f1 ‘1" I MIMI W IJHIIIHWHIIIWHIWIHIIHIWWM 129 199 THS A METHOD it)? GENERATING WE‘EGRAL REPRESENTA'HONS ‘E‘hesis 50: fits Degree of Ph. D.- MECHEGAN $573.75 UNEVERSEY W’afifar W’iiiiam Turner 1963 111E515 LIBRARY, Michigan State University This is to certify that the thesis entitled A METHOD OF GENERATING INTEGRAL REPRESENTATIONS presented by Walter William Turner has been accepted towards fulfillment of the requirements for Doctor‘s degree in Philosophy J flwwmflm “fajor professor Date May 7) 1963 MSU LIBRARIES -_. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES wilI be charged if book is returned after the date stamped below. A METHOD OF GENERATING INTEGRAL REPRESENTATIONS By walter William.Turner AN ABSTRACT OF A THESIS Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1963 Approved ABSTRACT A METHOD OF GENERATING INTEGRAL REPRESENTATIONS by waiter William Turner Imposing the condition that the Schrodinger equation ‘VQU + ¢U - 0 be simultaneously separable in at least two coordinate systems sharing a coordinate, one Obtains functional equations whose solution completely determines ¢. The Special functions Obtained by the separated ordinary operators can be related through integral relations by using a well-known integral theorem, With this theorem one can predict the value of the integral involving special functions, and in this way some new integral representations are discovered, which contain as special cases some of the existing integral repre- sentations. Thus, a unified theory of these integral representations is obtained. A METHOD OF GENERATING INTEGRAL REPRESENTATIONS By walter William Turner A THESIS submitted to the School of 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 1963 ACKNOWLEDGMENTS The author wishes to express his most sincere thanks to Dr. Alfred Leitner whose deep interest, and devoted supervision have made this investigation possible. He also wishes to thank Dr. Charles P. Wells for his interest in this thesis, and also acknowledge the more indirect influence of the ideas of Dr. Josef’Meixner of Aachen, Germany. The writer is deeply indebted to the National Science Foundation for the financial support that has made this investiga- tion possible. DEDICATION To Patricia, Constance, Anne, Roberta, Nancy, and Rose. I. II. III. IV. VII. VIII. TABLE OF CONTENTS Int reduction 0 0 O O O O O O I O O O O O O O O O Simultaneous Separability . . . . . . Solutions of Schr6dinger Equation . . A.S(l,2) s.s(2,h).. C.s(5,6)............... D.s(i,h) E.S(2,6)................... A Method of Generating Integral Representations. . Integral Representation Using S(l,2) . . . . Integral Representation Using S(l,h) Equivalence of Form for the Integral Representation Using S(3,6) and S(l,2) . . . Conclusion . . . . . . . . . . . . . . . . . . . 11 ll 12 1h 15 2O 27 30 55 51 INTRODUCTION This represents an attempt to initiate a unifying concept for generating many of the existing integral representations, as well as generating some new integral representations for special functions of Mathematical Physics. Three main ideas are involved; (1) simultaneous separability of linear partial differential equations, (ii) solutions of the Schrodinger equation, (1) V2U + ¢ (ul, u2, ”3) U s O and (iii) a theorem concerning a definite integral representing the solution of a linear differential equation of two variables (Meixner-7) . In order that (1) be separable ¢(ul, u2, “3) has a definite form as exhibited in Table 1, depending on the coordinate system under consideration. Equation (1) can then be solved by