g a in“ A)? «Ti LIBRARY Michigan State University PLACE N RETURN BOX to move this checkout from your record. TO AVOID FINES retum on or befom dd. duo. DATE DUE DATE DUE DATE DUE HAVE“. l : l 00124 92%. ,. 353 ___J MSU Is An Affirmative Action/Equal Opponunlty Institution emu-us ._—__%7*——¥7 77W EXPANSIONS 0F PARABOLIC WAVE AND HARMONIC FUNCTIONS By Yousef Alavi A THESIS Submitted to the School of Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1958 ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to Professor J. Meixner and Professor C. P. Wells for their stimulating advice, frequent encouragement, and for their unfailing interest in this investigation, the results of which are herewith dedicated to them. Thanks are also due Professor A. Leitner for his assistance and the interest he has taken in this thesis. EXPANSIO'NS 0F PARABOLIC mm: AND HARMONIC FUNCTIONS BY Yousef Alavi ABSTRACT Submitted to the School of Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1958 Approved by ABSTRACT The problem studied in this thesis is the expansions of parabolic wave and harmonic functions. The wave equation separated in the coordinates of the paraboloid of revolution yields solutions referred boas parabolic wave functions. Series expansion for the parabolic wave functions in terms of the spherical wave functions has been obtained, with coefficients of the expansion explicitly determined. These coefficients have been given in terms of certain polynomials due to Pasternack for which the orthogonality relation is known.‘W1th this relation then the series expansion has been inverted to express the spherical wave functions, in integral form, in terms of the parabolic wave functions. Two methods have been developed to find the expansion for the parabolic potential functions. Further, the linear generating function for the Pasternack polynomials has been obtained in terms of a hypergeometric function. In addition a new derivation of the bilinear generating function in the continuous case has been given for the parabolic wave functions. Finally a second method for the derivation of the series expansion of the parabolic wave functions has been found. TABLE OF CONTENTS SECTION PAGE I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1 II. PRELIMINARY NOTIONS AND NOTATIONS. . . . . . . . . . III. EXPANSIONS FOR PARABOLIC wAVE FUNCTIONS. . . . . . . 8 Iv. INVERSION. . . . . . . . . . . . . . . . . . . . . . 16 v. SPECIAL CASES. . . . . . . . . . . . . . . . . . . . 22 _1. AN EXPANSION FOR THE FUNCTION Wmhf , A) 2. UERIVATION OF THE SERIES EXPANSION OF BOCRSTADT v1. THE EXPANSION FoR THE PARABOLIC. . . . . . . . . . . 28 POTENTIAL FUNCTIONS VII. GENERATING FUNCTIONS . . . . . . . . . . . . . . . . 36 1. LINEAR GENERATING FUNCTION FOR THE PASTERNACX POLYNOMIALS 2. BILINEAR CONTINUOUS GENERATING FUNCTION FOR PARABOLIC NAVE FUNCTIONS APPENDIX 0 O O O O O O O O O O O O O O O O O O O O O O O O 1+8 BIBLIOGRAPHY.......................5‘+ I. INTRODUCTION The parabolic wave functions have received considerable attention in recent years. This has been due, for a large part, to the interest in the physical problem of diffraction of waves both acoustical and electromagnetic, by a paraboloid of revolution. The diffraction problem has been basic in the work of Fock in some recent advances in the general theory of diffraction. 