EXPANSIONS OF PARABOLIC WAVE 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 to as 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. With 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 hypergeometrie 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. EXPANSIONS OF 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 ProQuest Number: 10008569 All rights reserved INFORMATION TO ALL USERS The quality o f this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest 10008569 Published by ProQuest LLC (2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6 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. TABLE OF CONTENTS SECTION I. II. PAGE I N T R O D U C T I O N ....................................... 1 PRELIMINARY NOTIONS AND NOTATIONS............ • • • • 3 III. EXPANSIONS FORPARABOLIC WAVEFUNCTIONS................ 8 IV. INVERSION......... .. ............................... 16 V. SPECIAL C A S E S ......... ............................ • 2 2 1 . AN EXPANSION FOR THE FUNCTION £ , X) 2 . DERIVATION OF THE SERIES EXPANSION OF HOCHSTADT VI. THE EXPANSIONFOR THEPARABOLIC....................... 28 POTENTIAL FUNCTIONS VII. GENERATING F U N C T I O N S .......... 36 1 . LINEAR GENERATING FUNCTION FOR THE PASTERNACK POLYNOMIALS 2 . BILINEAR CONTINUOUS GENERATING FUNCTION FOR PARABOLIC WAVE FUNCTIONS APPENDIX............ ^8 BIBLIOGRAPHY..................................... 5^ 1 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. Of 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 can be done and the coefficients determined. expansion 2 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 orthogonolity 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. 3 II. PRELIMINARY NOTIONS AND NOTATIONS We define the spherical coordinates r, 0 , 'f , and the coordinates of the paraboloid of revolution £ , , 'f , by x = r sin 0 cos ^ y = r sin 0 sin 'f - \ ^ z = r cos 0 = cos sin 'f §2 - ^ 2 ). The wave equation AU + k2U = 0 transformed to the coordinates _ 1_ " ( F f ^ , T , is x +. i_ 1 _ _°b _ ,( v °U % rt, d a uV s ? 2 + 'i2 _ i £2 2 >2 L _ a + k2n = o 2 K u u The method of separation of variables then admits solutions of the form where e im ^ (§>A) + and ^ ( *? of A , satisfies the ordinary differential equation + ( k 2 ? 2 - f 5 + A ) ^ = 0' satisfies a similar equation with the sign reversed. We shall refer to the product of solutions V' m ( £ » ^ ) y rn( a parabolic wave function. or ^ O . - ^ J a s From (1 ), y^C £ , /S) can be given in terms of a confluent hypergeometric function = el k ^ 2/2 -t-S-i-i ,m + l ; - i k £ 2). (2) For physical reasons we shall expect to have wave functions which are regular and single valued and thus assume ra to be integral. Similarly, the wave equation transformed to the coordinates r, 6, X * becomes 2 r sin 0 2 r2 sin 9 — :— ^ + 2r sin 0 b . ~ + cos 0 d-U . 1 2 ' ^ U and it has solutions of the form + sin 0 ■— jjj + k U = 0, jn (kr)P“m (cos 0 )e -im f Here jn (kr) is a spherical Bessel function and satisfies the differential equation d — j (kr) ..... . dr2 and P~ 2 sin 0 2 d j (kr) 11... r dr £i _ nfa+U 4. k2 K 2 r jn (kr) = 0, (cos 0 ) satisfies d2 P*m (cos 0 ) d P“m (cos 9 ) 2------ + sin 0 cos 0 — d0 d0 n(n+l) sin20 - ra2 P"m (cos 0 ) = 0 . (3 ) 5 The function P “ (cos 0 ) is related to the associated n Legendre function p“ (cos 0 ) by » P n ^ c o s •) - (-1)“ Here we choose Pnm (cos 0 ) so that we can absorb the factor / \| (n+m)i la^er our expansions. Various notations have been employed to designate confluent hypergeometrie functions and the parabolic wave functions, notably those introduced by Whittaker and Bucholz. v (x), is defined by [3 * page 10] The Whittaker function, * 2. v + 1 = e"3^ 2 x 2 ^(-K + ; v + lj x) *2 and for v integral, is a regular function of x. It satisfies the differential equation i x .2 dx .2 + y = 0. Our functions £ , A )> %>( ^ A ) can be expressed in terms of this Whittaker function, as follows; m+1 , A ) = (-ik> yrB( l . - A ) - c ik ) 2 - j - mi A Q W m+1 2 1 5k (-ik % 2 ), 9 ° » O k ’?