l Z 2 ^ [ B o 2 ( z 2 ) + z 2 / ( l + | z 2 | ) 0 ( u ^ ) ] unformly w i t h respect t o t as *•• oo, where = -2(2a-c), c f i x e d , z 2 = ( l ^ c o t h " 1 ^ ) , B(a) = ( r ( c - 2 a ) / r ( l - a ) r ( c - a ) ) ( 2 u 1 T r ) 1 / 2 , B o 2 ( z 2 ) = ( l A ) t 1 / 2 - (c(c-2) + 3 A ) t - 1 / 2 + Gm , x2 T 2< » and f i x e d . F i n a l l y , i f We now consider the i n t e r v a l 0 < 1^ _< t < 1, 1/2 -1 then = ( l / 2 ) t a n h ~ (./t). Analogous t o r e s u l t s obtained before, we now have t s ^ | = 8z^2 t a n h ( 2 z ^ / / 2 ) s e c 2 h ( 2 z J / / 2 ) and Y z ~ 2 + h^z^zl 1 = t 1/2 ^ t " 1 / 2 + t ^ q ( t [ Z l ] ) dz 1 - - ( V l 6 ) z " 2 + z ^ i h ^ ) + c ( 2 - c ) ) , where h ^ z - ^ = ( l / 4 ) s e c h 2 ( 2 z J / 2 ) + (c(c-2) + 3/4)cosech 2(2 z^ / / 2). Therefore, as before y = -(3/l6) such that n = 1/2 and 2 = u + c(2-c) = (2a-c) ; furthermore, i f again we repl 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 , then z± = ( l ^ J t a n h " 1 ^ / ^ ) , u^. = -2(2a-c) and f ( Z l ) = 4 h 1 ( z 2 ) = sech 2(2z 1)+(4c(p^2)+3)cosech 2(2z 1). I t i s seen t h a t f ( Z ; L ) = 0 ( | Z l | ) as | Z l | - », a > 0. As before, the reason f o r modifying the l a r g e parameter i s to make, f ( Z l ) = 0 ( | Z l | ~ 1 _ a ) as | Z l | - », a > 0, such t h a t the s o l u t i o n s Y 1 ( z 1 ) and Y 2 ( Z l ) of equations- (8) corresponding to z = z-j^  have asymptotic expansions v a l i d as z^ tends t o i n f i n i t y . 3T. This 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 (11) and (12) near t = l and us i n g w e l l known estimates 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^| tends t o i n f i n i t y , the constants c ^(u 1) and c 2 ( u 1 ) can be obtained. To f i n d these constants, we f i r s t have t o o b t a i n the f u n c t i o n B ^ ^ ^ . B 0 i ( z i ) l s n o w given by the i n t e g r a l B o l / z l ) = ( V 2 ) J Z l f ( s ) d s = 6 ( l / 4 ) t a n h ( 2 Z ; L ) - (c(c-2) + 3 / 4 ) c o t h ( 2 Z l ) + C 1 6 , 6 > 0 and f i x e d . L e t t i n g t - 1~ i n r e l a t i o n (10) and keeping u-^  f i x e d we see that c 2 ( u 1 ) = 0, 1/2 ^ ( " I K T J ! ) V z l > ~ e ( u l - 2 ) z l ( 7 r u 1 ) - 1 / 2 c 1 ( u 1 ) as \ Z l\ - . , | arg I _< 37r/2 - e, e > 0, uni f o r m l y i n arg z^; since i s a l a r g e parameter and B Q ^ ( z 1 ) remains bounded as \z-^\ - « . Prom the estimates (12) we now r e a d i l y get c 1 ( u 1 [ a ] ) = ( r(c ) r(c-2a ) / ( r(c-a)) 2)(2(c-2a)7r) 1/ 2. The constant c 2 ( u 1 [ a ] ) i s w e l l defined f o r Re c > 0 and Re(c-2a) > 0, since the in 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 2 F l ( a , a ; c ; t ) = # 2 ) V 2 t - = / 2 ( 1. t )(c - 2 8-l ) / 2 Y ( ! ) C (^) dz-^ we then have derived the f o l l o w i n g - r e s u l t . 32. For a and c r e a l , c > 0, c-2a > 0 and the t - l n t e r v a l 0 < T1 < t < 1 (15) ^ ( a ^ c j t ) = A ( a ) t ( - c + ^ / 2 ( l - t ) ( C - 2 a ) / 2 x x[zY2 V 2 ( u i z i ) [ i + ° ( u i 2 ) ] - ( z l / 2 u l 1 ) I 3 / 2 ( U l Z l ) [ B o l ( z l ) + z 1 / ( l + i z 1 l ) 0 ( u - 2 ) ] | uniformly w i t h respect t o t as u^ -» where u.