0f the many papers of Fock we refer only to reference [8] where other references can be found. The diffraction problem has also been studied by Hochstadt [10] to which we shall refer later. It is found in the approach to the diffraction problem that the relation of the parabolic wave functions to spher- ical wave functions is of considerable importance. In this thesis we study the problem of expanding parabolic wave functions in infinite series of spherical wave functions. This assumes as a heuristic principle, that a solution of the wave equation in some coordinate system can be expanded in terms of solutions of some other coordinate system. However the number of cases where this has actually been done and the coefficients explicitly determined, is very small. we shall show that in the present case, the expansion can be done and the coefficients determined. It is of interest to note that as a by product of the expansion, certain polynomials due to Pasternack [12] will play an important part. These polynomials are orthogonal and the orthogonality relation is known. Hence the expansion can be inverted and as a result spherical wave functions are then expressed, in integral form, in terms of parabolic wave functions. Further, we are able to find the generating function for the Pasternack polynomials as a hypergeometric function and finally we are able to give a new derivation of the bilinear generating function in the continuous case for the parabolic wave functions themselves. II. PRELIMINARY NOTIONS AND NOTATIONS We define the Spherical coordinates r, 9, ‘f , and the coordinates of the paraboloid of revolution 5 , 7] , f , by r sin 9 cos? =57 cos‘f r sine sin‘f =57 sin‘f’ §( 52 - 72). H II ‘4 ll z=rcose The wave equation AU 4- k2U = transformed to the coordinates E , ’7 , )0 , is ___i__ 3.1 62 f—E-(E—g-EL)+%- 7(7 a77) Fig—5% 1‘2”” The method of separation of variables then admits solutions of the form 1/! (5.)) Ilium-A) e‘imf , where W (E , A) satisfies the ordinary differential equation 1.2. 4.411 2 2-_m_2. A - (152 E +(kg £24- )V—O, (1) and W ( ’7 ,- A) satisfies a similar equation with the sign of A reversed. We shall refer to the product of solutions V’m(€,>x)\#m(’7.-)\)e’imY or V’m(£./\) VmMrA) as a parabolic wave function. From (1), 770m( 5 , A) can be given in terms of a confluent hypergeometric function 2 Vm(£,)\) = gm eikg /2 1Fl(- éké+m_;_-_l 3m + 13-11(62). (2) For physical reasons we shall expect to have wave functions which are regular and single valued and thus assume m to be integral. Similarly, the wave equation transformed to the coordinates r, 9,'r , becomes ____l__. .____g gr! ___2 r2 sin 9 2sin 9 + 2r sin 9 + sin 9 “a a2 2 - +cosO%-g-—l— sine 302 +kU-0, -m -im)‘ and it has solutions of the form jn(kr)Pn (cos 9)e . Here Jn(kr) is a spherical Bessel function and satisfies the differential equation 2 and P;m (cos 9) satisfies 2 d2 P;m(cos e) d P-m(cos a) sin 9 + sin 9 cos 9 3‘ d9 d9 n(n+l) sinze - m2 P;m(cos 9) = The function P;m(cos 9) is related to the associated Legendre function P:(cos 9) by P;m(cos O): ( -1)m fifi P§(cos 9). Here we choose P;m(cos 9) so that we can absorb the factor n+m : later in our expansions. Various notations have been employed to designate confluent hypergeometric functions and the parabolic wave functions, notably those introduced by Whittaker and Bucholz. The Whittaker function, MK,1(X)’ is defined by [3, page 10] 2 x/z-S-l 11F (-K + 1-5-1 3 v + l; x) "we = e '5 and for v integral, is a regular function of x. It satisfies the differential equation -&+E+ l_213 ygo, Our functions Vm( é , A ), Y’m( ’7 .- A) can be expressed in terms of this Whittaker function, as follows: .931 1,1/m(c';’.)\)=(--1k) 2 4—H A m {-1}: £2), E k 92-- 95; 2 30(7.-)\)=(1k) -J'-M)\ (1E7). m ’2 {ii—'22! If in the above Whittaker function, v is not a negative integer, the functionqu’ (x) is defined by Buchholz ‘1 2 [3. page 12] as MK v(X) ‘5 “MK:!(X) ru + v) In terms of these functions of Buchholz, our functions are - Qil 511mm .A) = m: (-ik) 2 é—MA Q91}: 5 2). k ’ 2 and -211. 2 ym(7,-)\)=m:(1k) 2-33-MA EURQ ). k ' 2 Still a further notation Miflht) , useful in applications, has been introduced by Buchholz [3, page 53]. It is a solution 2 of E- g; [-x-F'(X)] - (E - 7(4- T) F0!) = 0 and is given in terms of the above functions of Whittaker and Buchholz by M (x) l - % x:% (v) - ' 2 .. 