the method of separation of variables in various coordinate systems for various forms of the function ¢. It is well known that the scalar Helmholtz equation obtained by letting ¢ 2 k2 is separable in all eleven orthogonal coordinate systems involving ellipsoidal surfaces and their degeneracies, which are listed in Table l. The case ¢ , k2 is discussed quite thoroughly in the literature and the use of the integral theorem is summarized by Meixner (Meixner-7). M In this thesis we ask what is the most general form of ¢ in order that (I) be separable in two coordinate systems, and this is called simultaneous separability. This yields functional equations for each pair considered and these equations determine the form of ¢. we will impose the restriction that the pair of coordinate systems in which (1) is to be simultaneously separable share a coordinate. In that case equation (1) can be reduced in each of the two coordinate systems to a partial differential equation involv- ing only two variables, by separating out the common variable. The form of the integral theorem we wish to apply in this thesis requires that the above restriction be imposed (see page 27). When each of the reduced partial differential equations is solved by the separation of variables we are led to two special functions. The product of either pair will serve as the kernel of an integral. By using the theorem and integrating over suitable paths, integrals relating the special functions in the other pair are obtained. In such a way it is possible to obtain integral represen- tations. In 1958 A. Leitner and J. Meixner investigated cylindrical, spherical, and prolate spheroidal coordinates and obtained new inte- gral representations (Leitner-h). We will investigate other pairs of systems of coordinates to obtain new integral representations, which appear in the thesis as (A2) page 33, (A9) page A} and (50) page A6. Special cases of these integral representations are also derived and appear as (hi) page 5h and (51) page h7. we believe (A2), (A9), and (50) are new integral representations. II. SIMULTANEOUS SEPARABILITY The original idea behind this thesis was to investigate all possible pairs of coordinates in Table l which share a common variable but we soon realized this was too ambitious a project to undertake, and furthermore such a project would be of little practical advantage. we could not obtain solutions for a few of the functional equations encountered. Moreover consideration of all possible pairs of coordi- nates led to special functions that were not of the hypergeometric class, such as Mathieu functions, Lame' polynomials, Spheroidal wave functions and other functions whose theory is complex. We, therefore, limited our investigations to those pairs of coordinates which lead to classes of most interest, namely the hypergeometric class of special functions, and their confluences. The new integral representations we found contained as special cases many of the existing integral representations and in this way they are an attempt to initiate a unifying concept into the broad area of integral relations. Our first concern is that of simultaneous separability so we begin by considering pairs of coordinates in Table l which share a common variable. The systems all having 2 in common are rectangular, circular cylindrical, parabolic cylindrical, and elliptic cylindrical. Systems having 0 as a common coordinate are circular cylindrical, spherical, paraboloidal, prolate spheroidal, and oblate spheroidal. Only two systems have the variable r in common, namely, spherical and 10. ll. . Sphero-conal —.