Asymptotic Expansions of the Hypergeometric Function
ASYMPTOTIC EXPANSIONS OF THE HYPERGEOMETRIC FUNCTION
FOR LARGE VALUES OF THE PARAMETERS
iy
GERARD SIMON PRINSENBERG B.Sc., Victoria College, 1962
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THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ARTS
In the Department of Mathematics
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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 ii
ABSTRACT
In chapter I known asymptotic forms and expansions of the hypergeometric function obtained by Erde'lyi [5], Hapaev [10,11],
Knottnerus [15L Sommerfeld [25] and Watson [28] are discussed.
Also the asymptotic expansions of the hypergeometric function occurring in gas-flow theory will be discussed. These expansions were obtained by Cherry [1,2], Lighthill [17] and Seifert [2J].
Moreover, using a paper by Thorne [28] asymptotic expansions of
2F1(p+l, -p; 1-m; (l-t)/2), -1 < t < 1, and
2P1( (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 is fixed and 0 < a < 1. The : expansions are in terms of Airy functions of the first kind.
The hypergeometric equation is normalized in chapter II.
It readily yields the two turning points t^, i = 1,2. If we consider,the case the a=b is a large real parameter of the
hypergeometric function 2F-L(a,b; c; t), then the turning points coalesce with the regular singularities t = 0 and t = <*> of the hypergeometric equation as j a | ».
In chapter III new asymptotic forms are found for this particular case; that is, for
2^ (a, a] c;t) , 0 < T-^ _< t < 1, and
-1 2F1(a,a+l-c; 1; t ), 1 < t _< Tg < » , as -a^» .
The asymptotic form is in terms of modified Bessel functions of order 1/2. Asymptotic expansions can be obtained in a similar manner. iii
Furthermore, a new asymptotic form is derived for
2F1(c-a, c-a; c; t), 0 < <_ t < 1, as -a-»«, this result then leads to a sharper estimate on the modulus of n-th order derivatives of holomorphic functions as n becomes large. iv
TABLE OP CONTENTS
I 1. INTRODUCTION 1
2. RELEVANT PROPERTIES OF 2F1(a,b?c;t) 5 5. KNOWN ASYMPTOTIC RESULTS 8
a. Results obtained by Watson 8 b. Results obtained by Erdelyi 9 c. Results obtained by Hapaev 10 d. Results obtained by Khottnerus 11 e. Results obtained by Cherry; and Sommerf eld 12 f. Results derived from Thome's paper 15
II THE NORMALIZED HYPERGEOMETRIC DIFFERENTIAL EQUATION 20
III THE ASYMPTOTIC BEHAVIOUR OF ^(a^ajcjt) 22
and gF^aja+l-cjljt"1) 22
IV REFERENCES Z>6 ACKNOWLEDGEMENT
I wish to acknowledge the Invaluable guidance and assistance extended to me by
Dr. C. A. Swanson and also would like to thank him for his assistance in preparing the final manuscript.
The generous financial support of the
National Research Council and the University of British Columbia is gratefully acknowledged. 1.
CHAPTER I
1. INTRODUCTION
The hypergeometric function ^F^a,,}}', c; t) has been investigated by numerous authors, see for instance the references [5, 14, 21, 22, 25, 51]. It depends on the three parameters a,b and c and the variable t.
The integral representations of the hypergeometric function by Barnes' and Euler [5] have successfully been em• ployed by Knottnerus [15], Seifert [23] and Sommerfeld [26] to derive asymptotic expansions as one or several of the para• meters tend to OB. Both Seifert and Sommerfeld used the method of steepest descents [6] on a slightly modified form of the
Euler integral representation. The expansions are valid in a t-interval which does not contain a turning point [6],
Asymptotic expansions of the hypergeometric function occuring in gas-flow theory on the other hand are valid in regions containing a regular singularity and a turning point which in this case corresponds to the transition point of sub• sonic flow to supersonic flow. The method used by the authors
Cherry [1,2], Lighthill [.17].,. and Seifert .[2J] to obtain the asymptotic expansion is described in for instance, the references
[7, », 19, 20, 27]. This method concerns the asymptotic solu• tions of differential equations with fixed turning points, or 2.
singularities.
Some authors, notably Erdelyi [5] Hapaev [9, 10] and
MacRobert [l8] used well known expansions of the confluent hypergeometric function ^F^(a; b; t) [24, 31] to derive asymp•
totic expansions of the hypergeometric function.
In a similar fashion, asymptotic expansions as p -• « are derived for
gF^p+l, -p; 1-m; (l-t)/2), -1 < t < 1, and
2 2F1((p+m+2)/2, (p+m+l)/2; p+3/2; t" ), t > 1 where m = -(p + 1/2)a, a fixed and 0 < a < 1. The expansions
are derived from a paper by Thorne [28] on the asymptotic
expansions of the associated Legendre function for large degree
and order. The expansions are in terms of Airy functions of
the first kind [5].
In this chapter a description of the known asymptotic
expansions for gF^(a,b; c; t) will be given.
The expansions are valid as one or several of the para• meters approaches », under various restrictions on the (other) parameters and t. The following table summarizes the results. 3.
large Author Date parameters other restrictions
G. N. Watson 1918 c a,b fixed, |arg c j_< 7r-e,e > 0, |t| < 1.
G.N.Watson 1918 a,b, c 1) (a+b)fixed, c fixed, l-2t+2(t2-t)1/2= e±p 2) (a-b),(a+b-c) fixed [t-2+2((l-t)t"1)1/2]t"1= e±p