771‘ (x) - x Mng) - x -—r-—-H 1‘” Hence our functions can be expressed as I ’9 (m) y’m(€91\) = m: (-ik) 27%. A (-1k 5 2) tx— m WEN 7,-A) = m: (1k) 2mmA (ik ~72). _ R— Finally we summarize these notations by writing the parabolic wave functions i,lrm(£ , A ) Vm( 7 ,- A ) as follows: mm) mm-M = 4—HT“ M x [IN-11:52) 57' fife: - Mix m(ik 7 2). 1:17,: or , 2 -(m+l) (,A) (,-A)=5i'b-)—k (-11. 2) ’“m‘ V“ 6'? ”ESP 5 2 X (11! 7 )9 -M£§-.% 0r _ ( > Y’m<5:M Pmt 17.4) (m)2 H7713 {-1}: .52) air (In) [fit- I I I . EXPANSIONS We now attempt to expand the parabolic wave functions \Vm( 5 . 2‘) Ymfll ,- )~) in terms of Spherical wave functions Jn(kr) P;m(cos 9) and assume an expansion 00 V15; . MIA/147.4) =2 afi Jn(kr) Pgmmos a). (1+) n=m 2 2 ”1 with r=12‘(§2+7z2) and cosG=§§-§-;?. The coefficients a3 , which depend on k,)\, m, can be determined from the condition that the expansion 0+) must be a solution of the differential equation (1) for all values 0f )l . Since the particular value of 7 does not matter, we choose arbitrarily small 72, and examine P;m(cos 9) for Y( near zero. As 7 ———> 0, cos 9 ——> 1 and since m -g 5 as x—> l, we find that m P;m(cos 9) -——> £3- .2?“— . The expansion (h) thus becomes 00 Ym(§.%) Wm(T(.-7\) =78“! . '23-:- ° Zara Jn(kr). n=m It remains here to investigate Ymhl’- )x) for small 7 . , /\ Let 1w - 2E . Then 2 YEW-M =7... 61k 71 /21F1" 1'5 * ”El 3 m + 1; ~11: ’12) 2 '1‘}? =11!!! elkn /2 1 + (-11: 7(2) m+l ””11” ...] , But the lFl above approaches 1 for small ‘7 , so that we have Ym(VZ"M“’Zm and the expansion 0+) reduces to co YEP)” = if? .3; 5““ Jn(kV) n=m with \)= 552/2. We now determine the condition that CO m -m 2 2: a. g Lu: T m n=m satisfies the differential equation in (1). Let 15/ =2: .3; 5““ Ann: 52/2). n=m Substituting V” in the differential equation (1) leads to 10 co _ 2 . :EZ: a2 ‘ 51%— Jn(k 52/2) + (i-m) J; (k:;2/2) n=m + 0125?? + iw) 3n (kfiz/Z) ]= 0- If we put v =(k/2)§2 the last equation becomes m I! E a2 [v- Jn(v) + (I-m)3; - (n+1) 3n+1(v> . (7) The differential equation for Jn(v) is voJ;(v) + 23;}(v) + [v - n 3+ :IjnW) = O, 01‘ v v-j;(v) = [M - v] Jn(v) - 23n(v). If we now introduce v-J;(v) into (5), it becomes, after 8 implifying m A E a2 [$93135 + 1w] Jn(v) - (1+m) J;(v)}= 0. n=m Using the recursion formulas, (6), and (7), the last equation reduces to ll n=m a) _ :E:;a§ iw- Jn(v) + ERQELI [}n+l(v) + Jn-l(V)_J - é%1%li [:n Jn_l(v) - (n+1) Jn+1(V):l = 0. 01‘ 0° 4L)... -—°° m n "m m gan 2n + 1 3nd”) *' 2mm an iv Jn(v) 0. Now on the right-hand-side the dominant term for small r is the first term of the series for Jm(kr) =V2k: Jm + 12‘ (kr), which, after dividing by rm. becomes 2m+1 . ‘ m .Also, Pmm (cos 9) = §%¥-§ which, after dividing by sinme, 2 m: - becomes -l- . So that we have now for the right-hand-side 2m m: an 2m m m' l _ am m m (2m+l)! ° 2m m: ’ m (2m+l)! ' But the left-hand-side Ym(§,)\)\l/&(7z,-)\) 5"” ’Z‘” approaches 1 in the limit, and therefore we have for the a: m _ (2m+1)§ . a — m km 1% Therefore we have for the coefficients sg, substituting am into (11) , m am 3 (Zn-r1) , +m ' in-m Inur) . n km n-m 3 13.2“!) We have finally for our expansion 0+) on Vm(§’)\) Vm(79')\) =Zm 3:: Jn(kr) P;m(cos 9): . - Fm * inm—Bm,witbiw=§g. (l2) m .. an " (2n+l) n-m 3° km 1...mh") The coefficients a: can also be given in terms of F;m(w), if, for the moment, m is not an integer. This can be done if we let bi? = in B: . Then the recursion formula for the bi: becomes, with iw = )‘/2k, n+lm nm (n+l)(n-m+l) i Bn+1 + (2n+l)(iw) 1 En + n(n+m) in-l B:_l = 0, or (n+l)(n-m+l) BIS-*1 + (2n-I-l) w BE - n(n+m) Bil-1 = O, which is the same as the recursion formula -m -m (n+l)(n-m+l) Fn+1(z) + (2n+l) z PH (2) - n(n+m) F;m(z) = 0. So that now we have 15 a BUt n - - m me(w) Fn (w) _ F-m(w) F-m(w) am = Lgafill (2m)! in m +9----- -2-- is defined as m k approaches an integer, and thus our expansion (12) can be given as _g; Ym(§ ./\)\Vm(7(.