‘(($t+ swam) _Y__5'l TABLE 1 U. U2. U3 . Rectangular x y 2 . Circular f3 (.054: psmcp z Cylindrical . Spherical r 5m 9 cos? Y we smo moss . Parabolic g 'l V2614?) Z Cylindrical . Paraboloidal fi'lcoscb 5'1 SWAP A2524?) . Elliptic c“ C (5-0042) 2 Cylindrical . Prolate W cost AWHM 33'! Spheroidal . Oblate a (mo )cost aw/(sfilwii’ismv aS'l Spheroidal (“1me ElliPSOidal (sou-co (wow-s) («-XXP-I Parabolic ”mi-34D Vka‘sfitgg‘g ha}: b.” Ellipsoidal g-VWW‘: '1‘) ¢ 3V1U+¢U=O Is schvable 9.00 + R20!) + 93 (z) B.(pl + 8%?) + 83(2) L3<¢l Y 2 sun—T6 CI( )+___ _C____z(9l D_____.(‘§)+D(~i) 5 '1 “WW BONES) E'm + ‘( 1v9) f___(§)+E('D '5 11+? F; Fm SUN g.(§l *Gzoql ) 8055'?) + 63(z ”105%”le + H3“) flit '1’) #61004?) Le) +15?) Lu») d‘e‘m‘) a‘umu-n‘) sphero-conal. These can be considered pairwise in thirteen ways, but as we previously noted, not all of these combinations can be separated simultaneously. we will consider, as an example, the simultaneous separability of rectangular and cylindrical coordinates in detail and merely list the results of other pairs. Referring to Table l we see the forms of ¢ for rectangular and cylindrical coordinates are respectively: Al(X) + A20) + A3(2) B (o) and Bl(p) + ‘ié" + 33(2) Since 2 is the common variable we have A3(z) s 35(2), but other- wise arbitrary, and we must solve the functional equation 320) (2) 31(0) +"—;§— = Al(x) +‘A2(Y) we proceed by differentiating (2) with respect to x obtaining dAl . dBl _ag_ +_9__ (1”) do dBa do dx dp dx do 2 Now since p2 s x2 + y2 and 0 s tan-1—%-it is Obvious that 92.5. E,1 2,1 “.13.: dx 9 ’ dy p ’ dx p2 dy 02 we now use these relations to Obtain dAl =_§_ dBl + ~2x B '_y__ dBe dx 0 do ph 2 pk d¢ Now we take the derivative with respect to y to obtain 2 _ xd _l_dp (131 del do 2xd 1 pp 2de2a¢ °‘aspa§ap*sear‘db‘HaB'T—a* do o y o y dB dZB -_1____2__y L(IE>22_A XII—31°— ph (10 dop dy d0 d¢2 By 2 0.1g Era “Mag 13.2-1 same all do 2 2 2 d Td¢ die 2 o o do o » o o o o d0 daB dB d2B dB 2‘ 2 l 1 l 2 2 _5__ 6 cos¢ 0-fl _.._.__.._.___ + % BB - + —— _/.L_€__ 2 2 d 2 2 ¢ 1+ o do p p o M d o o Now we separate the variables to obtain 2 (3) ha231_ 31.818 dB2+6cos2¢-5 dB2_BB p 2 do 2 coso sin¢ d¢ 2 d0 d9 Now (3) yields two ordinary differential equations which are 2 dBl -_;_dBl A. 2 d a ’4 (19 O p p dzB 2 dB 2+6cosQ-l 2-8B -A (102 cos¢ sin¢ dd 2 " i The solutions to these equations are: 22 2_A_1_ Bl(p)='ap+k+8 2 -A %- hag %- 1472 and B (.) a — + _ + 2 8 2 cos 0 2 sin ‘9 This determines the particular form that (2) must assume in order that the SchrOdinger equation be separable in both rectangular and cylindrical coordinates simultaneously. B (d) 1 - hoe 1 - hra 2 2 22 h 1+ Bl(o)+ 2 ak-ao+ 2+ 2 9 cos ¢ sin ¢ S(l,2) l 2 l 2 A(X)+A(y)=k2-a2(x2+y2)+E—;£+h I” l 2 x2 y? The problem of solving the functional equation in other pairs of coordi- nate systems is similar to the one Just illustrated and the detailed calculations will not be given. The results will be found in Table 2. To denote variOus coordinate systems we shall use the numbers as they appear in the first column of Table 1. Whenever the same greek letters are used for two distinct pairs of coordinates, for reasons of tradition, we shall subscript the variables according to the numbering in Table 1. For example (:7, n7, 2) are the elliptic cylindrical coordinates. The solution to the problem of simultaneous separability of any given pair of coordinate systems is denoted S(i,J), where i, J a l, 2,... 