UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Extension of Airy's equation Headley, Velmer Bentley 1966

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1966_A8 H4.pdf [ 2.04MB ]
Metadata
JSON: 831-1.0080544.json
JSON-LD: 831-1.0080544-ld.json
RDF/XML (Pretty): 831-1.0080544-rdf.xml
RDF/JSON: 831-1.0080544-rdf.json
Turtle: 831-1.0080544-turtle.txt
N-Triples: 831-1.0080544-rdf-ntriples.txt
Original Record: 831-1.0080544-source.json
Full Text
831-1.0080544-fulltext.txt
Citation
831-1.0080544.ris

Full Text

AN EXTENSION OP AIRT'S EQUATION by VELMER BENTLEY HEADLEY B,Sc.(Hons^, University of London, 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OP THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OP BRITISH COLUMBIA March, 1966 In p resen t i ng t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and s tudy . I f u r t h e r agree that permiss ion fo r e x t e n s i v e copy ing of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n permiss ion Department o f Mathematics The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date March 16, 1966. i i ABSTRACT ¥e consi d e r the d i f f e r e n t i a l equation d u/dz - z u = 0 ( z , u complex v a r i a b l e s ; n a p o s i t i v e i n t e g e r ) , which i s the s i m p l e s t second order o r d i n a r y d i f f e r e n t i a l equation with a t u r n i n g p o i n t of order n. The s o l u t i o n s which we study, h e r e i n c a l l e d A n f u n c t i o n s , are g e n e r a l i z a t i o n s of A i r y f u n c t i o n s . Most of t h e i r p r o p e r t i e s are then deduced from those of r e l a t e d B e s s e l f u n c t i o n s of order , but i n the d i s c u s s i o n of the zeros i n s e c t i o n 3 , r e s u l t s are deduced d i r e c t l y from the d i f f e r e n t i a l equation. I t i s easy to see t h a t the A n f u n c t i o n s are s p e c i a l cases of f u n c t i o n s s t u d i e d by T u r r i t t i n [9] . The r e l a t i o n of the former to B e s s e l f u n c t i o n s , however,, enables us to use methods not a v a i l a b l e i n [9] to o b t a i n uniform asymptotic r e p r e s e n t a t i o n s f o r l a r g e z. ¥e o b t a i n new r e s u l t s on the d i s t r i b u t i o n of the zeros which extend a p r o p e r t y [ 6 ] of A i r y f u n c t i o n s , t h a t i s , of functions,, to a l l p o s i t i v e i n t e g e r s n. A s i m i l a r remark a p p l i e s to bounds [ 8 ] f o r A i r y f u n c t i o n s and t h e i r r e c i p r o c a l s . i i i TABLE OP CONTENTS Page Int r o d u c t i o n 1 1. The d i f f e r e n t i a l equation 3 2. Asymptotic behaviour 11 3. Zeros of the A functions 14 n 4. Bounds f o r A functions 2 9 n 5. Bi b l i o g r a p h y 34 i v LIST OP FIGURES Page 1. Sectors containing zeros of A ^ ( z ) , A^(z) 21 2. Sectors containing zeros of B ^ ( z ) , B^(z) 21 3. The closed path OABCDEO f o r n = 2, k = 1 22 V ACKNOWLEDGEMENT I wish to express my gr a t i t u d e to Professor C.A. Swanson f o r suggesting the t o p i c and f o r h i s help and encouragement throughout the subsequent i n v e s t i g a t i o n s . I also wish to thank Professor E.D. Rogak f o r h i s u s e f u l c r i t i c i s m s of the d r a f t form of the t h e s i s . The f i n a n c i a l support of the U n i v e r s i t y of B r i t i s h Columbia and the Na t i o n a l Research Council i s g r a t e f u l l y acknowl-edged. , AN EXTENSION OP AIRI'S EQUATION INTRODUCTION We s h a l l define a c l a s s of f u n c t i o n s , the A f u n c t i o n s , ' n ' •which are s o l u t i o n s of the d i f f e r e n t i a l equation d^u/dz^ - z n u = 0, (z, u complex variables? n a p o s i t i v e i n t e g e r ) . This i s the simplest example of a second order ordinary d i f f e r e n -t i a l equation -with a t u r n i n g p o i n t of order n. The d i f f e r e n t i a l equation has no f i n i t e s i n g u l a r i t y and an i r r e g u l a r s i n g u l a r i t y of f i n i t e rank at i n f i n i t y . The s o l u t i o n s to be considered are r e a l on the r e a l a x i s and a n a l y t i c i n the f i n i t e complex plane; one of them tends to zero as z approaches i n f i n i t y i n a sector containing the p o s i t i v e r e a l a x i s . Their p r o p e r t i e s can be deduced d i r e c t l y from the d i f f e r e n t i a l equation, as i s done i n the d i s c u s s i o n of the zeros i n s e c t i o n 3, or from the p r o p e r t i e s of r e l a t e d Bessel functions of order l/(n+ 2 ) . The A r functions are e s s e n t i a l l y s p e c i a l cases of func-t i o n s studied by T u r r i t t i n [ 9 ] , who obtained uniform asymptotic representations f o r large z. We derive f o r A n functions the analogues of his representations by using known pr o p e r t i e s of Bessel f u n c t i o n s . In a d d i t i o n , we obtain