-A) = 2_ Bi? (2m): 1mm :21! F'mw) _ ‘ -9-- ° Jn(kr) an (cos 9). (13) F;m(w) Now, in order to establish the convergence of our series expansions for the parabolic wave functions, we 'vnrite (12) as 00 n+m Fm(w) ( .A) < .-)x) = (2 +1) l— - -9—— (tr) Wm § Y“ )2 11;, n km Ffiw) J“ m Pn (cos 6). 1T0 investigate the behavior of E§(w) for large n, we can use the asymptotic expansion of a certain polynomial given by Rice [13]. He defines these polynomials as Hn((‘:2P:V) = 3F2('nsn+11:3 1191);"), andwith§=liyelfl andp=m+l and v-l these reduce to Pasternack polynomials. Hn(§9.p,l) however, behave as a power of n for large n. Hence the coefficients afi in the expansion behave as a power of n. This is also the case for the function PfiCcos 8). But jn(kr) behaves as C(%?) for large n, so that the series converges everywhere. 16 IV. INVERSION OF THE SERIES EXPANSION FOR PARABOLIC WAVE FUNCTIONS It is now proposed to invert the series expansion for ;parabolic wave functions and express the spherical wave :functions in terms of the parabolic wave functions. This can be done using the known orthogonality relation for JPasternack polynomials. The orthogonality relation is given 'by Bateman [l] as 1, 1 Fm (130?:m (- ix) 0' P g n' §.r(l+m)r(1-m) cosh 17 x + cos m w dx " 1/ 2n+l, p = n. (1) 'CD Our series expansion for parabolic wave functions is n- , Fm(-? Y. 0; NY. hi w {1-53 L-F-Fg—L .21.) n=m - Jn(kr) Pgm (cos 9), which can be rewritten as )\ :1“ L25.) ( )x) <,=-> )ooW—em—l-gs- Y5 Y7 “ m as . Jn(kr) P: (cos a) (2) 17 A For convenience, we let t = :21— , and restrict this substi- tution for this section only. Now, multiplying (2) by F:(-it) cosh w t + cos m n and integrating with reSpect to t from ~00 to co , and as it is permissible here, interchanging the summation and the integration, leads to (n Fm m(it) Fm(-it) ____..E._.____ (“flm km cos 17t+ cos m1r Ym (5’ A)\‘/m(y(’ )K)dt 'CD ‘99— Fm(it) Fm(-it) = Z in Jn(kr) P§(cos e)(2n+1) n P dt (3) cosh wt 4- cos m1r From Pasternack [12] we have, if m is not an integer, (1+ m) Fgmm ' my: Fm“): [’(n- -m+1) %§%}F’” (1+) 'Under this condition, introducing (h) into right-hand-side of (3) for F§(-it), and then applying the orthogonality condition (1), leads to n Jn(k1‘) 132(008 9) = 121(_1)m km It: 3:113]. “m 00 Ffi(it) Fm(-it) cosh 1rt +ncos m1r ° Ym(§ ’A) Ym(”l’-)\)dt° (5) “(1) 18 The factor in the integrand involving Pasternack's polynomials can be expressed in terms of hypergeometric and Gamma func- tions. We have F$(-it) = 3F2(-n,n+1. % +13 - l5: m+l:1;1)- Using the transformation (9) F d19d29°<3 3 1 _ ll-(g2) l—(B +52“ 0(1-«2- °< ) 3 2 91.82 W: s35r'<——_+B1+Bz-° -i = _ n | r= an (V): (1) mgfm n=m with v=k $2/2, and where m = (2n+1) n+m : 0 l2:2 o fair: , n-m). km F$(w) with iw = )\/2k. Now with Yfif .N = (-ik)-(m+l)/2 3%- M_! Q (mtgz) 2 ’ 2 (1) becomes (m+1)/2 00 Mill; (-lk§ 2): Him-1 Z of: jn(v), 2’ 2 ' 5 n=m or 00 ___ M-(m-l)/2 n' M_E m (_1k§2)= LEM (2n+1)1 2 ’2 nm= F :(w) 41—3 DW), (2) 23 where we have substituted for the coefficients a: , page 1h. The series expansion (2) can now be compared with an expansion for the Whittaker function given by Buchholz [6, page 128]. In this case, however, essentially the same Whittaker function has been expanded in an infinite series whose terms are a finite product of a sum of two Bessel functions of the first kind whose orders are half an integer. 2h 2. THE DERIVATION OF THE SERIES EXPANSION OF HOCHSTADT In [10] Hochstadt makes use of an expansion of parabolic wave functions in terms of spherical wave functions and we shall show that his result is a special case of ours. Hochstadt gives, for s and m integral. = r. +S+ ( ) n: (p) -JAa—I-nl m m (-21k§) 8+2 in-t-m (m) -im‘f.