*' )• * 2 If in the above Whittaker function, v is not a negative integer, the f u n c t i o n „ ( x ) is defined by Buchholz i\|V 2 [3, page 12] as M r ^ (x) „(x) r < i + ▼> In terms of these functions of Buchholz, our functions are _ ra+1 , A ) - mt (-lk) 2 4 - ^ iA m (-ik 5 2 ), * 5k > 2 and ra+1 y*-B < 7 . - M « »t d k ) " 2 2 ' Still a further notation 5k s (lk ^ ' 2 5> f \ (x) , useful in applications, has been introduced by Buchholz [3 > page 533* It is a solution of _ d_ j^x.F»(x) dx v 2 - (j^ - \,+ - - ) F(x) = 0 and is given in terras of the above functions of Whittaker and Buchholz by . 1 .1 = x = x * W X) r(i+v) 2 Hence our functions can be expressed as 7 - ( ) 2ln ^ _ £ , A ) = ml (-ik) (-ik £ 2 ) W ~ V^m ( >?,-A ) = ml ( i k f (ik >|2 ). Finally we summarize these notations by writing the parabolic wave functions fm ( f . A > l ^ . - A ) *A ) ^m ( = ) as follows! MlX Q * W m ( - i k £ 2) *2 ’ M iX W °r /m i\2 fm U , A ) -(m+1) M i x . 0r fm( ^ ’ A) rn(ik 7 2)> *2 ? m(-i k " A ) = (m!)2 k m ? n iA C-ik £ 2 ) (iX ^ 2 ). IS" > 8 III. EXPANSIONS We now attempt to expand the parabolic wave functions Y in terms o t spherical wave functions m( f > 3n (kr) Pnm (cos 0 ) and assume an expansion oo Y m ( ? . A ) Y B ( M , - A ) n J 2 •£ in (kr) pnm(cos 9)> 1 v n=m 2 with r = ^ ( ^ 2 + ^ 2) and (U) v, 2 cos 0 = 2 * The coefficients an , which depend on k , A , m, can be determined from the condition that the expansion (*+) must be a solution of the differential equation (1) for all values of . Since the particular value of 7^ does not matter, we choose arbitrarily small 7^ , and examine P~m (cos 0 ) for ^ near zero. As 7^ — > 0 , cos 0 — > 1 m m C < » as x — — and since p *'1 “ 3 b > 1, we find that m - m , _______ P n “ (cos 0 ) .__ 1____5 id: The expansion (*+) thus becomes oo Y b< £ . » • 4 r- It remains here to investigate * A j an V to>- for sma11 Then Y m<^ A) = elk r^2 / 2 1 P 1 (.- 2 + ^ T = y)B elk ^ / 2 S m + l! -ijt \ 2 ) _ a . atl 2 2 2 1 + ---------- (-ik ~Yi ) m + 1 ( + 0( 7)k ) above approaches 1 for small But the , so that we have and the expansion (if) reduces to oo Y with m( f . A ) = i s / - a “ 1 n=m g - J n( k W c V = ^/2. We now determine the condition that X I an n=m f " Jn satisfies the differential equation in (1). Let oo r = Z « : n=m Substituting jn ( k f 2/ 2 ). 1 in the differential equation (1 ) leads to 10 oo E k l a jn (k p / 2 ) + (1-m) J* (k p / 2 ) n =m fk c 2 + (f p If we put v = (k/2)^ + iw) Jn ( k p / 2 ) = 0, the last equation becomes oo ^ a” ^v * + + + iw)jn ^v )l = °> (5 ) n=m ii i in which we can eliminate the 3n (v) and j^Cv) by using the 'n differential equation for 3n (v) and the recursion formulas t2n v 1} 3n (v) = Jn+1(v) + jn_l(v)' (6) (2n + 1) j^(v) = n • Jn-1(v) - (n+1) 3n+1(v) (7 ) The differential equation for _ v*3^(v) + 2j^(v) + «Jn (v) is nCn+1), v 0n (v) = 0, or v j n (v) = n(n±U _ v V “ 2V v)# If we now introduce v*Jn (v) into (5)> it becomes, after simplifying oo ( Y*; jn (v) - (1+m) Jn (v) = 0. n-ra Using the recursion formulas, (6), and (7), the last equation reduces to 11 CD n~m an *] iw * J„<*> ♦ M _ jm±ljL 2n + 1 [ w * > n ^n-l^ ♦ V i ‘« ] ~ (n+1) ^n+l(v) or oo X .m n(n-m) < 4 - n 2n + 1 n=m * W , n .