new r e s u l t s on the d i s t r i b u t i o n of the zeros. These theorems extend a property [6] of A i r y f u n c t i o n s , that i s , of A^ f u n c t i o n s , to a l l p o s i t i v e integers n. The bounds we obtain i n s e c t i o n 4 f o r A functions 2 . i n zero-free regions present, i n e x p l i c i t form, r e s u l t s i m p l i c i t i n T u r r i t - t i n ' s work. The m o d i f i c a t i o n s made i n regions contain-i n g zeros are new f o r values of n / 1. These r e s u l t s also extend known pr o p e r t i e s [ 8 ] of f u n c t i o n s . Although we s h a l l not do t h i s here, the p r o p e r t i e s of A Q functions obtained below may be used to obtain uniform asympto-t i c expansions of the s o l u t i o n s of a more general second order equation with a t u r n i n g point of order n f o r large values of a ( r e a l or complex) parameter. In formulating a p r e c i s e r e s u l t the main problem would be to f i n d the most general domain of v a l i d i t y . 3. 1. The d i f f e r e n t i a l e q u a t i o n C o n s i d e r t h e s e c o n d 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 (1.1) - z n u = 0 dz . where z and u a r e c o m p l e x v a r i a b l e s and n i s a n i n t e g e r > -2. The p o i n t a t i n f i n i t y i s an i r r e g u l a r s i n g u l a r i t y o f r a n k •|,(n+2) f o r n e v e n , ^(n+3) f o r n odd, and t h e o r i g i n i s a t r a n s i t i o n p o i n t o f o r d e r n. A l l o t h e r p o i n t s a r e o r d i n a r y p o i n t s * LEMMA 1; A f u n d a m e n t a l s e t o f s o l u t i o n s o f ( l o l ) i s g i v e n by 1/ ~ (n+2)/o -, where ^ V ( C ) i s t h e m o d i f i e d B e s s e l f u n c t i o n o f t h e f i r s t k i n d , o f o r d e r v. PROOF; I n [lO,p„ 9 6 ( 5 ) ] r e p l a c e v b y u , C by z, c by 1, t h e n s e t n = 2q - 2 t o o b t a i n u + ( z ) = _2_ ( n + 2 ) A n+2 z 1/ h+2 - n+2 (n £ -2) I g n o r i n g a m u l t i p l i c a t i v e c o n s t a n t , we have /^2 T v 2 2 \ u+ 2 = z x+ - L - ( S+2 z } - z - n+2 A l t e r n a t i v e l y , t h e t r a n s f o r m a t i o n (1.3) T. = o)(C), u = t ep'(C)] y(C) sends the d i f f e r e n t i a l equation ( l . l ) i n t o 1= 0, (1.4) . d % +• [ - .p 'V + a£ d ! i Choosing cp so that (1.5) c p , 2 c p n = 1, dz cp = ( S ± 2 c ) n+2 t = cp V 2 we reduce equation (1.4) to 2 - i + t - ( nT2 ) 0. This may be w r i t t e n i n the form ? 1/1 1/ 1 + (n+2) 2C 2 - V 2 C y = o and thus possesses l i n e a r l y independent s o l u t i o n s [ l O , p. 77] y + ( C ) = C" l + x (C). .. ~ n+2 By (1.3) and (1.5) i t now f o l l o w s t h a t (1.2) gives s o l u t i o n s of ( l . l ) . . This completes the proof of the lemma. We s h a l l consider, i n what f o l l o w s , only p o s i t i v e i n t e g r a l values of n, although (1.2) holds f o r n ^ -2. In the case n = -2 the d i f f e r e n t i a l equation ( l . l ) reduces to an E u l e r equation, with a fundamental set of s o l u t i o n s (z) = (1 i S)/2 DEFINITION: The A functions of the f i r s t and second k i n d are — — — — — n — _ — — — _ ^ _ _ _ _ — _ — given r e s p e c t i v e l y by (1.6) A n ( z ) = ^ Z V 2 [ I ^ _ ( C ) - I _ ^ ( C ) ] n+2 n+2 2 s i n ( - ~ ) y n+2 »»<«> - <sfe ' V 2 [ J .jj«> + 1_^<c> . n+2 n+2 r _ _2_ (n+2)/2 ^ " n+2 Z In the s p e c i a l case n = 1, these reduce to the A i r y functions of the f i r s t and second k i n d r e s p e c t i v e l y . LEMMA 2: A n ( z ) and ^ ( z ) are e n t i r e f u n c t i o n s , and are a  fundamental set of s o l u t i o n s of ( l . l ) . PROOF; The modified Bessel f u n c t i o n of the f i r s t k i n d has the expansion ^ ( k ) v + 2 k (1.7) I V ( C ) = >_, -2 k=0 k l r^v+k+1) Appealing to (1.6) we have (1.8) A n ( z ) = a ^ f z ) - K y n M where (1.9) a; 1 =. ( n + 2 ) ( n + 1 > / < n + 2 > T <|g§) , ^ = (n + 2 ) < n + 3 ' / ' n + 2 > P'<£§) , and (1.10) \ y ^ U ) = £ a k ~ ( n + 2 ) k z k=0 a f 1 = (n+l) (n+2) (2n+3) (2n+4) . „, {k(n+2)-l} k(n+2), k > 1, a =1 •K. .. 0 ^ k=0 b" 1 = (n+2)(n+3) ... {k(n+2)]{k(n+2)+l}, k > l , b = l . i t o S i m i l a r l y , (1.11) B ( z ) = (n+2) 7 2 a v ( z ) + (n+2) By (z). *x . xi i i n 2 1/ IA Prom (1.7) i t i s c l e a r that•z I j_ (C)» z I x (C) are e n t i r e n+2 ~ n+2 functions of z , so that A (z) and B (z) are also e n t i r e functions, n n Furthermore, (1.12) A (0) = a , ' A' (0) = - 6 B n(0) = (n+2) \ , B^ (0) = (n+2) ^ . Prom Abel's formula [3, Ch. 3 (6. 5 ) ] , i f u(z) and v(z) are s o l u -t i o n s of the d i f f e r e n t i a l equation (l„l), then the Wronskian ^ [u,v] of u(z) and v(z) i s a constant. I t f o l l o w s from (l.'12) that - V (1.13) ^ T [ A n ( z ) , B n ( z ) ] = |(n+2) , 2 f l i n ( ^ 5 ) > upon use of the formula r1 (p) r*(l-p) = cosec(p 'n ) , ' 0 < p < 1 . 