T.M.MacRobert 1923 c a,b fixed,|arg c|<^+e, e > 0, |arg(l-t)| < V.i
H. Sommerfeld 1939 a,b a=ipv^ b=iv, c=l, c real and fixed, v real and v-^>, |t| < 1 .
»
1 H. Selfert 1947 a,b , c a+b=c-l-(Y-l)" ,ab=|c(l-c)(Y-ir 1 M. J. Lighthill 1947 a>b, c Same relations between a,b and c complex, j arg 11 T.M.Cherry 1950 a, b, c Same relations between a,b and c complex, | arg(c-l) |_<-^, |arg(l-t) |< 7T. A. Erdelyi 1953 c a,b fixed, | arg cj_Cn--e, e > 0, |t|>l, jarg(l-t)| < ir. A. Erdelyi 1953 b a,c fixed, c^0,-l,-^,..., 0 < jt| < 1 R.C.Thorne 1956 a,b, c a,b linear In c,|t| M. M. Hapaev 1958 a b,c fixed |t| ~ r^a|-1 U. J. Khottnerus 19b0 c a,b fixed, t-plane cut from 1 to 1 + ioe . 4. (continued) large Author Date parameters other restrictions U. J. Khottnerus lybO a, b, c (a-b),(a-c) fixed, 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, ac"1 fixed, jtj The hypergeometric equation (1) t(l-t)^-| + (c - (a+b+l)t)-^ - abx = 0 dt has three regular singular points, namely, t = 0, t = 1, and t = One solution relative to the singularity t = 0 is developable in a series (2) 1 + (ab/c)t + (a(a+l)(b(b+l))/c(c+l)2j)t2 + ... 00 = c n n I ((aJn^n/ ( )n -')t , c t 0, -1, -2, n=0 where (a)n is the Pochhammer symbol and a+n (a)n = r( )/ P(a) = a(a+l)...(a+n-l), n = 1,2,3, [5]. a b c The series (2), due to Gauss, is denoted by 2F1( ' » * The series converges for jtj < 1 and for t = 1 it diverges when Re (c-a-b) _< -1, it converges absolutely when Re (c-a-b) > 0 and it converges conditionally when -1 < Re(c-a-b) < 0 and the point t=l is excluded -{14]. If c = -n, n = 0,1,2,..., then a solution of equation (l) n+1 m which is regular at t=0 is t Y( (a+n+1 )m(b+n+l )m/(n+2)m ml )t m=0 (3) = tn+1 ^(a+n+l, b+n+1 ; n+2 ; t). Also if a = -n or b = -n, where n = 0,1,2,... and if c = -m, where rn = n, n+1,... , then we define [5] 6. n r (4) ^(-n.b; -mS t) ((-n)r (b)r /(-m)p r )t ; r=0 similarly for 2F1(a,-n; -mj t). Since (3) and (4) are solutions of equation (1) we see that the hypergeometric equation has a solution which is a polynomial in t whenever -a or -b is a nonnegative integer. The property F a b c; X = (5) 2 l( ' > ) f(c)r(c-a-b) /V(c-&)V(cr\>)) will be used later. It is assumed that c, c-a-b, c-a and c-b are not negative integers [14J. There are many results: relating the hypergeometric function and the Legendre functions PjJ(t) and QJj(t). Hobson's [12] definitions of the Legendre function will be used. These definitions are in terms of contour integrals and are valid for unrestricted values of m and p. For the t-plane cut from 1 to -» and |l-t|< z. (6) Pp(t) = (r(l-m))"1((t-D/(t+l))-m/2 ^(p+lrpjl-mjM/S) and for |t| > 1 m7ri 2 m m 1 (7) e- ^(t) = 2P(r(p+l)r(P+m+l)/r(2p+2))((t -l) /VP- - ) 2 x 2F1((p+m+2)/2,(p+m+l)/2; p+3/2; t" ) For all t we have the continuation formulae, which also can be found in Hobson [12], 7. («)• Pj(-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. Results obtained by Watson. Watson [29] investigated the behaviour of 2F1(a,b; c;. t) for large values of jaj, jb| and |c|. If a, b and t are fixed and |c| is large with the restriction |arg c| _< w-e, e > 0, then for |t| < 1 n 1 (10) pF-^a^b; c; t) = l+(ab/c)t + . .. +((a)n(b)n/(c)nni )t +0( | c" With a slightly modified expression for the remainder term Erdelyi [5] proved this remains valid even if |t| > 1, |arg(l-t)| < 7r, provided that Re c - ». MacRobert [18] proved (10) for a range of argument c which is larger than TT and valid for all t provided that |arg (l-t)| < ir. In the case where more than one of the parameters approaches » Watson [29] derived the following results. If we define ? by t + (t2-l)1//2 = and put 1 - e = (eb - l)e where the upper or lower sign is taken according as Im t >< 0, then for large |\| a x -1 (11) (l/2t - l/2)" " 2F1(a+X, a-c+l+X; a-b+l+2\; 2(l-t) ] = 2a+b(r(a-b+l+2X)r(l/2)X-1/2^ 0 1 1 x (1 + e-?) -^" /^!+ 0(X- )L where |arg x| _< v-b, 6 > 0 and also 9. (12) ^(a+X, b-X; C; - jjt) a+b 1 c = 2 - (r(i-b-rx)r(c)/r^)r(C-b+x))(i-e^)- 4 x c a b X b i7r c } v(l+e^) - - - \ x-'-2[e( - >* + ei ( " i .x. X+a ve-( )*][l + OdX"1!)] , where theLupper or lower sign is taken according as Im t >< 0 and where |x| is large, ?. = p + in , - ^TT -^TT 0 - w2 + 6 < argX < + w1 - 6 6 > 1 1 0 w2 = tan" (h/p), -w-j^ = tan" [ (h - TT)/ p] n > 1 _1 0 w2 = tan" [(h + 7r)/p], -w1 •= tan (n/p) n j< and tan_1x denotes the principal value of the function. b> Results obtained by Erd'elyi Erdelyi derived the following expansion for |b| large [5]. If a,c and t are fixed, c ^ 0,-1,-2,... and 0 < |t| < 1, and if |b| - * such that - < arg tb < , then (13) gF^b; cj't) - gF-^b; c; on n 1 =[)_' ((a)n/(c)n n.')