\ m . . , x v) + 2 _ an lw ^n(v) n~m oo + \ m (n+1) (n+ni+1) 1 /„> _ 0 Z _ an (2n + 1) •'n+l °* n=m This can be rewritten as 21 CD j n (v) (n+1)( n-m+1) m . .n(n+m) (2n + 3) n+1 + lw an + m an-l 0, n=m which can be reduced to a simpler form if we let .m a“ = (2n+l) b“ . This leads to oo 2 1 3„ 0. Now on the right-hand-side the dominant terra for small r is the first term of the series for m •,______ which, after dividing by r , becomes Also, P“ra (cos 9 ) = ~ n -"" m 2 ml becomes 2m ml ,m '(gm+i)'« ^ which, after dividing by sinm 0 , — -— • So that we have now for the right-hand-side 0m . 2 ml via i k ml 1____ _ ' 2 m ml m (2m+l)S ' m 2 m vm m k “ <2nrt-l}l * But the left-hand-side m »m m approaches 1 in the limit, and therefore we have for the am flia _ (2m+l) I . m ” kra lb Therefore we have for the coefficients a®, substituting ajj into (11), _ (2n+l) n " vm . (n+m) I ..n-m (n-m)I 1 -m, . * F^(w) m We have finally for our expansion (b ) oo ln t a > p;m (ccs 8), ( v n-m ( » ,n-m an = <2n+1) ( ^ ! - “J T \ F^(w) . with IW - s . (12) m The coefficients a^ can also be given in terras of F~m (w), if, for the moment, m is not an integer. let b*? = in B™ . n n This can be done if we Then the recursion formula for the b^ n becomes, with iw = /*/2k, (n+1)(n-m+1) in+1 B™+1 + (2n+l)(iw) in + n(n+m) in”^ b“ = 0, or (n+1)(n-m+1 ) B®+1 + (2n+l) w B® - n(n+m) b“_1 = 0 ; which is the same as the recursion formula (n+1)(n-m+1 ) F ^ z ) + (2n+l) z F‘m(z) - n(n+m) F”nl(z) = 0 . So that now we have 1? am = i i n ± H ( ii i_Jii n-B k" ro^ « . But F~m(w) _—jt>✓ % f ;“ (w is defined as m ) approaches an integer, and thus our expansion (12) can be given as oo Y B Y m (1? ^ i2a^ ^ ’- A ) = n=m k J i11'” F~m (w) * -?=--- * Jn (to) P‘ra (cos 6). Fffl (w) n (13) n Now, in order to establish the convergence ofour series expansions for the parabolic wavefunctions, we write (12) as yr”” Y m( f , A ) Y b O | . - A ) = 5 (2n+1) .n+m F?(w) — • ^ p“ (cos 0). To investigate the behavior of F^(w) for large n, we can use the asymptotic expansion of a certain polynomial given by Rice [13]* He defines these polynomials as Hn (^,p,v) = 3F2 (-n,n+l, £ ; l,p;v), and with » = - 2 an(* P s m+l and v = 1 these reduce to Pasternack polynomials. Hn (?f ,p,l) however, behave as a power of n for large n. Hence the coefficients ajj in the expansion behave as a power of n. This is also the case for the function P®(cos 9 ). But j_(kr) behaves as 0 (^t) for large n 11 11• n, so that the series converges everywhere. 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 Pasternack polynomials. The orthogonality relation is given by Bateman t13 as oo f“ (ix) p-m c-ix) r°. E_ , --- dx — a 2*p(i+m )p(i-m ) \ cosh 7r x + cos m it dx p ^ n V.2n+l, -oo p = n. (1 ) Our series expansion for parabolic wave functions is oo Y m >A ) T m (Y ' A ) (2n+1) jn (kr) 'f V (cos 6), which can be rewritten as 00 . Y m (f ^ \ • v .n-m F F“ (- pr(2n+1) n=m m' 2k ; i k ) 17 For convenience, we let t = , and restrict this substi­ tution for this section only. Now, multiplying (2 ) by Fp(”it) cosh v t + cos m tr and integrating with respect to t from -oo to oo , and as it is permissible here, interchanging the summation and the integration, leads to oo rn m k ^ \ \ ff(it) F^(-it) cgS - 7 t " PcoS mir Y m (f . -CD 00 OO , \ i ■ L M^r) n P®(cos 0) (2n+l) n \ \ „ m m / ..\ -,m F„(it) F “(-it) ^ - £ ~ — — £-------- dt cosh w t + cos mir ,,, (3 ) n-m -oo From Pasternack [12] we have, if m _-mr > -m( » Fn U J ■ (l-m)n 'n12' " is not an integer, T (n+m+1) T (l-m) f (n-m+1) r < l + m 5 -m, . V z' Under this condition, introducing (M-) into right-hand-side of (3 ) for F^(-it), and then applying the orthogonality condition (1), leads to .n . x -i/ tr, ^ m ,_m T (n+m+1) _ i ln (kp) Pn (cos 6) - 2 < - n k rfn-m +15 [ 1 00 Fg(lt) Q - l t ) cosh TTt + cos mrr -oo Im7 I ml 7 * W w 18 The factor in the integrand involving Pasternack*s polynomials can be expressed in terms of hypergeometrie and Gamma func­ tions, We have Fn (“it) = 3*2(-n,n+1, \ + f - ^ . m+l,ljl). Using the transformation (9 ) Pl+p2" Ov .3 3* °^2 r Fjj(-it) | (m+1) | (| - | m + . F2 (-n+m,n+*+l, ^ ^ m + 77 ; m+l,m+l;l) (6) Also, we have which, with the relation [l*f, page 282] ?F1 (a,bjc;l) = 2 1 leads to p L ^ if a > | (c-a) (c-b) (1 + I m- 11) F^(it) = tzr-2 2— -v ~ i ----- 7 7 — m P (m+1) (| - ^|) • (7 ) 19 Hence, from (6) and (7) we get [f (m+l)]2 p ( l - l m + ^ ) | (1.1m-i&) If we now use the functional equation | + z) r ( | “ z) = cos’V z » we can write for <1 la . 