^ = ( l / 2 ) t a n h ~ 1 , / i - , A(a) = ( r ( c ) r ( c - 2 a ) / ( r ( c - a ) ) 2 ) ( 2 u 1 7 r ) 1 / 2 , B o l ( Z l ) = ( l / 4 ' ) t 1 / 2 - ( c ( c - 2 ) + 3 A ) t - 1 / 2 + C T , T-j^  > 0 and f i x e d . Completely analogously t o the f o r e g o i n g complete asymp-t o t i c expansions can be obtained f o r 2 F ^ ( a , a + l - c j l ; t _ 1 ) and 2F^(a,a;c;t) as -a -• » , us i n g the complete asymptotic expansions of Y-^(z) and Y2(z) i n s t e a d of t h e i r asymptotic forms. • By employing the a l t e r n a t i v e forms of the hypergeometric f u n c t i o n and the expressions f o r the a n a l y t i c c o n t i n u a t i o n of the hypergeometric f u n c t i o n , i t i s p o s s i b l e to deduce v a r i o u s other asymptotic expansions f o r the hypergeometric f u n c t i o n . For i n s t a n c e , f o r 0 < t < 1 2 F 1 ( a , a ; c ; t ) = ( l - t ) ( c " 2 a ^ 2 F a ( c - a , c - a j c ; t ) ; so we have the f o l l o w i n g r e s u l t . For a and c r e a l , c > 0 , c - 2 a > 0 and the t - i n t e r v a l 0 < T x < t < 1 (16) 2 F 1 ( c - a , c - a ; c ; t ) - A ( a ) t ( - c + 1 / 2 ) / 2 ( l - t ) ( 2 a - c ) / 2 x x | z 1 / 2 I 1 / 2 ( u l Z l ) [ l + 0 ( u j 2 ) ] - ( z 1 / 2 u - 1 ) l V 2 ( u 1 z 1 ) [ B o l ( z 1 ) + z 1 / ( l + | z 1 | ) 0 ( u ~ 2 ) ] j u n i f o r m l y w i t h respect t o t as •* » , where u^,z^,A(a) and B Q l ( z 1 ) are given by ( 1 5 ) . As an a p p l i c a t i o n of the asymptotic form ( 16) l e t us consider the complex z-plane and suppose th a t F(z)=U(z)+iV(z) i s holomorphic i n |z| < R and continuous i n |z| _< R. Furthermore, f o r z = re 1"^ l e t us denote Max F ( r e 1 - e ) by 0<Q<2ir M(r;F), r _< R. Then Cauchy's formula 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 t h a t Provided F ( z ) ^ M(R;F) we then have from the Maximum modulus p r i n c i p l e the i n e q u a l i t y (18) |F n ( z ) | < %L- M(R;P) J - M } |z|=R |t-z n+1 , r < R. But f o r I z j = r < R the i n t e g r a l I I d t I .., can be 2T? J jz|=R | t - z | n + 1 worked out [11] and i s found to be 2 F ] L ( ^ ( n + l ) , ^ ( n + l ) ; l ; (-yr)2). We t h e r e f o r e have the f o l l o w i n g r e s u l t . For j z | = r < R and o < T x _< (£) 2 < 1 , 3 4 , (19) | F n ( z ) | < n< M ( R ; F ) 2 F 1 ( ^ ( n + l ) ^ ( n + l ) ; l ; ( | ) 2 ) , here 2 F 1 ( ^ ( n + l ) ^ ( n + l ) ; 1 ; (-|)2) f o r l a r g e n i s given by (16), where u x = 2w, c = 1 z x = ( l / 2 ) t a n h _ 1 ( | ) , A[a(n)] =r(n)(r(-|(n+l)))-2 ( 4 7 m ) 1 / 2 and B 0 l ( Z l ) = ( l / 4 ) ( r R " 1 + R r " 1 ) + C T , T 1 > 0 and f i x e d . A s i m p l i f i c a t i o n of n.'A(n) i s obtained by u s i n g the d u p l i c a t i o n formula of the gamma f u n c t i o n (20) T T 1 / 2 (2n) = 2 2 n _ 1 T(n) T(n+l/2). I t then i s found t h a t (21) n.»A[a(n)] = 2 2 n + 1 ( F ( ^ + 1) ) 2 ( n 7 r ) " 1 / 2 . I f we now expand the 