(bt) ][l + Odbl" )] n=0 and here the asymptotic formulas for the confluent hyper• geometric function of a large argument [31] give 10. (14) pF-^b- c; t) = e-i7ra i7ra a 1 = e- [r(c)/r(c-a)](bt)- [i-.-+ ocibtr )] + [r(c)/na)]ebt(bt)a-c[i + ocibtr1)] 1 •' "3 and similarly, if - ^ir < arg bt < ^TT , we have (15) 2Fi(a'b; c> *) i7ra a - e [r(c)/r(c-a)](bt)- [l + OClbtl"1)] + [r(c)/r(a)]ebt(bt)a-c[i + odbtr1)]. c. Results obtained by Hapaev We shall now refer to some papers by Hapaev [9> 10]. He derived an asymptotic expansion in terms of modified Bessel functions of large order for the confluent hypergeometric 1 function 1F1(a; cj t) as a -• », where |t| ~ laf and c is fixed. Provided these conditions are satisfied, he then derived [10] es (16) ^(a; -c; t) = (c)(at^"c)/2 JitaT1)^2 » ra=0 (m) =0 1 , ^n d n where Bmn = -,f (x,n) and f(x,n) - (l+(|)x+(|)x +.'.. ) , mn my x=0 Ix| < 1. 11. He then used this result to derive (17) 2F1(a,b;c;t)= es n 2n+m /2n (HcJ/Ra) y Bmn(t h nJ ) (r(a+in+2n-l)/r(c+nH-.2n-l)) n,m=D +2nj +2n 2t), 1F1(a-l+m c+m -l; _1 where the expansion is valid for |t| ~ |2a| and b and c fixed. In another one of his papers [11] he obtained asymptotic expansions of 2F1(a,b;c;t) and 1F1(a;c;t) where a=ae, C=Y£, a and y are constants and l -* eo, that is a/c is fixed and b is fixed. , .• Then for |t| < |c/a| he derives :L 1 2 (18) ^(aliynt) ~ e(a/Y)t(l+r1((aY"1t)2(a-1-Y" )2" K^- (...)+..} 1 _1 0,1,2, provided that |kt +Yl > |fcfcT 1, where k > 2n,n = Similarly, for Ikt^+Yl > Ifct"1! and |aY"1t| < 1 he derives (19) ^(a^cjt) ~ 1 (l-aY"1t)"b{l+t" (2-1b(b+l)(Y"1-o"1)(l-2t)(l-t)"2)+ *~2(...)+...}. d. Results obtained by Khottnerus Next we shall refer, to results derived by Khottnerus [15]/ For large positive values of r and the t-plane cut from 1 to 1+eoi he derived the following relation (20) gF^ajbjc+rjt) = 1 + (abt)r"1 + 0(r2). 12. The expansion is valid uniformly in the closed bounded region G: fRe(t) 2 1 + 6 > 6 fixed and 6 > 0, (21) < |t| < In. k fixed and k > 1 + 6; Im(t) _> G. Using the well known relation [5] c a b (22) gF^ajbjcjt) = (l-t) " " 2F1(c-a,c-b;c;t) valid for t in the t-plane cut from t=l to t=l+ooi he readily derived the following result. If Re c > min{Re(a),Re(b)} + 1 and t satisfies Re(t) > 1, Im(t) _> °J then for sufficiently large positive r c a b r -2 (23) ^(a+r^+r^c+rjt) = (1-t) - " - {1 + (c-*) + 0(r )} The expansion is valid uniformly provided that t lies in a closed bounded region of the t-plane cut from 1 to l+»i. e. Results obtained by Cherry and Sommerfeld First we shall consider the hypergeometric equation- occuring in gas-flow theory )y w §+ It is satisfied by X (t) = t^gF-^v-a^ v-b^; v+1;' t), _ n— where 2av = v + 6 + ^/v (1+26) + 6 , 2 2 28v = v + 6 - yv (l+26) .+ 6 , 13. and p = -(Y-1)"1; Y > 1, is the fixed adiabatic index. The turning point of equation (24) is readily given by t = (l+2p)_1. s The comparison equation 2 ' • / nr'\ dZ.ldZ. 2/n IN (25) —2 + - -j- + v (1 - -2)z = 0 dT T is satisfied by the Bessel function J,(VT), where T = (1-^—) (1-t )_1. s Then for v ~ « with |arg v| _< v - e and 0 _< t _< 1-e, e > 0, Cherry [1,2] derived 1 (26) Xv(t) ~ N(v/hv) /2[Jv(vT.)(1 + q2V-2+ ^v-4+ _} 1 5 + Tj^(vT)(q1v" + q^v" + ...)], 1 2 1/2 where tanh" ((1 - J ) ) - (1-T2)1/2 1 2 2 = tanh"1 ((l- t-)(i-t)1/2- t^tanh-^t^t^-tr ) / 1 4 s 2 1/4 N = (1-t) / - lp((l-T )/(i- |-)) and ' s hv = (r(av)r(l+v-bv))/(r(av-v)r(v)r(v+l)); the expansion is valid uniformly in t and arg v. To calculate the qn, n = 1,2,..., W = (1 - T2)Z, and we set w- ((l-|-)VV(l-t)1/*-l/2e)x,(t), 1 u-tarff ((1,-T2)1/2)- (I-T^ then 2 4 2 w = W(l + Q2v~ + Q^v"' + . .. ) - -^(QTV'" + Q5v"^ + . .. ) 2 1 2 where QX = q-^(l - T ) / 2 2 Q2 = q2 + T qx / (2(1-T )), 14. 2 2 % = % + T q3 /(2(1 - T )), ... and 2dQl ' , x 2Qg = ( 2Q3 = ( 2Q4 = ( 2 2 6 2 * = ((i-s ) / 4(i-tB) ){5?" - (i+6ts)r^(>4ts)F.- +(i-2tsxi-4ts);; S- = T.1/ ' 2 . Integration constants are found from the limiting form of the Xv(t) expansion (26) for t = T = 0, which then is equal to 1 v 2v 1 2 1/2 (27) (ev" ) r(v+l)((2Trhv 6 ) / 2TTV) ~ 1+Q1(0)v" +Q2(0)v" +. . ., where 6 = aa(l+a)_1"a and a = 1/2(1+2B)1//2 - 1/2 . We expand the left hand member of (27) by means of 1 ... (28) log(27rhv) ~ 2v log 6 + c-^v" + Q.j\T\ and Stirling series and equate coefficients to get ^(0), r = 1,2,.... Similar results were obtained by Lighthill [17] and Seifert [23]. ' The asymptotic behaviour of the hypergeometric function gF^-in, -ipw;l;t) occuring in wave mechanics has been investigated 15 by Sommerfeld [26]. For t = -4p(1-p)"2sin2(a/2), 0 < p < 1, he obtained 1 (29) -gF^-ivlpnajt) ~ e-7rpn(27ma)"1/2(iu;1+ e *^) where uQ = ((1-p)/2)(l+i cot(3/2)), cot (a/2) £ 0, f(u)= i log(up(l-u)"p(l-ut)"1) and -a=f"(u0); 5 If cot (a/2) - 0, then uQ = ( (1-p )/2) (l+i(7r-a)/2 + i(7r-a) /24 2 f(u0)=27rp - i(l+p)log((l+p)/(l-p))+(p/(l+p))i(7r-a /2 -p(l-p)(Tr-a)3(l+p)-2/6 + ... 