7. Since the r i g h t hand side of (1.13) i s non-zero, i t f o l l o w s that A n ( z ) and B n ( z ) are l i n e a r l y independent. The lemma f o l l o w s at once. I t should be noted that, i f n i s a p o s i t i v e integer,, the  f u n c t i o n s u k ( z ) = A n ( w k z ) , w = e x p ( ~ p - ) , k = 0, ±1, ±2,... are a l s o s o l u t i o n s of ( l . l ) . PROOF; By (1.6), the f u n c t i o n z u k ( z ) i s a l i n e a r combination of the f u n c t i o n s I . ( _2_ wk(n+2)/2 ( n + 2 ) / 2 ) _ T ' ,__2_ k TT i (n+ 2 ) / 2 v + -L-K n+2 w z ' ~ i+ n+2 6 z ' ~ n+2 - n+2 = e x p ( ± S+2 J * *+ -L_ ( c ) • - n+2 Ve s h a l l now show t h a t V 2 (1.14) A n ( ~ z ) = f J__L.'^> + ], (n odd) n+2 n+2 A (-z) = (n+2) / B ( z ) , Vn even). PROOF; Let 3- = ^ ( e 1 7 i z . ) , ( n + 2 ) / 2 . I f n i s odd, l e t n = 2m + 1„ Then, by the c o n t i n u a t i o n formulae f o r I v(C)» [Watson, p. 75], we have, p u t t i n g v = , Iv(e) = I v ( C e m l T l + •-!-) = e m v , , 1 I v ( C e — ) _ . 3V 77" j . . ^ 3 '47 1 mVTil 5" T { r \ I " ^ r - ^ —• \ = e e ^ J v ^ » V 2 A R & ^ E < " ) = e 8. upon using the relations [lO, p. 77] between and I . Similarly, n+2 H e n c e - v V 2 V2 By replacing e ^ z successively by e^"*z, " ^ z, . ,.s we can v e r i f y the result i n sectors which together cover the z-plane. If n i s even. l e t n = 2m. Then I v (e) = I v ( C e i m + l ) , T i ) = e ( m + l ) v T r i I v ( C ) e 7 2 I ,(C) v Sim i l a r l y Hence z X/2 [I JO + I ,(£) ] n+2 - v v w v - v2 = (n+2) B (z) This establishes the result, 9. LEMMA 3: If k i s any i n t e g e r , then v2 (1.15) A n ( v k z ) = ^ [ I_V(C) - v \ ( C ) h 2 77 i 1 . v = e x p ^ r » v = n+r » 2 / k N ( n + 2 V 2  PROOF; Let \ = ^ ^ z ) • T h e n = I v ( c ) . S i m i l a r l y , Hence k„^2 +2 V a z [I „ (c) - wk 1(c)] . - n+2 L -v X b / v k S i m i l a r l y , we can prove the analogous formula f o r B n(w z ) . Any three s o l u t i o n s of the d i f f e r e n t i a l equation ( l . l ) are l i n e a r l y dependent, and i n p a r t i c u l a r (1.16) (w s- w r ) A n ( z ) + ( l - w S ) A n ( w r z ) - ( l - w r ) A n ( w S z ) = 0 where r and s are i n t e g e r s . This i s a consequence of the r e l a -t i o n s ( l . l 5 ) ; f o r i f we assume a l i n e a r r e l a t i o n between A ( z ) , n A n(w rz) and A n(w Sz) with undetermined c o e f f i c i e n t s , an appeal to the l i n e a r independence [lO, pp. 77(3)] of I V (C) and I V(C) 10. y i e l d s (1.16). I t can also be shown i n an analogous manner that (1,17) (w S- w r ) B n ( z ) + ( n + 2 ) 1 / 2 ( l + v r ) A n ( w S Z ) V2 - (n+2) ( l + w S ) A n ( w z ) = 0 where, as before, r and s are i n t e g e r s . In the s p e c i a l case f o r which s = 1, r = -1 and n = 1, the equation (1.17) reduces to the well-known r e l a t i o n [8, (4.5)], [6, p» 364] between Airy-functions of the f i r s t and second k i n d . 11. Asymptotic behaviour as z — ». I t i s known th a t as Q - ee, [10, p. 202 ( l ) , ( 2 ) , (3) j p. 199 (1), (3)] (2.1) K v(C) = (^) 1 > f e e ~ C [ l + 0 ( i ) ] , |arg cl < f° (2.2) Iy(C) = i ^ ) 1 / 2 ( e C [1 •+ 0{\) + e ~ C + ( v 4 2 ) 7 ; i [ l + 0(±)]J, - 2 < a r 8 > I V ( C ) = ( 2?c) / 2 { e C [ l + 0 ( i ) ] + e " C - ( v + 2 ) l T i [ l + 0 ( i ) ] ) , 37T ^ . /T - — < arg C < 2 » V 2 (2.3) J v ( C ) = (f^) [cos(C - lvir.,177) + 0 ( r 1 ) s i n ( c 4 v f r 4 i r ) ] ' l a r g Cl <"» . Each asymptotic form holds u n i f o r m l y with res p e c t to arg z i n any c l o s e d s e c t o r w i t h i n the s t a t e d open s e c t o r , LEMMA 4: The f o l l o w i n g asymptotic: forms are v a l i d as z -* «; (2.4) A n ( 2 ) = (n+2)~ 1 / 2 Tr~1/2sin(-JL-)z~n/4 e~Q[l+0(^)]9 | a r § z ! < nT2 - 1 / 2 - - n A (2.5) A n ( - z ) = 2(n+2) TT c o s t ^ q O z [cos(C-+ 0 ( C - 1 ) s i n ( C - f ) ] , 12. (2.6) B n(z) = M " V 2 z " % e C [ i + 0 ( i )], | a P g a | < J L , . (2.7) B a(- 8) = 2 tT 1 / 2 s i n ( ^ ) z " n / 4 t c o s ( C + f )+0( C " 1 ) e 1 I m C l ] , I arg z | < , n odd. Each asymptotic form holds uniformly with respect to arg z i n a n y closed sector within the stated open sector. PROOF; The form (2.4) follows from (1.6) by direct substitution o f ^ = n+2 a ^ n + 2 ^ 2 i n the asymptotic form ( 2 . l ) , whilst the form (2.6) follows from (1.6) and the forms (2,2) by noting that e^ dominates e ^ i n the sector Iarg Cl < ^ 0 To obtain (2.5) we use the r e l a t i o n (l„14) and the asymp-t o t i c form (2„3). Sim i l a r l y we derive the form (2<,7), provided we note that |sin(C + v §.- f ) -sin(C " v f - J ) l = U cos(C - | ) s i n ( ^ ) | < 2 e | l m C ! sin(v §) and / . vff 77 \ / ,. vTT 'tis. . , 77\ . / V ^ \ cos (C + -y - 4) - cos ( C - — - 4) = -2 sm(C - 4) s i n ( — ) = 2 cosVC + 4) s m ( — ; „ This completes the proof of the lemma. The asymptotic behaviour i n those sectors of the z-plane not covered by the forms (2.4) - (2.7) can be deduced from the forms (2.2) by making use of the relations (1.15). The resulting asymptotic forms w i l l be v a l i d i n one of the sectors 13. ( 2 k-i)ff < < (2k+3) rr n+2 a r g 2 n+2 (2k-3)TT ( 2 k + l ) f -n+2 a r g z n+2— where k i s any i n t e g e r . Each of the sectors has an angular width of 4'!'"