11) !"<*♦* 2 (1 - 1 m - cos 7r(| + i (1 ^2 2 r<£+ ^ 1 7T 4, 1 m - it) 2 2 2m + itx ^2 ” 2 cos (cosh 7T t + cos mir) 2tt‘ (9 ) Substitution of (9) into (8) leads to F®(-it)F“ (it) = — p— ------ pn m 2tt [ P (1+ra)] i (l + 1 i m + ^ • (cosh irt+ cos rair) jF2 (_n+m,n+m+l,^ + ^ m - n i + 1 * - * fc- ; m+l,m+l}l) (10) 20 Equation (10) can now be introduced into (5)> and thus we get i" in (*r) P£(cos 9) = fe <-*)■> oo -00 ^ »A ) Y n ^ » “ ^ *3^*2(”n+ra,n+ra+1, ^ + | m - ~;m+l,m+l;l) This result, which now holds for integral values of m, (11 ) can be reduced to a representation due to Buchholz [2 , page 202] if we use his parabolic functions. Buchholz redefines the functions 7Y^ ??/(m)(z> = (fe)l / 2 M ( ID ) (z ) [2 , page 198] as (z). *2 K Using this definition and the expression relating our parabolic wave functions and the Whittaker functions, section II, we get - I k' * ? ? ? 2> w (m) (ik 7^ ). ■ x (12) dt 21 Therefore, for integers m, with - ^7 = s, and the relation (12), equation (11 ) reduces to the result obtained by Buchholz, and we get in 3n(kr) P“ (cos 6) = | (-i)m - ± - r " n w ( m t r (n-m)! i 00 2 s)T (2 * 2 -i 00 • ^F2(-n+m,n+m+l, ^ s ^ m+ s; m+l,m+l;l) s 22 V. SPECIAL CASES 1 . A SERIES EXPANSION FOR THE FUNCTION ^ By the use of the expansion (12), section III, we can now derive a series expansion for the parabolic f u n c t i o n ^ ^ ^ in terras of the spherical Bessel function jn (kr). It is then possible to give an expansion in terras of the function jn (kr) for the Whittaker function M_w 2 m (-ikp * p ). 2 We have already shown, page 9 , that oo 1 m; c n=ra 2 with v=k E /2, and where / , \, .n-m _m ✓~ . v (n+ra)l . i n - (2n+1) W T krn F*?(w) _n _ Fm (w) > with iw = A / 2k. Now with ^ t- M ? ^ ( - i k £ 2) 2 * 2 (1) becomes (ra+l)/2 2* 0r 2 m* * 00 n=m « -w 2*2 n=m • ? (n+ra)l Fn ^ _ * , \ t s ^ t r Fm (w) (2 ) <2) 23 where we have substituted for the coefficients a™ , page 1*+. 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. 2b 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, (P) , £ n ± p i ^ ( - ) (.2 i k p 1 s + “ 2~ 1 oo • V m) 1+m (2i 1s + , = H n=m (2n+l) (-1)r (m-n)p (m+n+1)r(r+m+i)(g-r^ (m+l) (s-r)! r! r r=0 .jn (kr) P“ (cos e)e'lra^ . (3 ) Our expansion is n-m m •jn (kr) P’m (cos 0), with (*+) iw s ^/2k, Using the notation of Buchholz for our parabolic wave functions (if) becomes (raS)2 C - i k ^ 2 ) - ^ ”^ (ik-y2 ) = _ .n-m i = “ (2n+l) F®(w) I m T T Fm (w) 3 n < t o ) P £ < c o s 0 ) > 25 or % < ■ > ( - i k e 2 )- ? » M v (lk» 2 ) . j r l a e i i in+m ~ ? “ o 1 n=m (in!) Fm (w) pm (w ) 3n (kr) pn ^cos 9 ^# Now let -w = 2s + 1 + ra, where s and m are integers, then (5) becomes i+m S + "-p“ c 2> - V m| 1+n ( « ? 2 > = f : s + — 5— 1 n=m (m!) f “ (-2s-l-m) - 5 - --------- 3_(kr) P®(cos 6). F“ (-2 s-l-m) n n in+a (6) We have F“ (-2s-l-m) = jljl) (1+s)m m (s+m)! _ r(s+m+1) m! s! “ m! si Also Fn (-2s-l-m) = ^F2 (-n,n+l,-sjm+1,1jl) (2 + 2 m+s+ j m + (2 “ 2 m+s+ 2 m P ^(-n+m.n+irrta.