1 2 -a = f"(u0) = 6p(Tr-a)(l-p)" (l+p)- + ... f. Results derived from Thome's paper The following result will now be derived from a paper written by Thorne [28]: 1 (30) (r(l-m))" 2F1(p+l,-p;l-m;(l-t)/2) ~ {r(p+l+m)/np+l-m)}1/2[(l-t)/(l+t)]m/2 v 2 2 x (42/(t -P ))V^(M(Y2;?z)>l/3^ Eg(z)f2A „ s=0 s=0 as p - » , where m = -ya = -(p+l/2)a, a is fixed and 0 < a < 1, 2 p. = Jl-a , 16. |z(t)3/2 = a cosh"1(|(t-2-l)"1/2)-cosh"1tB"1 and the functions F (z) and E (z) are given by the relations (40) s s and (43) determined later. The expansion is valid for the t-interval |t| < 1. Completely analogously to the foregoing, the following result was found for t > 1: 2 (31) 2F1((m+p+2)/2, (m+p+l)/2; p+ J ; t" ) ~ ,77". 1 1 7T2-P1 (2p+2)/((r(p+l)(n:p+m+l)r(p+l-m)) "5 ) e " -5 m 12 2 ± 1 +m+1 tP (i-t2y ^(4z/(t2_p2))7Ul(Y-3e- 3 z) Y"3X » _ 2 . 2 _ 2 . _ 5 « 2s 1 37ri 1 2s X }>s(z)Y" + e'^A^Y e" z)Y" XFs(z)Y" } s=0 s=0 as p-*», where m,B,Y* and z are defined by (30); the functions E v(z) and F (zv ) are identical to those of the former case as well, s ' s ' We shall now set out to prove the first result. Instead of an asymptotic expansion for P^(t) as p » , an asymptotic expansion for the associated Legendre function of the second kind Q^(t) was used to derive the second result. Asymptotic expan• sions for Pm(t) and <^(t), m = -(p+-|)a, as p->» can be obtained by a method employed by Thorne [28]. The associated Legendre equation (1 2 i\ d$y + (P(P 2 2 1 (32) -t ) -2t dt +D - m (l-t )- )y = 0 dt has a fundamental system of solutions consisting of P^(t) and 17. Q^t). If we use Hobson's definitions of these functions then they are single valued analytic in the t-plane cut along the real axis from 1 to -a>, and are real when t is real and t > 1. For z = x, where -1 < x < 1, the fundamental solutions of (32) are taken as Pp(x) and Qp^x) defined by (33) Pp(x) = e~ * P^(x + i.o) and m m7Ti tl m7ri (34) 2e Q*(x) = e^Q^x + i.o) + e ^ ^(x-i.o), where f(x + i.o) =. lim f(x + i.e), e>0. e 0 The functions Pp(x) and Qp(x) are generally known as Ferrers' functions and are real for x real and 0 < x < 1. Equation (32) is normalized by the transformation 2 1/2 (t -1) / y = Y, the resulting equation (35) = { p(p+l)(t2-l)-1+ (m2- l)(t2- I)'2} Y dt* is now satisfied by (t2-l)1//2Pm(t) and (t2-l )1//2Q^( t) If we set rn = -(p+1.2)a, 0 < a < 1 and a fixed, then for p = J1_a2 equation (35) reduces to (36) = {(t2- p2)(p+l/2)2(t2-l)-2-4-1(t2+3)(t2-l)-2} dt 2 1 which is now satisfied by (t -l) /2pa(p+l/2)^j &nd 2 1 2 2 (t -l) / ^(P+l/ )(t). It is possible now, according to Olver [19,20] or Thorne [27], to obtain asymptotic expansions of Pp(t) and 0^(t), which are valid uniformly with respect to t, as p -» » , for t 18. lying in a domain D^., say, in which the points t=l and t = 6 + i.o are interior points and which extends to infinity. The coefficient of ((t2- B2)(p+l/2)2(t2- l)"2)Y has double poles at the regular singularities t = +1 and turning points at the simple zeros t = +B. To obtain expansions valid at the turning point t=B+i.o we make the t-z transformation [19?20] (37) H = -(t2- D(t2- P2)-VV/2 , X = (^)_1/2Y and 1 .t - - -|z^2 = -j (s2- B2)(s2- l)_1ds = acosh~1a6"1(t"2-l) "2"-cosh"1tB""1, 8 where the lower limit of the integral is p + i.o. Then X(z) satisfies the equation (38) £\ = {(p+l/2)2z + f,(z) }X, dz x 2 _1 2 2 2 3 2 2 4 where f^z') = ( |5)z + 4 z(t - l)(t - B )~ {t (4a - l)+(l-a )} The comparison equation 2 (39) ^-i » (p+l/2)2zX -dz^ is satisfied by the Airy functions Ai(z) and Bi(z). Thorne [28] now showed that if (40) EQ(z) = 1, 1/2 1/2 P8(z) = |z- jV {f1(r) Efl(r) - E»(r)} dr, and W?> - - iFk^ + f fi where the sequence of a s _> 0, are integration constants, then the following results will be obtained: (41) e ^ Pm(t) ~ {r(p+m+l)/r(p-m+l)} ^(4z(t2- B2)"1) J x 2 _ 1 » 2 _ 5 » 1 3 2S 3 2S x {Ai(v z)Y" £ Es(z)Y" + Ai(Y z) Y" "3 YFs(Z)Y- ] s=0 s=0 and 1 1 .2nlri+ £i (42) e"2 * 2 - 27Ti - 1 08 - -TTi 2 - 27Ti - 5 °° 1 1 1 2s 3 1 3 2S x{Ai(Y e" z)Y ^Es(Z)Y" + e Ai^e z)Y" )\(Z)Y~ } s=0 s=0 where the integration constants a are specified by the relation s ' 0, 1 2 28 23 1 m (43) l(%y' + PgY" " ) ~J-m/tor (V2) r(-m) {rCp+nH-D/TCp-iw-l)}" s=0 where R = vV(l+v)*"1"v, v = -5(a-1 -1). The expansions (4l) and (42) are valid throughout the t-plane cut from +1 to -» except for a pear-shaped"domain surrounding the singularity t=-l and a strip lying immediately below the real t-axis for which |Ret| < B+6, 0 > Im t > -6, 8 > 0. In both these regions asymptotic expansions can be obtained by use of the continuation