/(n+2), and the angle between rays corresponding to neighbouring (n+2)-th roots of u n i t y i s 21</(n+2) ; thus the whole z-plane can be covered by these s e c t o r s . I f n i s even, we can deduce the asymptotic behaviour of A n ( - z ) from that of B n ( z ) by appealing to (1.14). A s i m i l a r remark a p p l i e s to B (-z). n The complete asymptotic expansions, v a l i d i n the appropri-ate s e c t o r s , can be deduced from those corresponding to the forms (2.l)-(2..3). ( 14. Zeros of f u n c t i o n s . THEOREM 1: The functions A ( z ) . A^U) have no r e a l zeros f o r n even, but f o r n odd these functions have an i n f i n i t e number of simple zeros on the negative r e a l a x i s with no f i n i t e accumulation p o i n t . Their non-real zeros a l l l i e i n the region S, , where k K S k i s the double sector 2k TT / (n+1) < I arg z l < (2k+l) TT/( n+2), and k runs over the integers i n the range 0 < 2k < n+1. PROOF; Since ( l . l ) i s a l i n e a r homogeneous equation, the zeros are simple w i t h no f i n i t e accumulation point [3, p. 208], [l,p.27-] I f z i s r e a l and n even. u"(z) and u(z) have the same sign Since A (0) > 0, l i m A (z) =0, l i m A (z) = + » , we see that z—* +°° z~* — 0 9 A n ( z ) , A ^ z ) can have no r e a l zeros i n t h i s case. I f z > 0 and n i s odd,, a s i m i l a r argument shows that there are no zeros ,in t h i s case. I f z < 0 and n i s odd, an easy a p p l i c a t i o n of the Sturm comparison theorem shows that any r e a l s o l u t i o n of ( l . l ) has an i n f i n i t e number of r e a l zeros. • • A , '" We now turn to the non-real zeros. Using the d i f f e r e n t i a l equation ( l . l ) we may v e r i f y that ' — ~ r — A (az) - a n + 2 z 1 1 A (az) = 0 d z 2 n n -4- A (bz) - b n + 2 z n A (bz) = 0 d z 2 n n where a and b are complex v a r i a b l e s . M u l t i p l i c a t i o n o f the f i r s t 15. equation by A n(bz) and the second by A n ( a z ) , subtracting and i n t e -grating from c to 1 gives (3.1) ( a n + 2 - b n + 2 ) z n A„(az)A (bz) dz n n c = [A n(bz) f _ A n(az) - A n(az) A ^ b z ) ] ' . c b n + 1 M u l t i p l i c a t i o n of the f i r s t equation by — • — A (bz) and the second n+1 a n by ,a' v ••; A (az) and subtracting gives b n , n+1 , ,n+l , 0 - h e V-. V-> k Vb2> - V V>*> k V->] n+1 n+1 , -(a . - b j f; A n(az) f - A (bz) dz n dz n which leads to (3.2) ( a n + 2 - b n + 2 ) f^A* (az) A' (bz) dz J n n c = [ a n + 1 A n(az) A^(bz) - b n + 1 A n(bz) A^(az)] CC where a prime denotes d i f f e r e n t i a t i o n with respect to the argument of the A functions, n Suppose now that a = re i s a non-real zero of -^(z) or A^(z). Let b denote i t s conjugate a? and c = 0. Then (3.1) and (3.2) become (3.3) \ z n A n(az)A n(az)dz = - n+2 V ° > A n ( 0 ) ' • ' o a - a C 1 n+1 - n+1 (3.4) i i ( a . ) i ; ( e . ) d » = - \ + 2 " ! n + 2 1^(0) ^ ( 0 ) . o a - a By the Schwarz r e f l e c t i o n p r i n c i p l e , the integrands become 16. and IA'(az)| r e s p e c t i v e l y , which are both non-negative f o r 0 < z < 1. Moreover, A (O)A'(0) < 0. Suppose without — n n l o s s of g e n e r a l i t y that •© i s in.the upper h a l f - p l a n e . Let be the sector (2k-l)7T /(n+2) < $ < 2kTT/(n+l), k=l ,2,. . . ,n/2 i f n i s even =l,2,...,(n-l)/2 i f n i s odd. Let T be the s e c t o r " ( 2 k +l)/(n+2) < € < 7T. that i s , T i s o max ' — ' ' o the sector (n+l)TT /(n+2) < 3 < ^ i f n i s even, and i s the sector nrr /(n+2) < S < IT i f n i s odd. I f G k S. , 0 < 2k < n+1, we s h a l l derive a c o n t r a d i c t i o n . I f 3 € T . n even, then (n+l)7T< (n+2)fc < (n+2)77 , and sin(n+2)0 < 0, which c o n t r a d i c t s (3.3). I f 6 f T^. n odd, then n'< e < (n+2)e, and we have the f o l l o w i n g cases! (a) n i < (n+2)Q < (n+l)^ whence sin(n+2 )e < 0. (n+l)77< (n+2)e < (n+2)7T , whence n""< (n+l) 27T/(n+2) < (n+l)#< ( n + l ) " , and therefore sin(n+l)S < 0. This c o n t r a d i c t s (3.4). If_JL£_I k> where (3.5) 0 < 2k < n+1, then ( 2 k - l ) r r < (n+2)$ < 2k" + 2klT /(n+l) = 2k T7 (n+2)/(n+l), and we have the f o l l o w i n g casess (a) (2k-l)fr < (n+2)S < 2k- , whence sin(n+2)G < 0. (b) 2kTT< (n+2)e < 2kT (n+2)/(n+l), whence 2kT (n+l)/(n+2) < (n+l)fc < 2k7T . 