-sjm+l,m+l;l) (7 ) 26 (-n+m)r (n+m+1)r (-s )^ _ r (s+m+1) s • ml * (m+l)_(m+l) *r! r=0 s = r (s+m+ 1 1 s" m * (m-n)r(n-fnn-l)r(-l)r s{ r = Q t™+l)r(in+l)r rS (s-r)S y ' Here we have used the transformation [9 * page U-99] oC 1 2, oC | ( $ 1 ^ T ^ l +^2" °^l“ °^2" 3 ;1 3F2 Pl»P2 c*3 ;1 Pi- 3F2 Pi >Pi +P2“ 0 V ~ (-l)I‘(n-m)r(r+nH-l)^s_].j Z. mJ i_ (ra+1) (s-r)J r2 n=m r=0 . jn (kr) pJJ (cos 0), which is the series expansion given by Hochstadt with the parabolic coordinates Z = f - 7{ . X =2 ^ cos'/5, Y =2 sin f , 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 Y"m (^ , A) is ,2 dT m ( ^ A ) ^ d^2 2 t2 + A 1 •'r m < f . *> = which becomes, as k — > 0, 2 „2 -ra I F d? • ♦ ( A 1/2p 2 . * > = o- (1) This is Bessel's differential equation with a solution regular at £; * 0 , Jffl( A 1/ 2 ^ ). For the equation involving 29 y ^ 9 the corresponding solution is Jm (i A 1^ 2 ^ ) . Now for the right-hand-side of the expansion III (12) we must find Pnm (cos 6) jn (kr), as k — > 0, while Fjj(w), Fjjj(w), and is independent of k. We have Fjj(w) = ^F2 (-n,n+l,| + ^ m + ^ wjl,m+ljl) ■z — (-n)p (n+l)r (^ + \ m + | w ) (l)_ (m+1),. r r r=0 ± r! 1 1 1 The last term in series (r~n) contains (2 + 2 m + 2 w ^n* This factor is of degree n in r < n are of degree less than k = 0, m , n with r = n and iis ~ 1_ 2k nt 1 • 22n w. n in All other terms of w* Therefore, near w = -iA/2k, we have in n ,.(2n)'t , 1_ .___ m{__ ' nt nj (n+m) S , , ,n 1n AfJ _ mi(2n)t in A n . 4 _J_ kn - (n+m),(nS)2 22n kn Similarly, near k = 0, we have zP( V 1*) 2k m! (2m) I im A m (2m)!(mJ)2 22m im A m i_ " m! 22m ’ k“ . 1 k: Then, near k = 0 , for the coefficients a™ we have 30 Qm ^ j_2n+l)(n+m)5 jn-m mi(2n)lin A n n km (n-m)5 1_ (n+m)5(nl)2 22n kn ml 22m km im A m or m (2n+l) (m2)2 / lxn-m vn-m n ^ (n-rnjT (n»}2 *f A 1 l? * The dominant term in the expansion of Jn (kr) = \ / J ' near k = 0 n+ is the leading term and hence for small k < n„ - w 2n(kr)” ni V KrJ~ (2n+l)l Thus the right-hand-side of our expansion III (12) becomes, in the limit as k — > 0, oo p 7 " {rMnlTnT 9 ). C n-m It remains now to investigate the left-hand-side of our expansion as k — -> 0. Y . C f .*> = ? ” *lk We have 1 ^ 1 F 1 <' S i T + Then m k — >0 ' c A . .M (m+1) 15 if . + (m+1)(m+2) (45') 22l 31 = mia (- r (# r m (/\) . m! m ‘ (m+1)! 1! + (m+1)! 2! ] V m _ m! 2 oo 575 / ^ - ^ 7 5 m+2p c-Dp f"(m+p+1) p! Jm < ^ P Similarly m ,-ik7? /2 e A) = a <+ y 1 + ^ m+i* approaches ^ 75- Therefore the expansion III (12 ) now becomes (m?)2 22m T , \1//2 r x T ?n I'm Jm ( A >* 1 /' 1 \ f Jm(i(i vA l / /^ 2'>^ " n—m 2_ . n-m «n x n •<-t> - T T Pn A (“ • « ) . (mj )2 rn (n-i>! nS v = V ,» ^ P ( 2 A ) n rn n! (n+m)l / n-ra •P® (cos 0 ). (2 ) which is the expansion for the parabolic potential functions. As mentioned above a different way to get this expan­ sion for the potential functions would be to repeat the process for LaPlace*s equation that we did for the wave equation. We assume 1 /2 Jm ( A with the expansion 1 /? f }Jm ( i A 7 ) = / f 2 + yi 2 QQ bS n=m $2 - (3 > r"pn(cos 0)* y 2 r = ---- 3---- , cos © * — ~---- r . Again we attempt I *f to determine the coefficients b® by substituting (3) in 1/2 the differential equation for A p ) for all 7^ • Since the particular value of 7^ does not matter, we choose arbitrarily small Y] , Then in the right-hand-side 33 of the expansion (3), for Pn n ( \ ^ | 22 + >J / (n-m)S _m mS - ? 2n JE__ 2n rn ^ and near zero, we have 1/2 For the function 1/2 Jnr JiA V 7)~ / A - , m/2 ^ m m?, Thus the expansion (3 ) for ^ 1/2 T / \ ' V m( ? } ^ = 0 , we have T|), near -- > 0 J°° X vm .