formulae (8) and (9). The asymptotic expansion of (p+l-m, -p-m; 1-m; (l-t)/2) is now an immediate consequence of (4l), (6) and relation (33), where we have to take the + sign. The expansion of o 2F1((m+p+2)/2, (m+p+l)/2; p+3/2; t ) follows immediately from the asymptotic expansion (42) and the relation (7)- 20. CHAPTER II THE NORMALIZED HYPERGEOMETRIC DIFFERENTIAL EQUATION The hypergeometric equation (1) t(l-t) A_| + [c - (a+b+ljtjg*- - abx = 0 dt is normalized by setting 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)/4t2(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 is a large real parameter. Then the turning points ti, i = 1,2, are the roots of the quadratic equation (3) t2 + [2c(2a-l) - 4a2]t + c(2-c) = 0 For a large 2 2 (4) t]_ ~ c(2-c)/4a and t2 ~ 4a . If we now make the substitution t = c(2-c)z/4a , then 21. equation (2) for a=b is transformed into 2 (5) + 4a4 f(c(2-c)(l-z) + f(z))/g(z) y = o, dz I ) where f(z) = (c2(l-c)(2a-l)z/2a2) + (c2(2-c)2z2/l6a4) 2 2 ? and g(z) = z (4a - c(2-c)z)~. For z bounded f(z) = ©(a"1) as a - Equation (5) can be written in Thome's form 2 2 2 2 (6) 1-| + ((2a ) z- (l-z)p (z) + z qi(Z)| y = 0 , dz 1 ) 2 2 where p-^z) = c(2-c)/(4a - c(2-c)z) , p-j_(z) does not vanish and is regular for z < 4a /c(2-c); furthermore 2 2 qx(z) = 4a f(z) / (4a2 - c(z-c)'z) , _2 for z bounded q-j_(z) = 0(a ) as a -* » . If we set y = c(2-c) and u = 2a, then 2 2 2 2 2 2 (7) ^-g + ((Y/4)(l-z)/z + [(Y/2) (1-Z)/U Z][(2-YZ/U )/(1-YZ/U ) dz ( 2") + q]_(z)/z jy = 0. Now Y(1-Z)/4Z2 is not the dominant term, the turning point t-^ and the singularity t, = 0 coalesce for _a large and z bounded. Numerical analytic methods can determine the values for 2F]_(a,a jc > t) in the region in which the turning point and the singularity t=0 coalesce. Furthermore, ^F-^a, a; c; t) probably is the simplest function demonstrating this behaviour. 22. ' CHAPTER III _1 THE ASYMPTOTIC BEHAVIOUR OF 2F-L(a,a;c;t) AND 2F1(a,a+l-c;l;t ) A solution of the normalized hypergeometric equation 2^ (1) ^-g + {(At2 + Bt + C) / 4t2(l-t)2) y = 0, ' r" ' ) p where A = l-(a-b) , B = 2c(a+b-l) - 4ab, and C =. c(2-c), is (2) |- ^(a^bjcjtJt^Cl-t)^^1-0^2, 0 < t < 1, and \ ^(a^a+l-cra-b+l^-^t^^^^Ct-l)^^1-0^ t > 1. Let us consider the case that a = b is a large real parameter. Then equation (l) has the form (3) ^ = [up(t) + q(t)ly, dt where p(t) = l/(4t(1-t)2), q(t) = (c(c-2)-t2)/(4t2(l-t)2) and u = 4a2 - 2c(2a-l). New variables Z and Y are introduced by the relations f1*'20! ts. Z = [(1/2)" [p(s)]1/2ds]2 J t , dt j-1/2 1 \ dz ; y * 23. Therefore, we readily obtain 1/2 t CO = (1A)|" = (l^tanh"1^) 1 "o sx/*(l-s) = (-l/4)ln((l- ft)/(l-hrt)), t < 1, and 1/2 » 1 (5) z2 = (-lA)f -TTp^ - (l/2)coth" (/t) t s ' (1-s) = (-i/4)m((./tu-i)/(./t+i)), t>i . Equation (3) is then transformed into (6) = juz"1 + YZ"2 + h(z)z"1| Y , where Y = -3/l6 and for z = z2 h(z). = (-3/4)-(l/4)cosech2(2z1/2)+(c(c-2)+3/4)tanh2(2z1/2). For t > 1 relation (5) implies that t = coth (2z2 ). This leads to the following results for t > 1 and z s z2: t = (||) = 2z"1/2coth(-2z1/2)cosech2(2z1/2), q[t(z)] = (l/4)[c(c-2)sinh4(2z1/2)tanh4(2z1/2) - sinh4(2zl/2)] and therefore (t)2q[t(z)] = [c(c-2)z"1][tanh2(2z1/2)]-[z~1coth2(2z1/2)], 2 yz' + h(z)z_1 =ti, (t"1/2)+(t)1/2q[t(z)] dz = (-3/l6)z"2+ z'H (-3/4)-(l/4)cosech2(2z1/2) + (c(c-2) + (3/4))tanh2(2z1/2)]. 24. Therefore, y = -3/l6 and if we set 2 u = 1 + 4Y = 1/4, then without loss of generality we can take p. = 1/2. If we now put = u + c(c-2) = (2a-c) then the basic equation (6) becomes 1 2 2 1 (7) ^| = (u^" + ((p - l)/4)z" + h1(z)Z" )Y, dz where h-^z) = (-l/4)cosech2(2z1/2)-(c(c-2)+3/4)sech2(2z1'/2). A pair of linear independent solutions of the comparison equation, that is equation (7) with h-^z) = 0, are 1/2 1/2 Y-L = z K (2(ulZ) )and I and K are modified Bessel functions of order (jt . 2 2 For convenience, if we now replace z, ^ by z and u.j/4 respectively, then equation (7) becomes (8) ^| = z"1 || + iju2 + Hz"2 + f(z)| Y, 1 where ^ = -2(2a-c) and for z s z2 = (l/2)coth~ (,/t) 2 2 2 f(z) = 4h1(z ) = -cosech (2z) - 4((c(c-2) + 3/4) sech (2z)). The function f(z) is an even function of z and is regular in an unbounded simply-connected open domain D, actually f(z) = 0(|z|"1_a) as |z| - • , where a is constant and a > 0. 