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) n»A[a(n)] ~ 2 ( 2 n ) ' n e " n ( m r ) 1 / 2 or (23) A[a(n)] ^ 2( n + 1/ 2) . Therefore, i n view of the approximation formulae of modified B e s s e l functions- f o r l a r g e values of the 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 : f o r |z| = r < R and o < T x _< ( r R - 1 ) 2 < 1 , 3 5 . (24) | F n ( z ) | < nJ M(RjP) p F ^ n + l )y|(n+l); l ^ ) 2 ) , here f o r n l a r g e n J ^ ^ n + D ^ C n + l ; ! ; ^ ) 2 ) ~ 2 ( 2 n ) n e - n ( n 7 r ) 1 / 2 x x ( r R - 1 ) - 1 / 2 ( l - ( r R - 1 ) 2 ) - n / 2 z J/ 2I 1 / 2(2nz 1) = ^ ( 2 n ) n e " n ( r R ' 1 ) " 1 / 2 ( l - ( r R " 1 ) 2 ) _ n / 2 s i n h ( n t a n h " 1 ( r R - 1 ) ) , since z 1 / / 2 I - L / , 2 ( 2 n z 1 ) = ( s i n h ( 2 n z 1 ) ) ( 2 n i r ) " 1 / 2 and z1 = ( l / 2 ) t a n h " 1 ( r R - 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 by n o t i n g that t a n h " 1 ( r R " 1 ) = \ l n ( 1 + r R 1 ) , r R - 1 < 1, and th e r e f o r e f o r r R _ 1 < l - r R " x 2 s i n h ( n t a n h - ^ r R " 1 ) ) = {±+I*±)*/2 . ( l ^ 1 n/2 ^ g o t h a t 1-rR l+rR-- 1 i n the estimate (24) f o r n l a r g e , |z| = r < R and 1 2 0 < T i S. ( r R ) < 1 we now have ( 2 5 ) n J 2 F 1 ( - ^ ( n + l ) ^ ( n + l ) ; l ; ( | ) 2 ) ~ (l/2)./2 ( 2 n ) n e - n ( r " 1 R ) 1 / 2 ( ( l - r R " 1 ) " n - (1 + r R " 1 ) ' 1 1 ) . The asymptotic form ( 2 5 ) y i e l d s a b e t t e r estimate then | F n ( z ) | _< 2nJ R(R-r )" n _ 1M(R;F), v a l i d f o r |z|=r, 0 < r < R. This 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 that (R-r)"" n"" 1 i s an upper bound f o r the integrand. 36. REFERENCES Cherry, 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 , 1 9 5 0 , 224 -257. Cherry, T.M., Asymptotic expansion f o r the hypergeometric  f u n c t i o n s o c c uring i n gas-flow theory. Proc. Roy. Soc. London, Ser. A 2 0 2 , 1 9 5 0 , 5 0 7 - 5 2 2 . Coddington, E.A. and Levinson N., Theory of or d i n a r y d i f f e r e n t i a l equations. McGraw-Hill Book Co., New York, 1955-Dettman, J . , The s o l u t i o n of a second order l i n e a r d i f - f e r e n t i a l equation 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. Monthly, 71 No. 4 , A p r i l 1 9 6 4 , 3 7 8 - 3 8 5 . E r d e l y i , A. et a l , Higher transcendental f u n c t i o n s , v o l . 1, McGraw-Hill Book Co., New York, 1953-Erde'lyi, A., Asymptotic expansions, Dover P u b l i c a t i o n s , New York, 1956. E r d e l y i , A., Asymptotic s o l u t i o n s of d i f f e r e n t i a l equations 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 . J ournal of Mathematical P h y s i c s , 1 9 6 0 a , 1 6 - 2 6 . E r d e l y i , A., Asymptotic s o l u t i o n s of 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 equations. C a l i f o r n i a I n s t i t u t e of Technology, 1961 . Hapaev, M.M., Expansions of hypergeometric and degenerate hypergeometric f u n c t i o n s i n s e r i e s of Bessel f u n c t i o n s . V e s t n i k Moskov. Univ. Ser. Mat. Meh. Astr . F i z . Him., 1 9 5 8 , No. 5 , 1 7 - 2 2 . (Russian) I 37-[10] Hapaev, M.M., Asymptotic expansions of hypergeometric and  confluent hypergeometric f u n c t i o n s . Izv. Vyss. Ucebn. Zaved. Matimatika, 1961, No.5 (24), 98-101. (Russian) [11] H i l l e , E., A n a l y t i c f u n c t i o n theory. 