___ ml m(211: ) Z (2 +1) 8 S ( -l)r(m-n) r(m+n+1) r(r+m+l)(:_ _:2 ‘:E%2 (m+l) :(s-r)£ r. '1 .Jn(kr) P:(cos e)e “‘79 . (3) Our expansion is °° . mF m(w) Ym(§’)"Ym‘7"’\’ ‘Z‘an Ht: 1.5.2— Fm( n=m Fm W) .Jn(kr) pg“ (cos 9), (N) with iw =zs/2k. Using the notation of Buchholz for our parabolic wave functions (k) becomes (m: Zk'm -ik 2-) ik 2): E 2n+1 ) 777‘; ( g 777:”;(7 n__m( > gin-m Fm(w) m km ;§?;- Jn(kr) Pn(cos 9), - (-1)m 25 01‘ (m) (m) - 2 . 2 .. Emu n+m 77?-E(1k§)77z_1(1k7)-:(m3)2 1 2 2 Fm(w) NIL-J (kr) Pm (cos 9). (5) Fm(W) New let -w = 28 + l + m, where s and m are integers, then (5) becomes 777(m:L"-§l (—-21k§) Min—(11:72) -Zi2ritliin+m n=m (m!)2 P: (-2s-l-m) F: (~23-l-m) Jn(kr) P§(cos 9). (6) We have m Fm (-25-l-m) gimn=fl§1r£ll (7, Also F:('2S’1'm) = 3F2('n,n+1,-s;m+1,131) - naamswa re - a a m «a mm .3F2(-n+m,n+m+l,-s3m+1,m+l;l) 26 (mm) (n+m+l)r(-s)l: S r] + +1) 3 m Z (m+{7r(m+l)ror: s! m! r=0 (m-n) (n+m+l) (-l)r s! s - O ._ O r=0 m+1 r(m+l)r r. (s r). s' m. Iiere we have used the transformation [9, page #99] F C"1,"‘2."‘3;1 F‘Bl’l-‘51*52'°‘1‘°<2'°(3) 3 2 B B ‘ rr151-Bz-°‘F5“§7 L 1’ 2 B1- °<1951- 0‘29 d331 - 3F2 o< o< L 91'91*92' 1’ 2 l Hence we have with,some simplifying, ( -1)r (n-m)r(n+m+1)r(m+1+r) Fm(- 25— l-m)= r=0 (”1): r! (3-1-7: Lu). (8) Ilrltroducing (7) and (8) into (6) leads to 77:11)]?! (11:; 2-)-:m;)"}?] (11:72) 2 + 1n+m m; §§ n- (m!) (s+m+1) -m °° (-l)r(n-m)r(n+m+l)r(m+l+r)I _ 2 ':E:: (n+1)r r! (s-rF! n=m . Jn(kr) P2 (cos 9), 27 or (m) + + 2 rgszmllm:m:l§@(-1k§2o) 3+1§Q<1k7> =i in’mggg+n i (-1)r(n-m)r(r+m+l)(§_:2 m! (m+1)r(s-r)! r! n=m r=0 m . jn(kr) Pn (cos 9), ‘ihich is the series expansion given by Hochstadt with the parabolic coordinates X=2 J § 7? cos f, Y=2 4/; sin f, 2’- S '7 o 28 VI. THE EXPANSION FOR PARABOLIC POTENTIAL FUNCTIONS The parabolic potential functions are the solutions of LaPlace's equation A U = 0 in parabolic coordinates. The expansion for these functions can be derived in two ways. Directly, by repeating the process for LaPlace‘s equation which we did for the wave equation, and this will be shown later in this section. Secondly, the expansion for these functions can be derived by taking the limit of the expansion for the parabolic wave functions, (12), section III, as k ——> 0. Then the parabolic wave functions become Bessel functions in the limit, as is the case if we separate the LaPlace's equation in the parabolic coordinates and find its solution. To show this, the differential equation satisfied by \{fm( 5 , A) is 2 W+lm< 'N. k2 2 23— 2x 2 g a; § ‘ 2 "’ d§ § 0Ym( g ’ A) = 0, which becomes, as k ———> O, 2 a2 < m) d < ,N s ”Lg—se— W, d§ F .Ym(§, A) = o. (1) This is Bessel's differential equation with a solution regular at $- = 0, Jm()\1/2§). For the equation involving 29 71, the corresponding solution is Jm(i Al/Zl'l). Now for the right-hand-side of the expansion III (12) we must find FEW), F$(w), and jn(kr), as k ———> 0, while P.m (cos 9) is independent of k. n We have F§(w) = 3F'2(-n,n4-l,‘;-+ +315 m + 32‘ w;l,m+l;l) =:(-n)r (n+1) (112“? +%m + % w) . .l_. (11)r(m+1)r ‘ 1‘3 r=0 The last term in series (r=n) contains (1%- + J5 m + -2- w)n. This factor is of degree n in w. All other terms of r < n are of degree less than n in w. Therefore, near k = o, with r = n and w = -1)\/2k, we have n3 n3 F: (- $25}; - (-1)n n3 5313-11 - L'T—T‘nm: n 1'1 1'1 L273 (-1)n ini‘lzfig-MLA—b— . .1125 (n+m)3(n3)2 22n Similarly, near 1: = O, we have A ~W.L Fivérr) m m ~ (2m)3(m3)2 2‘?m km _ 1’” N" , 1 m3 22m km 0, for the coefficients a: we have Then, near k 3C m~ (2n+l)(n+m)3 in-m m3§§n23in)\n L a ~ km(n-m)3 (n+m)3(n3)2 22n kn :3 , 22m km 1m Am 0 r 2 ~ 2+ ' n- - , ag~figfij§9 &) m Anm(2n).