17. But 2k(n+l)/(n+2) = 2k - 2k/(n+2) > 2 k - l , by hypothesis (3.5), and t h e r e f o r e s i n ( n + l ) 6 < 0. I f 0 < e < 7f/( n+2). l e t c = ». The i n t e g r a l i n ( 3 . l ) e x i s t s , s i n c e A n ( z ) i s an e n t i r e f u n c t i o n with asymptotic r e p r e s e n t a t i o n (2.4). Then (3.1) gives ( a n + 2 - a n + 2 ) j " z111 A n ( a z ) | 2 d z = 0< Since z > 0, and | A n ( a z ) | i s not i d e n t i c a l l y zero f o r 1 < z < », n+2 — n+2 t h i s i m p l i e s that a - a = 0> that i s , at l e a s t one of - - - n+1 a = a, aw, ..., aw must hold. This i m p l i e s t h a t a i s r e a l or arg a - arg a i s a m u l t i p l e of 2' /(n+2). This c o n t r a d i c t s the hypothesis 0 < arg a < "/(n+2). Hence there are no zeros of A n ( z ) i n t h i s range. S i m i l a r l y , i f a i s a zero of ,A^(z) with 0 < arg a <77y(n+2), the i n t e g r a l i n (3.2) e x i s t s f o r c = «, since (n+l)/2 , , r r A^(z) = -M z K x_ v ( C ) , M = 2 i r - 1 ( n + 2 ) ~ 1 s i n ( ^ ) , v = J a n < ^ ^ ] _ _ v ^ ) ^as ^ e asymptotic r e p r e s e n t a t i o n (2.1). Then (3.2) gives f |A'(az)| 2dz = 0, which i s a c o n t r a d i c t i o n . Hence A^(z) has no zeros i n the s e c t o r | arg z| <^/(n+2). This completes the proof of the theorem. I t should be noted t h a t A (z) and A'(z) do have zeros i n S,. n n j£. 18, (k+ h m PROOF: Set C = Z e ^ . Then I V ( C ) = e ( k + l ) v 7 r i I V ( Z e " ^ ) = e ^ ^ J (Z), (- |*< arg Z <77) and Using the r e l a t i o n J _ V ( Z ) = Y V ( Z ) s i n v/T+ J y ( Z ) cos , where Y V ( Z ) i s Weber's Bessel f u n c t i o n of the second k i n d of order v, we see that I_ V ( C ) - I V ( C ) = e ( k + % > ^ i [ - l v ( z ) s i n v?T + J v ( Z ) c o s vrr] - e ( k + ^ » f ± Z) (- | < arg Z < /? ) This can be w r i t t e n i n the form J (Z) cos a' - Y (Z) s i n a' v v provided that s i n v77 tan a 1 = Z ( 2 k + l i v / T i cos v i / - e I t i s well-known that tan a' takes every value once and only once with two exceptions: tan a ' - i .' • These e x c e p t i o n a l values can; only occur i f cos (2k+l) v/T= cos vTT and sin(2k+l) vn" = + s i n vTT ; that i s to say (2k+l)vTf= 2l~ + vTT, (l = 0 , 1, 2, . . . ) . But 0 < 2k < n+1, and t h e r e f o r e these 1 9 . exceptional values cannot arise. Hence a' i s uniquely determined. We now use the fact that the function J^(Z)cos a' - v v ( Z ) s i n a' has zeros with positive real part. T; This implies that A q ( Z ) has zeros i n the sectors ( 3 . 7 ) . Similarly, since A»(z) = -M z ( n + l ) / 2 K ^ U ) , M = 2(n+2)- 1Tr- 1 s i n ( ^ ) , we can show that A'(z) also has zeros i n the stated sectors, n REMARKS. By using the methods of Theorem 3 below, we can fin d the number of zeros of A n ( z ) , A^(z) i n a certain closed subregion of S^. If n = 1, A n(z) i s the Airy function of the f i r s t kind, the zeros of which are a l l real and negative. The assertion i n Theorem 1 concerning non-real zeros i s thus vacuously true i n this case. THEOREM 2; The functions B^(z). B ^ z ) have no real zeros i f n i s even, but for n odd these functions have an i n f i n i t e number of  single zeros on the negative real axis, with no f i n i t e accumula-tio n point. Their non-real zeros a l l l i e i n the region U s - , where i s the double sector ^ ^ +2^ " ^ a r g z ^ ^ ^ n+1 ^  ' and  k runs through the integers i n the range 0 < 2k < n+3o PROOF; If z i s re a l , arguments similar to those i n Theorem 1 apply here. i1© Suppose that a = re i s a non-real zero of B (z) or B'(z), n n 20. Then the r e l a t i o n s (3.3) and (3.4) remain v a l i d when F^(z) i s . replaced by F ~ ( z ) . In t h i s case, however,; B (0) B'(0) > 0. Ve suppose without l o s s of g e n e r a l i t y that 6 i s i n the upper h a l f - p l a n e , since we may use the r e f l e c t i o n p r i n c i p l e . Let T£ be the sector ^ 2 k ~ ^ 7 7 < 6 < ^2^2 " ' k = 1 > 2, • • • , U+2)/2 i l n i s even, k = 1.2..... (n+l)/2 i f n i s odd. Let T^  be the sector (2k ' - l ) ^ "/(n+l) < :G<TT; that i s T' i s the ray Q = TT ' i f n i s even, and i s the sector nf^/(n+l) < 0 < 7 7 i f n i s odd. I f O k S• , 0 < 2k < n+3, we s h a l l derive a c o n t r a d i c t i o n . I f ^ g T1 , n even, t h i s reduces to the r e a l case which o ' has been t r e a t e d e a r l i e r . I f 6 g TV. n odd, then n(n+2) <i"/(n+l) < (n+2)G < (n+2)TT and we have the f o l l o w i n g cases! (a) (n+l)TT < (n+2)S < (n+2)7T, whence sin(n+2)6 > 0. (b) n(n+2) V /(n+l) < (n+2)£ < (n+l)F , so that n T < ( n + l ) e < ( n + l ) 2 /T/(n+2). But (n+1) 2/ (n+2) < (n+1) 2/.(n+1) = n+1 and n(n+2)/(n+l) > n(n+l)/(n+l) . = n . Hence sin(n + l ) 6 < 0 and sin(n+2)$ < 0 . If. 3 € T£ , where 0 < 2k < n+3, (2k-l)7T (n+2)/(n+l) < (n+2)e < (2k+l)77 . (3.6) then 2 1 . But (2k-l)(n+2)/(n+l) = 2k-l+(2k-l)/(n+l) = 2k+(2k-n-2)/(n+l) < 2k , by hypothesis (3.6). Hence we have the f o l l o w i n g cases: (a) 2k7f < (n+2)© < (2k+l)TT , whence sin(n+2)© > 0. (b) (2k-l) 7T (n+2)/(n+l) < (n+2)© < 2k TT , whence s i n (n+2)© < 0 and (2k-l)TT< (n+l)© < 2kif (n+1 )/(n+2) < 2k77, so that sin(n+l)© < 0. In the case 0 < © < 7T/(n+2), we have 0 < (n+l)© < (n+l)TT /(n+2) < 7 7 and 0 < (n+2)© < 77 , so that sin(n+l)© > 0 and s i n (n+2)© > 0. Thus i f © k S k f o r any k, 0 < 2k < n+3, e i t h e r sin(n+2)© > 0 or sin(n+l)© and sin(n+2)© have the same sign. This c o n t r a d i c t s the analogue of (3.3) or (3,4), and completes the proof of the theorem. Figures 1 and 2 below i l l u s t r a t e the case n = 4. Figure 1. > Figure 2. 