-m > T bn 1 (-1} n=m reduces to (n+m)! A -m/2 £ ? We now determine the condition that - m /2 oo 2 _ < i_m < ^ m « « ? n=m satisfies the differential equation (1 ). Z bn n=m This leads to ^ °° i \ bn (n-m); n (f 9n A n=m which can be simplified to j- 2n-m+2 c% = 0, 31* oo 2 n=m+l 2n-m • f = 0 (If) 2n-m Hence for (h-) to hold the coefficient of |f must vanish, and thus we have the two-term recursion formula 0n ,m bm = - l2±»trUi .1 n (n-m-l)S 0n-l 2 V i v . {n-m)! / (n+ra)i ^•nCn-m) or ,m _ n (2 *+n(n+m) ,m n-1 / c\ * ^ The solution of (5) gives, for in terms of b™ , bm _ (-l)n~m (2 A ) n~m mi (2m)£ bm n *+n~m n! Now the coefficients from expansion (3 ). (m+n)! b® m can be given if we evaluate This can be done by dividing (3 ) by ^ in ^ m m _ | ^ = r sin 0, and taking the limit when ^ , 1) This gives for the right-hand-side <-l)m ( 2 m)i bm 2m m! The left-hand-side is b“ 1/2 _ m I v 1/2 -^ra T — > 0, 35 and this becomes i 22m (mi)2 3S f * ^ b ^ an<^ therefore we have for the b m = (-l)m im A m m 2m mi (2m)! Finally, we have for the coefficients .m / 1 nn _ 1 ( ~ t \ n “ n! (m+n)i .m bm n \m (2A) The expansion for the parabolic potential functions is therefore 1/2 Jm (/^ im (- h 1/2 £ ) J ra( l A 7 } = / n=m ( 2 A ) “ rn nt (n+m)l • P® (cos 0). This is exactly the same expansion as in (2 ), and it can be rewritten as 1/2 Jm(A v 1/2 J,(iA P ^ \ } lm (- r)” / _ nt-reSTT" n-m It follows immediately that the series is convergent for all ^ and 7^ . 36 VII, GENERATING FUNCTIONS 1 , Linear Generating Function for Pasternack Polynomials, In order to find this generating function we first consider the representation B(p, * - p) 2F1 (c*,ps /;x) = u^"1 ( 1 - u ) * " ^ 1 (1-xu)” (1 ) du, which is an integral representation for the hypergeometric function [11 , page 12 ]. Let ©C = m + 1/ 2 , p = (m+l+w)/2 , = m+1, a n d x - -*ft /(1-t)2 , then (1) becomes b 1 u m-l+w 2 ■> - 7 ^ 7 - > m-l-w (1-u) 2 1 + *ftu du, (2 ) d- t ) and 1 u 2 - m m-l-w m-l+w (1-u) 2 l+t2-2t(l-2u) I ] 0 - I 2 du. (3) Now the generating function for the Gegenbauer polynomial ,v+l/2. -v-l/2 , and we have [5» page 1751 37 -y - 2 oo v + 2 = ^> Cn (z)hn = n=m (l-2hz+h2 ) v 00 - H a n-m 2 1 5 which is convergent for | h | < z ± (z -1) ‘ c . <*>»"• < « 1/2 Substitution of (V) into (3 ) leads to, with v=m, h=t, and z=l-2u, 1 rt/m-KL+w m-H-vK f -+^ 2 nH"l 2 9 2 ) 2 Fl^m (1-t) i m+l+w 2 ' 2 oo m m ml \ *n-m 0m-l = (-1 ) ( 2 i 7 T / 1 2 n-m -z 2 • t 1 m+w-1 2 _ *+t x ' m+1»“ d - t )2 “l+z' 2 f /, 2* \ (1"z } 2 Dm/ v n m-w-1 2 dz. (5 ) where we have applied the relation Pnm (x) = (-l)m [n+m]i • p J U ) . 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 r 0 tr <««.-f* r< 1+m-w ) 38 P /1+m+w ( 2 (6) » — 1>— ljl)* ^^C-n+nijn+m+l> ^ 2 ^ If we put ^anS^ = y i ^ § » z = cos "f in (6), we get m-w -1 m-HW"l l-z 1+z dz P?(«> 11 * / t 2" v 1-z -1 (l-z> 2 (1+z) 2 (1+z) -1 (1-z) •p” (z)clz ,m-l m m+w-1 2 2 t o (1-z2) 2 -1 m-w-1 2J l±zl 2 P®(z)dz, (7 ) which except for a constant factor is the integral in (5)« Introducing (7 ) into (5) leads to* from (6), i _ B(- t o , - t o } 2Pi(m+l , - t o ;m+1;. bt (l-t) (1-t) oo .n-m (-l)m (n+m)I r~/m+l-wx (mi)2 I ( 2 J m mi = (-1 ) (2m) ! nsm |“ (— t o ) ^f2(-n+m,n+m+1 * ; m+l,ra+lj 1), or .m (l-t) P . ( m + i , ■*$** , m+i ; 2m+l 2 1 * ^ !tL) = / i_+^ (l-t) 39 2mF^Z n=m 3F2(“n+m»n+m+1» J IS±i±S£ ; m+1,m+1; eL 1)* tn Using ~w for w, we get p (m .1 m+l-w _ 21 2* 2 * — $01 d - t ) 2m+1 . _ _ 4t . i " d - t ) 2 " 2m{ oo (n-mli 3F2^_n+m»n+m+1; m+1, m+1; l)*tn (8) n=m But we have m+l-w ^ A mi 1 (~ - g m + ~) •jF~ (-n+m,n+m+1, -— ; m+1,m+1; 1) = — ;■ ■ 1 f 3 I (| + | m + f ) • f “ (-w ), (9) and f(i + | m + f) O w) - :,-'r A g i - 2 w; • ml | (j ' J " + 2' <10> Introducing (9 ) and (10) into (8) leads to (l-t)2m+1 2Fl(l" + 2 ’ ^ 2 ”^* “+li " (l-t)2* 11 TSnTT . y l _ n=m {a± 4 f (n-m)! F _m,(w)« . tn . nr But from Pasternack [12 , page 212] we have F™(-z) = (-l)n f£(z), so that (11) becomes (id (-1)“ (p-nq); . V .w? tn (n-m)l -m. * z ' n=m nr ' or, multiplying by (ik)m , lifflll . t _ , (ik)m d-t)2m+12 1 00 - = y .(-;)n P?(ir) laa o t f n i l i tn I— (ik)m T^iTT Fm( . n=ra K The coefficients of mv } ; tn bt . (i-t)2 1 m+l-w. m+1 . 