-i; . The dominant term in the expansion of Jn(kr) =\/-7-27j-k-1: J (kr) n+1, near I: = O is the leading term and hence for small k Thus the right-hand-side of our expansion III (12) becomes, in the limit as k —-> 0, CD 2 Z {Air—21m <- W A” at? «as 9» n=m It remains now to investigate the left-hand-side of our expansion as k ——> 0. We have 2 / 2 Ym(§ ,2») = 5m eik 5 2 11:10. if + fl‘i’lmumkg ). Then m 1 (£5? Iii—so vm<§,>\>=§ PM”??? 1 (“75-5) - 00.... + (m1) (m+2)“ "—21" 31 I {:ls' m a Similarly m 722 A \VmWwA) = 7? e"1k /2 1F1 (+ h— + 19?; mi; 41:72) approaches a—Vm' 2’” a 1m A 2 ° Jm(i 72). Therefore the expansion III (12) now becomes ~(-r-n--:-n)13--—-)\2‘1:Jm(Al/2;)'J(i)\ 71) -Z MH— =m .(-&)n-m 2n /\ n Am m (cos 9), 32 01‘ (n Jm (Al/2 g) Jm(i )‘1/272 ) =2 Fig-'(ZAfl n:m1:(-&)n =11} .n-m r Pn (cos 9) 00 n n z in, <- 3;) (ZAL r“ 3 (n+m)3 . PE (cos 9). (2) which is the expansion for the parabolic potential functions. As mentioned above a different way to get this expan- 81on for the potential functions would be to repeat the process for LaPlace's equation that we did for the wave equation. We assume the expansion 1/2 1/2 °° . .1ch f) Jm(i )x )2) = Z bf; r“ pg (cos 9). (3) n=m 52+n2 §2_722 with r: COSG= 2 ’ 2 2 f *7 tC) determine the coefficients bg by substituting (3) in . Again we attempt 1/2 tile differential equation for Jm(}\ g.) for all 7? . Since the particular value of 7 does not matter, we Qhoose arbitrarily small 7? . Then in the right-hand-side 33 of the expansion (3). for (7 near zero, we have 2 2 m 7? N 9 Pg ET “I’m (131%)? ° 321?: ’ 1/2 For the function Jm(i A 7), near 71 = 0, we have 1/2 1‘“ Am/Z’Zm Jm(i)\ 7)z 'r m: frhus the expansion (3) for 7K -——9 0 reduces to M2 §2n-m 5,1.1(A§)Zbim(1)%- n=m We now determine the condition that 00 -m/2 _ m +_ . )\ 2n-m E b2 i m (-1) m ""'"""""" f n-m 3 2n n=m satisfies the differential equation (1). This leads to Zb‘" n H)“; «m - km» :2” n=m +an%%A§2n-m+2=-O O, n=m which can be simplified to 3h 00 . m n+m ' _1_ 2 _ m 53-]+m)3 ] j:E::i bn n-m 3 2n (kn hnm) + bn-l n-l—m)3 2n-l )\ ] :n:m+l 2n-m - 5 = o (u) 2n-m Hence for (1+) to hold the coefficient of g must ‘vanish, and thus we have the two-term recursion formula , 1 2n bm m ._ _ n+m- . 0m ' .- bn (n-m-l)! 2n-l ‘)\ (n+m)3 En(n-m) ’ or m - _ 2 )‘ m ‘ bn ‘ n n+m bn-l ° (5) The solution of (5) gives, for b: in terms of b: , n hn-m n3 (m+n)3 m Now the coefficients hi can be given if we evaluate b m m from expansion (3). This can be done by dividing (3) by m m m m g )2 = r sin 9. and taking the limit when {, 7? ———> o, This gives for the right-hand-side The left-hand-side is 1/2 1/2 EEK f) Jguk 7]) 5m (“1 35 and this becomes 1121 Am 22m (m!)2 as , -———9 O, and therefore we have for the bm m - ‘_]2m 1m /\m bm m 2m m3 (2m): Finally, we have for the coefficients b3 . n bm-lm ( _&)n(2)\) n ' n3 (m+n)3 ’ {The expansion for the parabolic potential functions is ‘therefore n n 1/2 1/2 °° 1mc- ) (2A) n Jm()\ f) Jm(i A 72) = E n: ($4,110: I‘ n=m m Pn (cos 9). {This is exactly the same expansion as in (2), and it can Tae rewritten as n n )\1/2 23/2 °° im(- (1;) )x Jm( §) Jm(i )2) =2 n3 (n+m)! n=m 2 2 2 n )ngz 72 {+72 ITt follows immediately that the series is convergent for all é and 77 . . (5:2 +77 36 VII. GENERATING FUNCTIONS 1. Linear Generating Function for Pasternack Polynomials. In order to find this generating function we first consider the representation 3(B: K 'B) 2F1(0(933X31)= 1 J 11B"1 (1_u)b’-B-1 (l-xu)-O( du, (1) 0 which is an integral representation for the hypergeometric function [11, page 12 ]. Let o( = m + l/2, B = (m+l+w)/2, 3’: m+l, andx= 4+1: /(l-t)2, then (1) becomes m+ + + - + was. exam 12*. 9%” z -;—%§,—— > 1 m-lfig 31-1-1 -m -% = S u 2 (l-u) 2 [1 + 439-5- ] du, (2) 0 (140 land + - + h (1-t)2m+1 NEE-gm . 