22. Notation. Let OA, OC, and OE be the rays arg z = (2k-2)TT/(n+2), arg z = (2k-l)i7 /(n+2), arg z = 2k7T/( n+2) r e s p e c t i v e l y . Let AB and DE be arcs of the c i r c l e |z| = R, with R a r b i t r a r i l y l a r g e , and B,D the points R exp[(2k-l)7f /(n+2) +6]i, 6 being an a r b i -t r a r y number i n the range 0 < 6 < / ' / ( r i+2). Let the curve BCD have the equation Im C = constant = ( - ) k + 1 2 ( n + 2 ) - 1 R ( n + 2 ) / 2 c o s ( n / 2 + l ) 6 , Pigure 3 i l l u s t r a t e s the case n = 2, k = 1 Let Apg arg f ( z ) denote the increase i n arg f ( z ) as z traverses the arc PQ. Let x = z on OA. o n = 2, k = 1 Pigure 3. 23, THEOREM 3; I f j i s a s u f f i c i e n t l y large i n t e g e r , the closed path  OABCDEO (defined above) has i n s i d e i t j zeros of B ^ z ) i f R = [(§ + l ) ( j - J)tTsec(| + 1 ) 6 ] 2 / / ( q + 2 ) ^ n d j zeros of B ^ z ) i f R = [ ( f + l ) ( j - ^ s e c ( ^ + l ) 6 ] 2 / v * + 2 ) PROOF; Ve examine the changes i n arg B n ( z ) and arg B^(z) as z traverses the closed path OABCDEO. Along OA ve use t h e , r e l a t i o n s (1.15) with k replaced by k-1 to obtain BnK e(2k-2)Tfi/{Q+2)} = (n+2)"2 X q 4 [ l _ v ( ? 0 ) W^^) ], where ? Q = ^  x < n + 2 ) / ' 2 , v = l/(n+2). Appealing to the asymptotic form (2.2) we deduce L0A arg B n ( z ) = a r g ( l + d*'1) + o ( l ) and so (3.7) A Q A arg B n ( z ) = - ^ 1 ' + o ( l ) . 2kTTi/(n+2) e - -2— x(n+2)/2 Along AB and DE we put z = x e ^ K i M n + ^ j , 5 - n + 2 x and use the r e l a t i o n (1.15) to derive 1 1 , P 2 ( z ) = (n+2)"2 x 2 [I_v(§) + u K I v ( 5 ) ] . From the asymptotic forms (2.2) i t now follows that (3.8) A A B arg B j z ) = -\( -~ - 6) - Im §| + o ( l ) ADE a r g B n ( z ) > -4 (n^2 " 6 ) ~ I m ? IB + o ( l ) , 24. upon noting that e ^ i s dominant i n the sector < arg 5< and e i s dominant i n the sector < arg ? < ^ • Moreover, y2  rv2 _ 2 - % ( l l + 2 ) - V 2 x -%, Prom (1.15) and (2.2.) we deduce (3.9) A E Q arg B n ( z ) = - ^ + o ( l ) . On BCD we put z = ^ U k - l ) " i/(n+2) , § 1 = -JL x ^ * ^ . Then I V(C) = I v ( ? x e ( 2 k " l ) , 7 r i / 2 ) = e k v 7 r i 1 ^ e~ r i / 2 ) = e ( k - ! ) v f r i J v ( § 1 ) , (-f < a r g ? 1 <TT) S i m i l a r l y , i_v(c) = e - ^ - i ^ 1 J _ V ( § ! ) , <-| < « * 5 X < r ). Therefore (3.10) J (k-i)vTTi -(k-i)vT»i , 1 K B n ( z ) = (n+2) 2 x 2 e 2 [e 2 [ e " ( k ~ 2 ) v f f l J ^ ^ ) + • ( k " : 2 > V F F I J v ( 5 1 ) ] , arg ?, < TT Hence the asymptotic form (2.3) y i e l d s (3.11) > B C D a r g B n ( z ) = -J(26)+ A B C D a r g [ e " ^ v f T i c o s (§ 1 + J^) . (k-^) v ^ i /_ v?T 7T\-i , /,\ +e 2 cosl^j^- J + o i l ) . 25, We nov put § = a +iB, (a,8 r e a l ) . Then, noting that x x= xe 7 7 1^* 1 4" 2^ and therefore §^ = ^e^"""^2, ve see that ?^ = i a -'6. ' Hence c o s ( V f - J ) - . . c o s ( § i _ | ) ; . e ( k - | ) v a = c o s ( 6 + f ) [ e a c o s k ^ + e ^ c o s - i ^ ^ l ^ i s i n ( P + f ) [e acos ^ -a (k-l)<7" v -e c o s - ^ ^ - J , a f t e r some s i m p l i f i c a t i o n . 77" We now choose. R so that B + ^ i s a m u l t i p l e of /T, say (*) 8 + 4- = - j " " , ( j a p o s i t i v e i n t e g e r ) . This i s always p o s s i b l e since (3.12) B = Im.-5 = (-) kIm C = R ( n + 2 V 2 Cos(§ + 1)6 along BCD, so that B i s larg e and negative. With t h i s choice of R, the expression i n brackets on the r i g h t side of (3.1l) i s r e a l and one-signed along BCD, so that ABCD a r g = - ^ r + . ° ( l ) ' Combining t h i s with (3.7), (3.8) and (3.9), we f i n d , upon use of (*), AOABCDEO a r g B n ( z ) = _ 2 p ~1 + o ( l ) = 2 ^ o ( l ) . Now l e t C (§) = (n+2)- V 2 u>^[l (5) + u>k I ,(?)]. -v v Then B n ( z ) = B n ( a ) k x ) = x 1 / 2C v(§). 26. y-Hence 0) >kB'(z) = \ x - 1 / 2 C v ( ? ) + x % C ; ( 5 ) - x n / 2 = 2 x " 1 / 2 x ( n + l ) / 2 t V l ( ? ) - v 5 " \ ( 5 ) ] = x ( n + l ) / 2 c ^ ( 5 ) which leads to (3.13) B«(z) = (n+2)~ 1 / 2 ^ 2 x ( n + l ) / 2 [ ( ? ) + W k I v _ 1 (?) ] . Noting that ?" 1 / 2 x ( n + l ) / / 2 = 2~1/2(n+2)^2 x^ 4, we deduce from (2.2) that 1 A B arg B n ( z ) = - 6) - Im §| + o ( l ) , D E.apg B'(z) = | ( ^ 2 - 6)- Im *>y o ( l ) -k/, . k\ . k/T AE0 a r g B n ( z ) = -arg W'k(1+CJk) + o ( l ) = ^ + o ( l ) . Replacing k by k-1 i n (3.13) leads to L0A a r g BA ( z ) = a r g ^" k + 1(l +" k" 1) + o ( l ) = --^M^ o(l)-Using the recurrence r e l a t i o n s we can deduce from (3.10) that (3.14) B' (z) = (n+ 2 ) - V 2 e - ( k - V 2 ) v n ( n + l ) / 2 [ e ( k - ^ ) v ^ i J ( } n 1 v-1 1 (-| < arg ^ < T ). 