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 Press, 1931. [13] Hochstadt, H., D i f f e r e n t i a l equations, a modern approach. H o l t , Rhinehorst and Wiston, New York, 1963. [14] Ince, E.L., Ordinary D i f f e r e n t i a l Equations. Dover P u b l i c a t i o n s , New York, 1956. [15] Khottnerus, U. J . , Approximation formulae 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. Walters, Groningen, the Netherlands, i960. [16] Kummer, E.E., Uber d ie hypergeometrische Reihe. J. Reine Angew, Math. 15, 39-83, 1836, 127 -172. [17] L i g h t h i l l , M.J., The hodograph tr a n s f o r m a t i o n i n t r a n s - s o n i c f l o w I I . A u x i l i a r y theorems on the hypergeometric f u n c t i o n s Yn(Y)» Proc. Roy. Soc. London, Ser. A, 191, 1947, 341-351. [l8] MacRobert, T.M., Functions of a complex v a r i a b l e . 4th ed., MacMillan and Co. L t d . , London, England, 1954. [19] Olver, F.W.J., The asymptotic s o l u t i o n of l i n e a r d i f f e r e n -t i a l equations of the second order 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. 38 • [20] Olver, F.W.J., Uniform asymptotic expansions of s o l u t i o n s  of second order d i f f e r e n t i a l equations f o r l a r g e  values of a parameter. P h i l . Trans. Roy. Soc., A 250, 1958, 479-517. [21] Poole, E. G. C. , I n t r o d u c t i o n to the h i s t o r y of 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 Press, Oxford, England, 1936. [22] 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 MacMillan Co., New York, i960. [23] S e i f e r t , H., Die hypergeometrischen D i f f e r e n t i a l -gleichungen der Gasdynamik. 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 Press, i960. [25] Snow, C., The hypergeometric and Legendre f u n c t i o n s 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  theory. Applied 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, D.C., 1952. [26] 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, Brunswick, 1939, 800-806. [27] Thorne, R.C., The asymptotic s o l u t i o n of l i n e a r second  order d i f f e r e n t i a l equations 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. Report 12, Dept. Math.,Cal. I n s t , of Techn., 1956. [28] Thorne, R.C., The asymptotic expansion of Legendre f u n c t i o n s of la r g e degree and order. Techn. Report 13, Dept. Math., Cal. I n s t , of Techn., 1956. [29] Watson, G.N., Trans. Cambridge P h i l . Soc. 22, 1918, 277-308. 39. [30] Watson, G.N., Theory of Bessel f u n c t i o n s . Cambridge U n i v e r s i t y Press, 1944. [31] Whittaker, E.T. and Watson, G.N., A course of modern a n a l y s i s , 4th ed. Cambridge U n i v e r s i t y Press, 1952. 

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