25. Let D' be any simply-connected domain lying wholly in D, the boundaries of which do not intersect the boundaries of D. Let 6 > 0 be an arbitrary real point in the sector |arg z| < TT/2, then the domain comprises those points z of D' which can be joined to 6 by a contour which lies in D1 and does not cross either the imaginary axis or the line through z parallel to the imaginary axis. Next suppose d to be an arbitrary point of the sector |arg z| < TT/2, which may be at » , and e to be an arbitrary positive number. Then D2 consists of those points z of D' for which |arg z| _< J>ir/2, Re z < Re d and a contour can be found joining z and d which satisfies the following conditions [19,20]5 (i) it lies in D'; (ii) it lies wholly to the right of the line through z parallel to the imaginary axis; (iii) it does not cross the negative imaginary axis if TT/2 _< arg z _< J>TT/2. and does not cross the positive imaginary axis if -37r/2 _< arg z _< -TT/2; (iv) it lies outside the circle |r| =, e|z|. A pair of linear independent solutions of the comparison equation (9) are now Y-L = zl (u-jz) and Y = zK u z 2 M( l )* The basic equation (8) has solutions Y-^(z) and Y2(z) such that for Re \x > 0 26. (i) if z lies in Dx, M-l 2s 2M Yx(z) = zl^^z) V As(z)u" + 0(u" ) U=0 fM-1-l1 2s 2M £ Bs(z)u' + (z/(l+|z|)0(u- ) ls=0 as u^ -* <», uniform with respect to z, (ii) if z lies in B>>, M-l \ 2s 2M Y2(z) = zyulZ) I As(z)u" + 0(u" ) ^s=0 M-l 1 2s 2M - (z^ )^!^)! £ Bs(z)u" + (z/(l+|z|) 0(u" ) (s=0 i as u^ - <*>, uniform with respect tc.z. Since f(z) = 0(|z|~1_a) as |z| - oo and a > 0, the asymptotic expansion is valid for z tending to infinity. The sequences of functions A (z) and B (z) are given "by the relations AG(z) = 1 1 2Bs(z) = -Ai(z) + J^f(t)As(t) - (2p+l)t- As(t)j dt, 1 2As+1(z) = (2n+l)Z- Bs(z) - Bl(z) + J* |f (t )Bg(t )jdt + Cfi , where C6 is a constant and 6 > 0 is fixed. Asymptotic forms, that is M = 1, of these solutions are (i) if z lies in L^, 2 Y1(z) = zl(j(u1z) ^1 + 0(u" )^ + au 1 I (u z) B (z) + Z 1+ Z 2 ( i ) n+i i ^ o ( /( I D) °K ^ 27. as - », uniform with respect to z in D^, (ii) if z lies in Dg, 2 Y2(z) = zK^(u1z) | 1 + 0(u" ) 2 - (zu-^K^^z) |B0(Z) + (z/(l+|z|)) 0(u~ ) as u-j^ - <*>, uniform with respect to z in Dg. / The function Bq2(Z) S BQ(z) is now given by the integral Z Bo2(z) = (1/2) J f(s)ds, 6 > 0 6 z = (-1/2) J cosech2(2s) 6 + 4(c(c-2)+3/4)sech2(2s) ds = (1/4)coth(2z)+(c(c-2)+3/4)tanh(-2z)+C26, where C2& is a constant and 6 > 0 is fixed. Since any solution is linearly expressible in terms of two linear independent solutions and since' 1 Y(t[Zl]) = (^)- /2p(a,a;C;t)tc/2(1.t)(2a+l.c)/2 x dz£ 1 0 < t < 1, is a solution of equation (8), it follows that (10) Y(t[z1]) = c1(u1)Y1(z1) + c2(u1)Y2(z1), and similarly, since 1 2 1 c 2a 2 2a+1 c 2 Y(t[z?]) - ( ^g)" / 2PT(a,a+l-c;l;t- )t( - )/ (t-l)( - )/ , dzj • 1 t > 1, is a solution of (8), it implies that 28. (11) Y(t[z2]) = d1(u1)Y1(z2) + d2(u1)Y2(z2). Now, fix and let t -» 1, that is z-^ and z2 approach infinity. It is well known [30] that 1/2 z +1 2 iri z I (z) ~ (27rz)- (e + e±^ / ) e" ), |arg z| < 3^/2 - e, e > 0, as |z| -» » uniformly with respect to arg z; the upper sign applies to the range -TT/2 + e _< arg z _< 3T/2 - € and the lower to the range -3t/2 + e _< arg z _< TT/2 - €. Also, 1/2 z Kfj(z) ~ (V2z) e" , |arg z| _< 3ff/2 - e uniformly with respect to arg z as |z| -» ». Near the regular singularity t=l [p(t)]1/2 ~ l/(2(l-t)) and therefore according to equations (4) and (5) e-4zx ~ 1-t , 0 < t < 1 e ^z2 ~ t-1 , t > 1. Applying these estimates to the solutions (2) for a=h we get (12) y(t[ ]) ^Xl ^D 2 2 Zl -^(a.ajc -e ^te-Sjt ^)/ 2(2a+i c)z ~ (r(e)r 2 4 2 y(t[z2]) ~ ^(a^+LcjUf^l+e^^li^ ^/^ ^)^ ) ~ (r(c-2a)/(r(l-a)r(c-a)))e-2(2a+1-c)z2 as t 1+ and -a is large. Letting t -» 1+ in equation (11) and using the approxi• mations for the modified Bessel functions as |zP| - », we readily see that u 0 d2( i) = and since u^ is large and Bq2(Z2) remains bounded as | j -» • /2Y d u d 1 I z u 2 z 1 2 l( l)(- |- ) ( 2) ~ e( l- ) 2(7ru1)- / d1(u1) , dz2 as |z2| -» OB , | arg z2| <_ 3T/2 - e, € > 0, uniformly in arg z2 Prom relation (12) we then find that 1 2 (13) d1(u1[a]) = (r(c>2a)/(r(l-a)r(c-a)))(2(c-2a)7r) / . The constant d^u-Ja]) is well defined for Re(c-2a) > 0 since the inverse of the gamma function is entire. Because 1 pMa.a+l-Cilit- ) = (-SS-)l/2t(2a-c)/2(t_l)(c-2a-l)/2 x x dZ| x Y1(z2)d1(u1[a]), we then have derived the following result. For a and c real, c-2a > 0 and the t-interval 1 < t _< T2 < « (14) gF^a+l-cjljt-1) - B(a)t(2a-c+1/2)/2(t-l)(c"2a)/2x Z /2 u z [l+ U 2 ] 2 /2u 1 I (u Z [B z x| 2 Il/2( i 2) °( l ) " ( 2 l ) 3/2 l 2^ o2( 2) + z2/(l+|z2| )0(u^)] unformly with respect to t as *•• oo, where = -2(2a-c), c fixed, 1 z2 = (l^coth" ^), 1/2 B(a) = (r(c-2a)/r(l-a)r(c-a))(2u1Tr) , 1/2 1/2 Bo2(z2) = (lA)t - (c(c-2) +3A)t- + Gm , x2 T2< » and fixed. Finally, if We now consider the interval 0 < 1^ _< t < 1, 1/2 -1 then = (l/2)tanh~ (./t). Analogous to results obtained 2 //2 2 //2 before, we now have t s ^| = 8z^ tanh(2z^ )sec h(2zJ ) 1 1/2 2 + h^z^zl t ^ 1/2 