27. Now c o s L ^ - ( v - l ) |" - j] = - sin(§1- -y - ^) (3.15) c o s t ^ - (-v+l) f - -j] = s i n ( ? 1 + y -and (3.16) - e ( k - 2 ) v , r i s i n ( ? 1 - ^ f - f ) - e " ^ ^ v 5 f i s i n ( ? 1 + ^ - £) = - B i n C B + f j C e S o s ^ + e - a c o s - ^ | ^ ] iaJ!'w a k 'T _ a ( k - l ) / ^ n - l cosip+^JLe c o s j ^ - e cos n + 2 — a f t e r some s i m p l i f i c a t i o n . I f B i s chosen so that 8 + ^  i s an odd m u l t i p l e of ^  , say, 6 + ^  = (-2j + l ) ^ , ( j a p o s i t i v e integer) then the r i g h t side of (3.16) i s r e a l and one-signed on BCD. I t then f o l l o w s from (2.3), (3.14), (3.15) and (3.16) that A B C D arg B^(z) = S§ + o ( l ) f o r the above choice of R. Hence ^0 ABC DEO a r g B n ^ z ^ = " 2P + f + o(l) = 2j77+ o ( l ) . This completes the proof of the theorem. Remark; Theorems 2 and 3 include as a s p e c i a l case (n=l) the d i s t r i b u t i o n of the zeros [6] of the. A i r y f u n c t i o n of the second k i n d . Theorem 3 shows that the A functions of the second k i n d n do have zeros i n S,1 . k I f n i s even, the f r a c t i o n V of the plane containing 28. 2 zeros of F ^ z ) or F«(z) i s SpCn+l) (n+2) = i - 4(n4) (n+l) > so that A_ < v < — 12 - n 4 ' n - l I f n i s odd. V n - 4 ( n + 2 ) » s o t h a t 0 < V < 7 . — n 4 S i m i l a r l y , i f i s the f r a c t i o n of the plane containing zeros of F 0 ( z ) or F l ( z ) , then Hence ., _ (n+2) n 4(n+1) n 4(n+2) 7 < v ' < k 4 n — 3 1 < V' <• i 6 — n 4 f o r n even f o r n odd f o r n even f o r n odd. ~ \ Bounds f o r A fu n c t i o n s . . n • .LEMMA 5: The f o l l o w i n g functions are bounded ( f o r f i n i t e n ) ; n n ^ (4.1) , (1 + | Z | 4 ) A n ( z ) e C , ( l + | z | 4 ) " A ' ( z ) e C , |arg z |< - E U > 0) n n ^ (4.2) (1•""+ | z | 4 ) B n ( z ) e ~ C , ( l + | z | 4 ) " B^(z) e" C, | arg z |< ^  - e 7 k+1 7 - 1 k+1 (4.3) (1 + | z | 4 ) A n ( z ) e x p ( - ) C , U + Ul ) A ^ z J e x p t - ) C , (2k-l)7T + £ < l a r g z l < (2k-H)~ _ n + 2 e ^ lar g z | ^ n + 2 e , i . (k = 1, 2, . . . ) , (0 < e < ~ 2 ) 7 k+1 7 -1 k+1 (4.4) ( 1 + | z | 4 ) B n ( z ) e x p ( - ) C, U + IzI ) exp(-) C, ( k - i , 2 , . . . ) , i 2 ^ * e - 5 « g . < ^ k ± U 2 : n+2 - 6 " n+2 n PROOF: This i s t r i v i a l f o r the functions (l+1 z l,4)A ( z ) e ^ and n (1+[z|4 ) B n ( z ) e ~ , being a d i r e c t consequence of the asymptotic forms (2.4) and (2.6). To prove the corresponding r e s u l t f o r the functions i n v o l v i n g d e r i v a t i v e s , we note that I A n ( z ) = M z 2 K v(C), M = 2(n+2)- 17T~ 1sin(7| 2 -) Hence , , n A ^ z ) = M[| z \ ( C ) + z2K;(C)-z2] 1 (n+l)/2 -1 M[| z 2K v(C) + z {vC K ^ O - K ^ U ) } ] -M z ( n + l )/ 2K v _ 1 ( C ) . 30. Use of the asymptotic form ( 2 . l ) y i e l d s the des i r e d r e s u l t f o r A ^ ( z ) . Moreover, by d e f i n i t i o n , B (z) _ . _, 7 . w_ Hence 1 A v ( c ) , i i 1 n • »("*1>/aov.1(c),: upon use of the recurrence r e l a t i o n s c i;(c) + v iv(c) = c * v _ i ( c ) c rv(c) + vi_v(c)=ci_v+1(c) . I t now fo l l o w s from the asymptotic form (2.2) that the f u n c t i o n (l+|z| '*r\(z)e-(> i s bounded i n the sector |arg z|.< n+2 * To see that the functions (4.3) and (4.4) are bounded i n the stated s e c t o r s , we use the r e l a t i o n s A n U ) . (n+2)-V [ l . v ( ? 2 ) - r k + 1 I v ( ? 2 ) ] , YM(m2)/2 1 1 ?2 = ii+2 x 2 B n ( z ) = ( n + 2 ) " 2 x 2 2 [ I _ v ( ? 2 ) + o ) k + 1 I v ( ? 2 ) ] . • "?2 As noted i n s e c t i o n 3, e i s dominant i n the sector — < arg ?2 - ? • This i m p l i e s , upon use of the asymptotic n/ ^ forms (2.2), that the functions (l+|z|. *)A (z)e 2, ( l + | z | n / 4 V A n ( z ) e ? 2 , (l+| 2 r / 4)B n(z)e ? 2, (l+1 z | ^ PVU) e f 2 are bounded i n t h i s sector. Hence the functions (4.3), (4.4) k+1 are bounded i n the sta t e d s e c t o r s , since = (-r^ c. This completes the proof of the lemma. This r e s u l t extends known pr o p e r t i e s [8] of A i r y functions. LEMMA 6; The r e c i p r o c a l s of the functions ( 4 . l ) , (4.2) are  bounded i n the sectors I arg z( < n"£Y - £> I arg z| < ~ j - e r e s p e c t i v e l y where 0 < £ < — . I f k i s any i n t e g e r i n the  range. 