2’ 2 ’ (12) • in the expansion (11 ) are proportional to the coefficients of the expansion in section III. In fact if we rewrite (12 ) as LjmLl . ____ _____ dic)m ( i - t )2 m + 1 2 1( -Z_ it is seen exactly the p ( in"m km + 1 2’ (n+m)! Fn ^ (n-m)! ^ 2 ’ ’ d - t )2 5 .n .n at once that the coefficients b™ \ mtlrw . m+1. _ of section III, page 12 and I1*-. intn here are That the coefficients in (12) are proportional to the coefficients b^ can also be seen by considering the differential equation ? . H (t -l)f + (m+3)t+2w+ ^ f* + (m+1 + |)f = 0, (13) hi 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 oo f = £ n=m ^ upon introducing this series into the differential equation, we get oo _ qp n(n+ra+2)+(n+l) 1 cn intn + n=m w(2n+l)cn intn"1 n=m oo + ^ __ n(m-n)cn intn'2 = 0, n=m which after simplifying becomes oo 'y in"^sn'"^ (n+l).(n-nn*l)cn+^ + iw(2n+l)cn+n(n+m)cn_1 = 0. n=m YS Hence for (1*0 to hold, the coefficient of t 1 must vanish, and thus we are lead to the following three-term recursion formula for the cn (n+l)(n-m+l)cn+1 + iw(2n+l)cn + n(n+m)cn-1 » 0. With {lb) iw = A / 2k, this is exactly the same recursion formula, equation (9)> page 11, satisfied by the coefficients Thus the cn must be proportional to the coefficients b® . b“ , and hence the function on the left-hand-side of the equation (12) is a generating function for the coefficients bjJJ . k2 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 Erdelyi O , page 66] as 1/2 (try) e 1+t (-2+X.1dk) 2 1+t ; 1/2 1 2(txv) 1+t 27ri r (i - K + |0 f ( i * K + |0 .K “ k , , (x) [ I (2|1 + 1)]' (y) a . where L poles of (1) is a path from -i oo to + i oo, separating the I (i + K + ji) from those of (i - K + y). 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 [*+], can at once be given. We start with (Appendix, equation *f) Jm (z sin o C sin p)eiz c o s cos P = J~ i n “ ra (2n+l) n=m ' (n-m^l V 2) C Now let (cos =*> Pn”(cos p). ? 2+7?2 f 2 -'*?2 a k, and cos p = — a , z = kr = then + V. ? we have f 2- ? 2 ik --- 5 Jm (k f 1? s i n o l ) e cos o£ i""1" (2n+l) 2 n-m (n+m)! . .. . _ m (n^mTT V k r > Pfi (cos e) Pn - with ? 2 + C r = ---- 5--- , f 2->?2 and cos 0 = — 5---- f If we now substitute cos oC = 1—t . + f v , sin ©<. = 1/2 2t — in (2), we get t1/2 f V V 2k V W f ' <2n+1> W f . lk ? 2^" •l?t , ^ 5 ^ 2 1 1+1 ‘ ^ Jn'm n=m Jn C (cos e> C O • (3> Now we make use of a certain integral involving the Pasternack polynomial F^(x). Pasternack gives [12 , page 216] +00 |"~(m+1 ) e “ sech x P~m (tanh x) = ^ ^ eixz F^(iz-m) - oo • sech(^j 7r z) dz . If we substitute tanh x = - l-t » so that sech x = 1/2 — 1/2 and x = In t , and replace 1/2 r ( m + i) f i _ P-- (1 ^ ) = iz for z, (+) becomes p* 03 m+z t 2 Fm (.8.B) iV A go • sech(^ 7T i z) dz or 2t ~ 00 1/2 \ r(m+D , m+z * 2 FS <-*-»> -i oo • sec (| 7T z) dz . Let w = = “ (m+z), then (5) becomes ra+i oo 1/2 rc^i) f r r C f O = - 4 J w * m-i oo • sec ^ 7r(-m-w) dw, which can be written as 2 Fn (w) * m+ioo w i+t 11 W ^ W ) 2 \ * „nii ^ in ra-ioo i '2 ' T " (i : I ^2 ? 2' I m _ W) • 2 2' 2 p iu o/ -j 7 7 1 + 1 m . w , sec 2 *<-“-*> dwI ^2 2 2' <6) But we have (| “ 2 m ” f) * T (| + 2 m+ 2> = n sec 2 7r(in+w) * so that (6) becomes m+ioo 1/2 t -m , 1 ^ n vl+t _ _ 1 +7ri w 2 \ m-100 Trl x ^ Trl + I m I (2 + 2 m + 2 ‘ 1 (2 2 f “ (w ) - -2____ dw. [ f (m+l)] Returning to equation (3 ) above, if we multiply it by 1/2 V" t— 1+t and introduce (7) there, we get 1/2 hr .1/2 * v « ■ 1 r ^ i lk 5 T 2L > • V kr) Pn“ 2 . 1=1 ^ r (*■♦*— g) f| (m+X)]2 o«) (8) P^(w) W" ~m-ioo We can interchange the summation and integration here, and then use our expansion 111(12) for the parabolic wave functions, thus (8) becomes 1/2 1/2 Ik ? 2- * 2 . £§ i + r Jm <2 k ? ¥ hr e m+ioo km 2 \ K iri m-ioo /I . 