and Yz~ = t" + t^q(t[Zl]) dz 1 - -(Vl6)z"2 + z^ih^) + c(2-c)), 2 2 2 //2 where h^z-^ = (l/4)sech (2zJ/ ) + (c(c-2) + 3/4)cosech (2z^ ). Therefore, as before y = -(3/l6) such that n = 1/2 and 2 = u + c(2-c) = (2a-c) ; furthermore, if again we replace z^ 2 2 and by z^ and u.j/4 respectively, then z 1 ± = (l^Jtanh" ^/^) , u^. = -2(2a-c) and 2 2 2 f(Zl) = 4h1(z ) = sech (2z1)+(4c(p^2)+3)cosech (2z1). It is seen that f(Z;L) = 0(|Zl| ) as |Zl| - », a > 0. As before, the reason for modifying the large parameter is to 1_a make, f(Zl) = 0(|Zl|~ ) as |Zl| - », a > 0, such that the solutions Y1(z1) and Y2(Zl) of equations- (8) corresponding to z = z-j^ have asymptotic expansions valid as z^ tends to infinity. 3T. This also makes it therefore possible to compare relations (11) and (12) near t=l and using well known estimates for the modi• fied Bessel functions as |z^| tends to infinity, the constants c^(u1) and c2(u1) can be obtained. To find these constants, we first have to obtain the B z ls now function B^^^. 0i( i) given by the integral Bol/zl) = (V2)JZl f(s)ds = 6 (l/4)tanh(2Z;L) - (c(c-2) + 3/4) coth(2Zl) + C16 , 6 > 0 and fixed. Letting t - 1~ in relation (10) and keeping u-^ fixed we see that c2(u1) = 0, 1/2 ^("I TJ! ) z u 2 z 1 2 K V l> ~ e( l- ) l(7ru1)- / c1(u1) as \Zl\ - . , | arg I _< 37r/2 - e, e > 0, uniformly in arg z^; since is a large parameter and BQ^(z1) remains bounded as \z-^\ - « . Prom the estimates (12) we now readily get 2 1 2 c1(u1[a]) = (r(c)r(c-2a)/(r(c-a)) )(2(c-2a)7r) / . The constant c2(u1[a]) is well defined for Re c > 0 and Re(c-2a) > 0, since the inverse of the gamma function is entire. Because # )V2 -=/2 -2 )/2 ) (^) 2Fl(a,a;c;t) = 2 t (1.t)(c 8-l Y (! C dz-^ we then have derived the following-result. 32. For a and c real, c > 0, c-2a > 0 and the t-lnterval 0 < T1 < t < 1 (15) ^(a^cjt) = A(a)t(-c+^/2(l-t)(C-2a)/2 x x[zY2 V2(uizi)[i + °(ui2)] z /2u 1 I U Z )[B z + 2 - ( l l ) 3/2( l l ol( l) z1/(l+iz1l)0(u- )]| uniformly with respect to t as u^ -» where u.^ = (l/2)tanh~1,/i- , 2 1 2 A(a) = (r(c)r(c-2a)/(r(c-a)) )(2u17r) / , 1/2 1/2 Bol(Zl) = (l/4')t -(c(c-2)+3A)t- + CT , T-j^ > 0 and fixed. Completely analogously to the foregoing complete asymp• totic expansions can be obtained for 2F^(a,a+l-cjl;t_1) and 2F^(a,a;c;t) as -a -• » , using the complete asymptotic expansions of Y-^(z) and Y2(z) instead of their asymptotic forms. • By employing the alternative forms of the hypergeometric function and the expressions for the analytic continuation of the hypergeometric function, it is possible to deduce various other asymptotic expansions for the hypergeometric function. For instance, for 0 < t < 1 c 2a 2F1(a,a;c;t) = (l-t)( " ^2Fa(c-a,c-ajc;t); so we have the following result. For a and c real, c > 0, c-2a > 0 and the t-interval 0 < Tx < t < 1 c+1 2 2 2a c 2 (16) 2F1(c-a,c-a;c;t)-A(a)t(- / )/ (l-t)( - )/ x 1/2 2 x |z I1/2(ulZl)[l + 0(uj )] 1/2 1 2 - (z u- )lV2(u1z1)[Bol(z1) + z1/(l+|z1|) 0(u~ )]j uniformly with respect to t as •* » , where u^,z^,A(a) and BQl(z1) are given by (15). As an application of the asymptotic form (16) let us consider the complex z-plane and suppose that F(z)=U(z)+iV(z) is holomorphic in |z| < R and continuous in |z| _< R. Furthermore, for z = re1"^ let us denote Max F(re1-e) by 0 Provided F(z) ^ M(R;F) we then have from the Maximum modulus principle the inequality (18) |Fn(z)| < %L- M(R;P) J -M} , r < R. |z|=R |t-z n+1 But for I z j = r < R the integral I Idt I .., can be 2T? J jz|=R |t-z|n+1 2 worked out [11] and is found to be 2F]L(^(n+l),^(n+l);l; (-yr) ). We therefore have the following result. 2 For jz| = r < R and o < Tx _< (£) < 1 , 34, n 2 (19) |F (z)| < n< M(R;F)2F1(^(n+l)^(n+l);l;(|) ), 2 here 2F1(^(n+l)^(n+l);1; (-|) ) for large n is given by (16), where ux = 2w, c = 1 _1 zx = (l/2)tanh (|) , A[a(n)] =r(n)(r(-|(n+l)))-2 (47m)1/2 and 1 1 B0l(Zl) = (l/4)(rR" + Rr" ) + CT , T1 > 0 and fixed. A simplification of n.'A(n) is obtained by using the duplication formula of the gamma function (20) TT1/2 (2n) = 22n_1 T(n) T(n+l/2). It then is found that (21) n.»A[a(n)] = 22n+1(F(^ + 1) )2(n7r)"1/2 . If we now expand the gamma function in a Stirling series, we get for n large (22) n»A[a(n)] ~2(2n)'ne"n(mr)1/2 or (23) A[a(n)] ^2(n+1/2) . Therefore, in view of the approximation formulae of modified Bessel functions- for large values of the variable we can restate relation (19) as follows: -1 2 for |z| = r < R and o < Tx _< (rR ) < 1 , 35. (24) |Fn(z)| < nJ M(RjP) pF^n+l )y|(n+l); l^)2), here for n large nJ^^n+D^Cn+l;!;^)2) ~ 2(2n)ne-n(n7r)1/2 x 1 1 2 1 2 n 2 2 x (rR- )- / (l-(rR- ) )- / zJ/ I1/2(2nz1) = ^(2n)ne"n(rR'1)"1/2(l-(rR"1)2)_n/2sinh(n tanh"1(rR-1)), 1//2 1 2 since z I-L/,2(2nz1) = (sinh(2nz1)) (2nir)" / 1 -1 and z1 = (l/2)tanh" (rR ). A further simplification is obtained by noting that 1 1 1+rR -1 _1 tanh" (rR" ) = \ ln( 1), rR < 1, and therefore for rR < l-rR"x 1 2 1 2 sinh(n tanh-^rR" )) = {±+I*±)*/ . (l^ n/2 ^ go that 1-rR l+rR--1 in the estimate (24) for n large, |z| = r < R and 1 2 0 < Ti S. (rR ) < 1 we now have 2 (25) nJ2F1(-^(n+l)^(n+l);l;(|) ) ~ (l/2)./2 (2n)ne-n(r"1R)1/2((l-rR"1)"n - (1 + rR"1)'11). The asymptotic form (25) yields a better estimate then |Fn(z)| _< 2nJ R(R-r )"n_1M(R;F), valid for |z|=r, 0 < r < R. This relation follows readily from (l8) upon noting that (R-r)""n""1 is an upper bound for the integrand. 