1%”) 21"1 (m + 12‘ . Liz‘flsm‘tli- fl) min 214- - W 112‘ :5 u 2 (l-u) 2 [1+t2-2t(1-2u)] du. (3) 0 hWowthe generating function for the Gegenbauer polynomial -v-1/2 Czfl/Zu) is (1-2hz + hz) . and we have [5. page 175] I‘lll'uucn.l‘flnll|llll!1£\l J... I 37 -v-J2‘ oo v+32* (l—2hz+h2) =E on (z)hn = =m % [— + + -v ZVfiéi-y 2D T—fiu (1- -z 2) Pn+v (z)hn . (1+) n=m 2 1/2 which is convergent for |h | < |z 3; (z -1) . Substitution of (14») into (3) leads to, with v=m, h=t, and z=1-2u, “"‘1‘27-7' B( (221:: m)2F1(m+12. Max:521: m+1;- W2). (1- t) (l-t) oo 1 _ £3 = (--1)m (53%;: tn-m 2m'IS (1-22) P332) n=m -1 m+!2-1 m-gz-l . [1&4] [131] dz. (5) w -m = _ m -m ' here we have applied the relation Pr1 (x) ( l) n+m 1 PER), started the summation from n=m, and, as it is Permissible here, changed its order with the integration. Now from Buchholz [2, page 202] we have Tr W I (tan 325-) pg (cosf) a)“ ~(m:)2 444% [Vi-m) O 38 u I f we put tan ‘J;=vl:5 2 1* z = z , cos 70 in (6), we get -1 1-22 . m Pn(z)dz l ..m. m:u;l = m-l - 2 3 (1-22) 2 1:3- 2 m- - -1 2 [1:2 2 . 2 Pm(2)d n Z: (7) ept O a O f 1' CO 1 I dUC1n8 (7) t0 9 rom (6) 9 1 m 3 ! (i—t)2m+1 B(W24. ' + - ) 2 2p1(.m.% , 22:12.! __L_ 2 ;m+l;- t (1-t)2 ) CO = (-1.)m “" ‘ZE:: 1-1l—.{—-%s 2.1.x (2m): tn-m - m n+m ' n=m (m3)2 n-m 3 [-( +2- ) ... , 01' tm (1-t)2m+1 2F1(m'"2 ' 22112 2 ; “1+1; -_—&-t-—) (1-t)2 = 39 oo 0 2&1 :E:. 2:: . 3F2(-n+m,n+m+1,'gi%:! ; m+1,m+1; 1). tn n=m ° Using -w for w, we get m L mzlzxm . - “t - _;_ (l_t)2m+1 2F1(m‘*2' 2 ' “+1, 2) ' 2m: a) .E 2n+m ' 3F2(-n+m,n+m+l;‘m:%:E;m+l,m+l; l)°tn (8) n-m 2 n=m But we have m!_(L-l‘m+!) 3F2(-n+m,n+m+l,‘m:%:x’; m+1,m+1; 1) = __ 1 2l 2 2 (2 + 23m +'§) m . Fn(-w)’ (9) and (l +‘l m +‘!) F$(w) = 2 1 2|; 2 w . (10) m. I—‘-2m+-2-) Introducing (9) and (l0) into (8) leads to t” 1 m+]-n at _ F.(m + , 3 m+1; - ) - . (l_t)2m+1 2 i 2 2 (l-t)2 2m . 00 +m 0 F$(-W) n .ZE::§n-m;! m ' t ' (11) n=m Fm(w) But from Pasternack [12, page 212] we have m _ _ n Fn(-z) - < 1) F§. so that (11) becomes MO tm _LJ.vm*-. .-...ltL. -.i. (1_t)2m+1 2F1(m +‘%’ 2 ’ m+i, (1-t)2 ) - (2m)! Fm(w) (- l)n n: ;.._Q__ tn, or, multiplying by (1k)m 2m ' tm 1 m+1-g ht - F ( + -. 3 m+1;- (1k)In (1-t)2m+1 2 m 2 2 (1—t) . FE(w) n =ZE:_(;k)m n-m 2 Fm(w) t ° (12) n=m m The coefficients of tn in the expansion (11) are proportional to the coefficients of the expansion in section III. In fact if we rewrite (12) as 2m ' , m m ; ”1+1; - ht (ik)m (1-t) ‘l 2m+1 231‘m * 2’ 2 Fm(w) i... inieii._n___. =Zk n‘m 3 Fm(w) in tn ’ n=m it is seen at once that the coefficients of intn here are exactly the b2 section III, page 12 and 1h. That the coefficients in (12) are proportional to the coefficients om can also be seen by considering the differential equation n (ta-1)f" + [(m+3)t+2w+9§l:l r' + (m+1 + %)r = o, (13) kl satisfied by the function on the left-hand-side of (12). However, as t=0 is a regular singlular point of (13), if we assume a solution of the form 00 f =:E: cn intn, =m upon introducing this series into the differential equation, we get 00 a) E [n(n+m+2)+(n+l);] cn intn + E 'w(2n+1)cn intu-l n=m ‘ n=m 00 + E n(m-n)cn intn"2 = 0, =m which after simplifying becomes so :5: in-lsn-l [(n+l)(n-m+1)cn+1 + iw(2n+1)cn+n(n+m)cn_1] = O. n=m Hence for (in) to hold, the coefficient of tn"1 must vanish, and thus we are lead to the following three-term recursion formula for the cn (n+l)(n-m+1)cn+1 + iw(2n+l)cn + n(n-‘-m)cn_1 = O. '(lh) With iw =JA/2k, this is exactly the same recursion formula, equation (9). page 11, satisfied by the coefficients b: . Thus the cn must be proportional to the coefficients bi, and hence the function on the left-hand-side of the equation (12) is a generating function for the coefficients b: . #2 2. Bilinear Continuous Generating Function for Parabolic Wave Functions and Whittaker Confluent Hypergeometric Functions. The bilinear continuous generating function for the Whittaker functions has been given by Erdélyi [h, page 66] 83 _l. ' 2ni v2(-%X%fi) 2 V2 1+1: e J2“ 1+1; ] J K Fé-x+»>ré+x+u> ' t MK, (x) L [ |_(2u + 1)]2 p .MK.’ (Y) dK, (1) where L is a path from -1 co to + 1 oo, separating the poles of f-(% + K + p) from those of r-(% - K + p). The purpose of this section is to derive the bilinear continuous generating function for our parabolic wave functions using our series expansion for these functions and some properties of Pasternack polynomials. Then with the relation between our parabolic wave functions and Whittaker functions already established, the above formula, a proof of which has been given by Erdelyi [k], can at once be given. We start with (Appendix, equation k) Li3 (I) Jm(z sin o< sin meiz cos d cos B =Z in-m (2n+l) n=m n+m ' -m -m n-m 3 Jn(z) Pn (cos c><) Pn (cos {3). 2 2 2 2 N _ __ g +72 - g - )7 ow let 2 - kr - 2 k, and cos B - -§--§ , then € ”2 we have ik Jmu: 57 sin one 2 =2 1” (”am-1) n=m (n+m): -m -m 73:57? jn(kr) Pn (cos 9) Pn (coso<), (2) g 2 + )7 2 g2- )7 2 with r=--———, andcosG=—————. 2 f2! ‘?2 ... 1/2 If we now substitute cos 0( = fl , sin 0< = %§-€— in (2),we get 2 2 WM ik--—g 2»)? H Z” t + - n-m Jm(2k ‘ 1+t ) 9 ~' 1 n=m o(2n+l) {3% Jn(kr) p;m(cos e) aha-E) . (3) Now we make use of a certain integral involving the Pasternack polynomial F§(x). Pasternack gives [12, page 216] +oo {-(m+l) emx sech x P;m(tanh x) = % S. eixz F§(iz-m) -s . sech(% n 2) dz . (M) 1/2 If we substitute tanh x = iii—E , so that sech x = ‘31-." , -l/2 and x = in t , and replace 12 for z, (k) becomes 1/2 -1 03 m+§ 2t -m 1-12 __ _i_ 2 m __ _ Hm“) l+t Pn (1+t) ‘ 2 g t Fn ( z m) la) - sech(% n i 2) dz , 2.12.— ‘m Lfi - l. 2 - .. I—(mi-l) 1+t Pn (14-12) 21 t F12: ( z m) -i a) . sec (% w z) dz . (5) -i)\ Let w = 2k = -(m+z), then (5) becomes m+icn 1/2 E -m l-t _ _ l ' 2 m f—(m+l) 1+t Pn (1+t) ' E: t Fn(W) m-icn ' sec %'n(-m-w) dw, which can be written as #5 / m+lC0 1 2 t P-m(_l_.;§) _ 1 l t‘ 1'2’ F13”) l+t n l+t El (m:)2 FEW) m-ioo resin-a r , . l l K . l 1 ll sec21r(-m-w) dw. (6) {—(2’2'1‘1'2) l—(2‘“T2"11+ 2) But we have l (% - -1n- 2)- of—(% %+ng) = v sec % n(m+w), so that (6) becomes m+ioo w 1/2 -m -t _ l "2' t pn (fi)-'Evi S t m-ioo [(1. 1m, 12!) r(.1é.+il2.m-fl) F2 on 2 - dw. (7) [l—(m+1)]2 F: (w) Returning to equation (3) above, if we multiply it by 1/2 f3t- and introduce (7) there, we get 2_ 2 3.11.2. t1/2§77 11‘ 5 271 ' 1";t: Em _ n-m 1+t Jm (2k 1+t ) e - i (2n+l) n=xn +m . n- m 2 3n (kr) P; “(cos 9)- 57%f #6 m+ioo {gW 32:2,, (8, H {MD}2 F:(w) m-ioo We can interchange the summation and integration here, and then use our expansion III(12) for the parabolic wave functions, thus (8) becomes €242 1/2 1/2 1k ——2—- 0 % -t-——-J (2k 3— e l+t m §7 l+t m+im km "'2' = - E‘ni t m-ia: I‘<%+%m+§>r<%+%m-%>Y( N [Hm-1)]2 m g, .Ym(72,-/\) dw. (9) This is the bilinear generating function for our parabolic wave functions. We have already shown that Vm(§.)\) Ym(7’(,-)\) = ?? k’mfl) M u {-ik 52) - 2, NIB - M (ik72), ’ : wit: NIB “7 1.5. With w = " 2k 9 so that for the Whittaker confluent hypergeometric functions (9) becomes, after multiplying by f 7 , 2 2 1/2 1/2 migLug 1+1: k§7i§r~1m<2k§7izfle m+ioo ._ _ 1 ...; I<%+%m+¥>l‘<é+%m-§> ”m t [|()Cn (cos B). (2) 1+9 here JV (2) is the Bessel function of the first kind and CX(x) is the Gegenbauer polynomial. Equation (2), with v: m +% , becomes 1 2m + (sinoé sin (3)-m Jm(z sinoc sin {3) eiz c0504605 B = 2 5 2 - L 90— in n!(m+n+l) [l—(m+%)] (1rz)2'>_ 2 J n=o r(2m+n+1)""' m+n+£ (z) m+% III-I’é' ‘ Cn (cos (X) Cn (cos [3). (3) Substituting for Gegenbauer functions in terms of Legendre functions [5, page 175] 1 .. l m + m - m Ch 2 (x) g 2'- Em-PI [_l-nix-{Z (1-x2) 2 Pfifmu‘” in (3) leads to . 2 1 (sin °< sin (3)-m Jm(»z sino< sin [3) e12 cos°< cos B = 2 m+ 2 oo — 2 '- l in n2(m+n +l) -[ (+1) ( > Z-Z— '2—J .22In l m 2 J "2 n=0 |—(2m+n+l) m+n+% (Z) -2 I. + I- 2 - - . [T’f-STIIH'T . Tfihyllj (sinxsin B) m Pngm (cos oé) . ngm (cos B). 50 This simplifies with ;_ _ lJ'n . 2m 3 I—(m+2) " 22m m! ’ to . 00 J(z sin