1 * Wv r r- + I m . H x (2 + g m * 2 ) I (2 g_____ gj_ y ^ (e ^ ^ [ r(m+l)]2 ,\j/" ( 7^,- A) dw. (9) This is the bilinear generating function for our parabolic wave functions* Y m(£,>) We have already shown that = ^ k' (m+1) M_ | ^ - ik ? 2) « wm - 2*2 1 *+7 with iA w = - 2k so that for the Whittaker confluent hypergeometric functions (9 ) becomes, after multiplying by £ ^ , f2 » 2 1/2 lk ? ~ ( . Izt t > _1K 2 1+t l/2 V 2k £ ^ l+t-) e 1 Im FT m+ioo f \ -j _ 1 i * i i n ? m + 2> r < 3 - 2 m - roK m [ r becomes (sin ex' sin p)~ra Jm (z sin oC sin p) eiz cos . 2 - i X in ni(m+n + i) [ r (■ + 2)] ( ™ ) • z _ r(2m+n+1) n J 1 ^ (2 ) m +4 m +i * C cos P = 2 (cos oC ) C n (3 ) (cos p). Substituting for Gegenbauer functions in terms of Legendre functions [5> page 1751 m +2 , x 2m F (m+l) P(n+2m+l) /- 2n n (x) = r(2n*l) r ] (-2) -2r(2m^ i r - Jm + n + i _ ■ m+n . 0 n-0 2 2m 2‘ ^ - 2 P (m+1) # 1 (n+2m+l) [~(2m+l) r(n+l) (sin c*. sin p)” r P~ (cos n+ra P”™ (cos p). n+m v r ) 50 This simplifies with Rm+i) = . I2aii 2^ m ^ m* , to 00 J (z sinoi sin P) e1 z cos * cos P = \ f ^ V 2z * ^2m»2n+l)(n+2m)1 Z I in n=0 T , . p-m , m *.* m+n + l?> ( ’ n+m < c o s o i J "5 oo • p;:m (coS p> - z : in-m L 2 & n=ra y ' ^ * P~m (cos © 0 P“m (cos p). (*+) Now, comparing (1 ) and (*+), we can substitute (*f) into (1) if we put p = *f , iz cos oi. = - -K3! + a2 ^ > z sin oC = t v'r~a^~~a^ 2 “ ^ . and Thus al " a2^ a-, 3q --- - -----al “ a2 cos Hence (1 ) becomes, with p = m, and we can change the order of integration and summation here, 1/2 (a,a0) M (e-it)* x,§ 1 (aofc) = • -i. -- “ ® x,f 2 ♦ *) t rr--- r < H “ - *) 51 00 y in-m I2n»l)(n+m)! / (n-m)J n=m r it , *■ 2 ' l „ a2'J 7T • c (i£f) •f ~m (cos ^ K t a n h £-) Of ‘'o 1/2 I (ala2) oo t * : r^l+m ♦ x) r ^ - *> .n-m *\Z. 1 n-m -i\Di 7r • 3n [i|-(a1-a2)] F-“ ( P“ (ccs f ) •(tan 1 )-2x d f . (5 ) Using equation (6 ) section VII. 1 , (5 ) "becomes CP 1 (a a )l/2 t lala2 ; r • -A^ m (alt)' x,2 ,P2 (-n+m,n+ra+l,-x + in~m m (a2tJ = ^ 2 *2 n=m k ; (2n+l) i m+l.m+l;1) (6) 52 Now, as before, we have ^2 C-n,n+m+l,-x + ^5^; m+l,m+l;l) rai R | - 2 m+ f£ (-2x ) |“ (| + § ® + x) F” (~2 x ) F™ m (-2x) so that (6) becomes 00 l /2 . (a^ g ) ' t -M- m^al ^ * x, 2 M. .n-m ffl(a2t) = / i (2n+l) x,2 n=m al+a2 al“a2 r\ Now in (7 ) we put and a,-a2 i t = k al+a2 al-a2 and get X = - 'if , a^t = -ik ^ = c 2 »2 f + V ? 2.— f 2 - yl 2 — 2----- 2 = kr = cos G » , a2t = ik 7^ (7) 53 2 / | \ km+i t Yj > ( M, w ra ^-lk £ 2 ' 2 .n-m -in m /L_ n=m ?? (w) . (2rH°l)(n»m) i kin w m (ik y 2 ) “ 2*2 _n (n-m5T ■.m F„. W , • jn (kr) P~m (cos 9 ) . (8) We have for our parabolic wave functions 2 • x * « < i k r ” 2*2 > • h H s ( i k 7 2 > . " 2 * 2 Introducing this into (8) gives at once our expansion (12) for the parabolic wave functions in terms of the spherical wave functions in section ill* BIBLIOGRAPHY Bateman, H. An Orthogonal Property of the Hypergeometric Polynomial. National Academy of Sciences Proceedings 28 : 371-375* 19*+2 . Buchholz H. ii Integral-und Reihendarstellungen fur die verschiedenen Wellentvpen der mathematischen Physik in den Koordinaten des Rotationsparaboloids. 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Pacific Journal of Mathematics, volume 7 1* 1365-1380, 1957. 11 . Magnus, W. and Oberhettinger, F» it Formeln und Satze fur die Sneziellen Funktionen der Mathematischen Phvsik. 12 . Pasternack, S. Springer-Verlag, Berlin, 19^ 8 . A generalization of the Polynomial Fn (x). London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 7th Series, 28 : 209-226, 1939. 13 * Rice, S. 0 . Some Properties of ^F 2 C-n, n+l,£ jl,p;v). Duke Mathematical Journal. 6 : 108-119* 19^ 0 . 56 lU. Whittaker, £. T. and Watson, G. N. Analysis. A Course in Modern MacMillan, New York, 19*+7