0 < 2k < n+1, the r e c i p r o c a l s of the functions (4.3)!.are bounded i n the sectors ^^"^" + ' e — ' Ia r g z I - n^ +1 ~ e'' v h e r e 0 < e, e' < 5 f o r a l l other admissible values of k, they are bounded i n the same sector as (4.3).. The r e c i p r o c a l s of the  functions (4.4) are bounded i n the same sector as (4.4) provided 2k > n+3; f o r a l l other admissible values of k the r e c i p r o c a l s * (A *\ U J J - XU x (2k-rl)7T . _ I I A 2k+l) ~ of 14.4) are bounded i n the sectors 1 n+\ + e S Iarg z| < n+2—" PROOF; I t was shown i n s e c t i o n 3 that the zeros of A n ( z ) , A n ( z ) l i e oh the negative r e a l a x i s (when there are r e a l zeros) or i n the region ^ S^, where i s the double sector 2 k j f ^ I - I ^( 2k+l) ~ , ,• ' . .. , * < I arg z| < n ' ^ 2 — a n (* k takes i n t e g e r values i n the range 0 < 2k < n+1. It.was also shown there that the zeros of B ( z ) , n B n ( z ) l i e on the negative a x i s ( i f there are r e a l zeros) or i n the region ^ S£, where S£ i s the double sector (2k+l)7T | - ( 2 k - l ) ~ , , . . ' . . .. ' 1 n + 2 ^ I arg z| < ' n + i — a n ° - k takes i n t e g e r values i n the range 0 < 2k < n+3. The lemma now f o l l o w s from these r e s u l t s and the preceding lemma. 32. Bounds f o r sectors containing the rays l a r g z| = ^ 2 k~-0 ; / (k = 1, 2, ...) are given below. L E M M A 7: The f o l l o w i n g functions and t h e i r r e c i p r o c a l s are bounded i n the stated s e c t o r s : n 1 (1 + | Z | 4 ) C | A n ( Z ) | 2 f | B N ( Z ) | 2 ] 2 exp ( - ) k C , (4.5) n 1 i 4 \ - l r I (_-Vl2 . in,-/_M2i2 \k, n (1 + U l V ^ U ' t z ) ! 2 .+ | B ' U ) | 2 ] 2 exp ( - ) k C l ^ - e S l a r g z I s i ^ , (,=1,2,...) PROOF. Let z = ^ exp[ < ], ? 1 = ^  x ^ . This s u b s t i t u t i o n leads to the r e l a t i o n s (4.6) I v ( C ) = e ( k 4 ) v ' r i J v ( ? 1 ) , I _ v ( C ) = e - H ) v / r i J _ v ( § 1 ) , (-| < arg 5X <7T). n+2 (2k-1) If i 2 2 ~ 2 Since %^ = ^ 2 z e , these r e l a t i o n s hold f o r the sector - ^ f ^ < a r g » < ( 2 k f f f . Now l e t T?(~\ J. lE ^ \ -(k--)vTTi /_ v£ ff\ (k-^)v/Ti n z j = c o s ^ S ^ y - j je 2' - c o s t f ^ - —£ - 7/e 2 re \ (a ^  v/T '"r \ -(k-^)vTTi , v?T /T. (k--^)vfi G(z) = cos(? 1+ y - 7 )e 2' + cos(§1~ y - 7)e* 2' D i r e c t computation shows that 33. F(z)=sin£ . [ s i n j ^ exp ( - ) k C - s i n % ^ e x p ( - ) k + 1 C ] (4.7) - i C O S 4 * t s i n n 7 2 ' e xp ( ~ ) k C + s i n - ^ ~ ^ - e x p ( - ) k + 1 C ] . G(z)=cos 4-[cos^^ exp ( - ) k C + c o s " % f 2 ^ ~ ~ e xP(~) k + 1£] +i s i n ^ - [ c o s k ~ exp ( - ) k C - c ° s ^ ^ " ' e x p ( - ) k + 1 C ] . ¥hen arg z = a r S £ = ( 2^~^) ^  , s 0 that e~^ = e - 3' *ra ^ nA, _ „ 1A, + and therefore the functions (l+|z| j [ | A (z)|- + |B (-z) r] e~Q are bounded on the rays arg z = ^ 2 k+2^" ' (k=l»2,...), upon use of (4.6), (4.7) and the asymptotic form (2.3). R e c a l l i n g the no t a t i o n of s e c t i o n 3, ve note that exp ( - ) k + ' ' " C (=e~^) i s dominant i n the sector ^ ^ " ^ ^ < arg z < ^ ^ " j ^ ^ . Hence the functions (4.5) are bounded i n the stated s e c t o r s . Moreover, A (z) and B (z) have no common zeros, t h e i r ' n n ' Wronskian being non-zero, so that the r e c i p r o c a l s of the functions (4.5) are bounded i n the same sectors as the functions them-selves. This completes the proof of the lemma. COROLLARY; The functions (4.5) and t h e i r r e c i p r o c a l s are bounded  i n the zero-containing regions. PROOF; Choose e so that the domain of d e f i n i t i o n of (4.5) includes the zero-containing regions. This i s alvays p o s s i b l e , since the S^, are r e s t r i c t e d by (3.5) and (3.6). We have therefore obtained bounds f o r A functions on the n e n t i r e z-plane. 34 BIBLIOGRAPHY 1. J.C. B u r k i l l , The theory of ordinary d i f f e r e n t i a l equations. O l i v e r and Boyd (1956;. '. ' ~ 2. T.M. Ch erry. Uniform asymptotic formulae f o r functions with t r a n s i t i o n p o i n t s , Trans. Amer. Math Soc. 68 (1950), pp. 224-257.. 3. E.A. Coddington and N. Levinson, Theory of ordinary d i f -f e r e n t i a l equations. McGraw-Hill (1955). 4. R.E. Langer, On the asymptotic s o l u t i o n s of ordinary d i f -f e r e n t i a l equations, with an a p p l i c a t i o n to the Bessel  fu n c t i o n s of l a r g e order. Trans. Amer. Math. Soc. 33 (1931) , pp. 23-64. 5. R.E. Langer, On 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, with an a p p l i c a t i o n to the Bessel functions  of l a r g e complex order. Trans. Amer. Math. Soc. 34 (1932) , pp. 447-480. 6. P.V.J. Olver, The asymptotic expansion of Bessel functions of l a r g e order, P h i l o s . Trans, Roy. Soc. London Ser.A r247 (1954), pp. 328-368. 7. F.W.J. Olver, 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 containing  one t r a n s i t i o n p o i n t , P h i l o s . Trans. Roy. Soc. London Ser.A, 249 (1956), pp. 65-97. 8. C.A. Swanson, P r o p e r t i e s of A i r y Functions, C a l i f o r n i a I n s t i t u t e of Technology. Pasadena (1956). 9. H.L. T u r r i t t i n , Stokes m u l t i p l i e r s f o r asymptotic s o l u t i o n s of a c e r t a i n d i f f e r e n t i a l equation. Trans. Amer. Math. Soc. 68 (1950), pp. 304-329. ~" 10. G.N. Watson, Theory of Bessel f u n c t i o n s . Cambridge (1952). 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080544/manifest

Comment

Related Items