36. REFERENCES Cherry, T.M., Uniform asymptotic formulae for functions with transition points. Trans. Amer. Math. Soc. 68, 1950, 224-257. Cherry, T.M., Asymptotic expansion for the hypergeometric functions occuring in gas-flow theory. Proc. Roy. Soc. London, Ser. A 202, 1950, 507-522. Coddington, E.A. and Levinson N., Theory of ordinary differential equations. McGraw-Hill Book Co., New York, 1955- Dettman, J., The solution of a second order linear dif• ferential equation near a regular singular point. The Amer. Math. Monthly, 71 No. 4, April 1964, 378-385. Erdelyi, A. et al, Higher transcendental functions, vol. 1, McGraw-Hill Book Co., New York, 1953- Erde'lyi, A., Asymptotic expansions, Dover Publications, New York, 1956. Erdelyi, A., Asymptotic solutions of differential equations with transition points or singularities. Journal of Mathematical Physics, 1960a, 16-26. Erdelyi, A., Asymptotic solutions of ordinary linear differential equations. California Institute of Technology, 1961. Hapaev, M.M., Expansions of hypergeometric and degenerate hypergeometric functions in series of Bessel functions. Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him., 1958, No. 5, 17-22. (Russian) I 37- [10] Hapaev, M.M., Asymptotic expansions of hypergeometric and confluent hypergeometric functions. Izv. Vyss. Ucebn. Zaved. Matimatika, 1961, No.5 (24), 98-101. (Russian) [11] Hille, E., Analytic function theory. Vol. II. Ginn and Co., 1962. [12] Hobson, E.W., Spherical and ellipsoidal harmonics. Cambridge University Press, 1931. [13] Hochstadt, H., Differential equations, a modern approach. Holt, Rhinehorst and Wiston, New York, 1963. [14] Ince, E.L., Ordinary Differential Equations. Dover Publications, New York, 1956. [15] Khottnerus, U. J., Approximation formulae for generalized hypergeometric functions for large values of the parameters. J.B. Walters, Groningen, the Netherlands, i960. [16] Kummer, E.E., Uber die hypergeometrische Reihe. J. Reine Angew, Math. 15, 39-83, 1836, 127 -172. [17] Lighthill, M.J., The hodograph transformation in trans-sonic flow II. Auxiliary theorems on the hypergeometric functions Yn(Y)» Proc. Roy. Soc. London, Ser. A, 191, 1947, 341-351. [l8] MacRobert, T.M., Functions of a complex variable. 4th ed., MacMillan and Co. Ltd., London, England, 1954. [19] Olver, F.W.J., The asymptotic solution of linear differen• tial equations of the second order in a domain containing one transition point. Phil. Trans. Roy. Soc, A 249, 1956/57, 65-97. 38 • [20] Olver, F.W.J., Uniform asymptotic expansions of solutions of second order differential equations for large values of a parameter. Phil. Trans. Roy. Soc., A 250, 1958, 479-517. [21] Poole, E. G. C. , Introduction to the history of linear differential equations. Claridon Press, Oxford, England, 1936. [22] Rainville, E.D., Special functions. The MacMillan Co., New York, i960. [23] Seifert, H., Die hypergeometrischen Differential- gleichungen der Gasdynamik. Math. Ann. 120, 1947, 75-126. [24] Slater, L.J., Confluent hypergeometric functions, Cambridge University Press, i960. [25] Snow, C., The hypergeometric and Legendre functions with applications to integral equations of potential theory. Applied Series 19, U. S. Government Printing Office, Washington, D.C., 1952. [26] Sommerfeld, A., Atombau und Spektrallinien, 2nd Vol., Friederich Vieweg and Sohn, Brunswick, 1939, 800-806. [27] Thorne, R.C., The asymptotic solution of linear second order differential equations ir a domain containing a turning point and regular singularity. Techn. Report 12, Dept. Math.,Cal. Inst, of Techn., 1956. [28] Thorne, R.C., The asymptotic expansion of Legendre functions of large degree and order. Techn. Report 13, Dept. Math., Cal. Inst, of Techn., 1956. [29] Watson, G.N., Trans. Cambridge Phil. Soc. 22, 1918, 277-308. 39. [30] Watson, G.N., Theory of Bessel functions. Cambridge University Press, 1944. [31] Whittaker, E.T. and Watson, G.N., A course of modern analysis, 4th ed. Cambridge University Press, 1952.