10. SO since rectangular coordinates in Table 1 correspond to the number 1 and cylindrical to the number 2, we denote the solution of the problem of simultaneous separability in rectangular and cylindrical coordinates by S(l,2). Likewise S(2,6) refers to the solution of the prOblem of simultaneous separability of cylindrical and paraboloidal coordinates and so forth. Furthermore we found that the same function of O arose as the solution to the problem of simultaneous separability for pairs of coordinates more than ones. In fact we found that S(l,2), S(l,7) and s(2,7) have the same solution, as do s(2,3), s(2,8) and s(3,8). ‘we denoted these triplets by S(l,2,7) and s(2,3,8) respectively in Table 2. Table 2 has been so arranged as to call attention to the fact that ¢ has the same fbrm when the table is read horizontally. This symnetry differs from the one discussed above, since the previous symetry deals with the same function of space whereas this new symmetry involved equality of form and the variables are different functions of space. This equality of ferm.can easily be explained by geometrical considerations. Consider (2,3,8): when the azimuth o is held fixed in each of these systems, we Obtain three two dimensional coordinate graphs which are illustrated in Figure l. we Obtain the same graphs when we hold 2 fixed in (1,2,7). (a) (b) (c) Figure 1. Simdlarities between (2,3,8) and (1,2,7) .Azimuth 0 held constant in (2,3,8) and 2 held constant in (1,2,7) (a) Picture of lines of constant 9,2 (or x and y) (b) Picture of lines of constant e, r (or o, p). (c) Picture of lines of constant :8, "8 (or :7, n7) NW NW “N93“ N? NW +~W III N? +NW. «o +7.3“ 9 +I|I| red +w. 3.5+me ONE-m N} b I. d» _ v+¢moun + Iml +~al§~v +SU $.me 3.. w "rem that- 7% ratio: I .emu +32 + w I £33135 3.3 m cfvcmyso scare o n. + a .I. «Tow Illmll | U + Clur+NWvu N fiSNI + ”Wt—I— oacfi a» 0 «81 |fl||+IIN~IN|+ syn. +¥| IAQMWU +CV_U m Wing? en! :2? 3 «mm 6 65 m N @493. «e. M so: lull~>+~m I. «r +~W Lei; -weflm + a + My. .. Erica Genoa at Im... at 29 11.022“. + m +egu 3&5...th Esm been, I «r +«w NrNWNN II flAvaWVNW fl a. In .v I :‘QNO4 AWV.O er die: «x + :3 y. n Sea. + 26 r. em ANfI-v CMMVwU Nu: II I U | I. NY+NW AP¢IF b¢I AA. «gnaw mI Arveolwig 0&2an 4 NwOU Ni 0.4.5 III II I + sucrIix +sb¢I£ + . IeI IAeIwolsm a; x ex j + HALE? s. s 233.2 10 Figure 2 indicates why 8(l,h) and s(2,6) likewise yield the same functional form for ¢ even though the variables are not the same functions of space. Figure 3 indicates why s(2,h) and s(3,6) also yield the same functional ferm.for ¢. Also notice that in Figure 3 6 corresponds to 90° -0. we have not been able to solve the functional equations for the coordinate pairs (6,8) and (lb?) but they too have geometrical similarities. Y z nhel n6a-1 ‘ "6.1 X D gh.1 £6... 1 £631 Plane of constant 2 Plane of constant 0 Figure 2. Similarities Between (1,1I) and (2,6) nu'l Ill y a Plane of constant 2 Plane of constant 0 Figure 3. Similarities Between (2,14) and (5,6) III. SOLUTIONS OF'THE SCHROIINGER EQUATION In this section we will solve the various forms of the SchrOdinger equation, which are Obtained when we use the results of the various simultaneous separabilities. Usually we will indicate two or more solutions of the ordinary differential equations that occur after we assume a separated solution of the partial differen- tial equation. Finally we will summarize the results of this section in Tables 3 through 5. A. s(1,2) The simultaneous separability form of this equation is: 1 2 l 2 2 2 —-1I —-1It2 122+ _8_2IJ_+ —a—g—+ 322(x +y2) +——— a +L—2———+ A3(z) U: 3x By 32 x2 y we assume a solution of the form U’s k1(x)k2(y) em2 to Obtain 2 _l_ 2 dkl+k2_ 2_ -a2x2+‘LL—'f£—k-O 2 “ co 2 1‘ ’ dx x 2 1 2 d d k2 + c - a2 2 + h - hT k - 0 an 2 o y 2 2 - ' dy Y These two equations can be reduced to the form of Whittaker's differential equation. Thus we can find k1(x) and k2(y) to be: 2 (h) kl(x)=-:-L—W (ax2)ori-MV ax),V=____ 3 v; v; m (5) k 2(y)= =\/_. W7 1 (By2 ) o§/::2M7 r (aye) ’ 7 3 EE— 12 In cylindrical coordinates the same lequation is: ' 2 l is an 1 _a__20 figs "rho The _ (p )+— 3,229 + + +A(z)U=O p'Eb 5p 02 372 32 2 p2cos2¢ o2s1n2¢ 3 we assume a solution of the form U = gl(p)g2(¢)eipz to Obtain 2 d s 8 1+1 d 1+ 2 2 2 2 1 ~———— k - u - a o --——— g = o , dP2 +p do 2 1 deg -1—- If —1-- use 2 h II and—2+ °i+—2—— +“—12—“ 82-”- d¢ cos ¢ sin ¢ The first of these equations can be reduced to the form of Whittaker's differential equation, and has as a solution (6) 81(9) =-1— w (ape) or 1 (ape), e = —.j— p v+7,e E'Mv+7,e The second equation can be transformed to a generalized hyper- geometric function (Leitner and Meixner - 5). The solution is AL .. - -i. (7) 82 (¢) = (1 - c052°)h ¢ 21’ 2° 2 (cos °) 26 -»le 2 s<2.h) In parabolic cylinder coordinates the equation is: -l—— _§E§_ aZU +BZU+ + -——- + -E— §2_q2 + d + B ( ) U-O 2 2 2 +_ 2 2 k222812 2 2 2 2 2 2 ' 3 z ‘ i +n 8: an 52 e +n 5 n e +n i n we assume a solution of the form U = kl(£) k2 (n) e1“z to Obtain l + k2 + 2a - co - u?§2 +.Qih. k = O , dg g2 1 l3 '2 2 d + b + co - u n + 2 k2 = O and 2 dfl n Both of these differential equations can be reduced to the form of Whittaker's differential equations, 2 l 2 1 2 . k +2a-cQ ‘Vl+hb-hd (8) kl(§) +%+12-- 12(5 +n > g n Case(l)d=0 T=O b=—i—- a In this case for rectangular coordinates the SchrSdinger equation becomes: 2 l in. sin a_2_u 2 1'“ ”3(2) he 2 k + 5x2 5y 2+ oz 2+ x We assume a solution of the form U = kl(x)k2(y)eiuz to obtain deki 2 2 2 i" “2 + k - u - 7 +'——2—— kl == 0 , dx x d2k 2 and 2 + 7 k2 = O . div The first equation can be reduced to the form of Bessel's differential equation, 1 (17) 1‘1 (x) = x2 Zo- (i V3402 x) , a = V J‘s-kg , where Za is any cylinder function. The second equation is easily solved. (18) k2(y) = sinyy or cos7y For parabolic cylinder coordinates we have: 1_2 2 -a 520 5211 1 1 (U+—) 22+k+————(—+—)+A(z)U=o ’2282 an2 +225 “2112;? n2 5 17 We assume a solution of the form U = gl(§)g2(n)eiuz to obtain 2 l 2 d g -a 2 2 2 h 21 + C1 + (k '“ ) 5 +'__—§_- 81 = O ’ d: E deg .1_a2 and 2 + -c + (ke-ue) n2 + h g o . (in2 1 2 2 Both of these equations can be reduced to Whittaker's differential equation, (19) glm ’17:" w (61:2) “vi: M a (an 2), a = VIP-k2, «231—, 6’2 ’2 (20) 32(n) =— 3(an2 ) or-—-M 0(an2 ) WW“ V'n— ‘32 Case (2) a = 0 cl = o b = 1 - 1602 In rectangular coordinates the Schrodinger equation becomes: 2 aeu + 520 + 320 + k2 + 1-166 _ hTeyz _ T2x2 + A (2) U a o 2 2 2 2 5 dx By 52 x we assume a solution of the form U a kl(x)k2(y)eiuz to obtain 2 V d k 2 1 2 2 2 2 - 6 + k- -u- -72- T x +— 1 1 a , k = O , 2 2 1 dx . x 2 d k and 2 + 72 - hT2y2 k = o 2 2 dy The first equation can be reduced to Whittaker's differential equation 18 (21) k (x>= w__v “NF—1662-3 m2) orim_v_i€3r—1502-3 (132), Vac; 2 V; T , 2 u k2 72 V=_. T ’ hr e = The second equation can be reduced to the parabolic cylinder equation (22) k2(y) = D 1 (-2 V373» or D (2V1? y). l V-2 v-72— The differential equation for k2(y) can also be reduced to the confluent'hypergeometric equation of Kummer, and another choice for k2(y) would be 2 (23) k2(y) = e 11y 1F1(:g+%3%32i1y2) or e iTyg'lF lC—-+E;§3211y2). For parabolic cylinder coordinates we have: § 6 6) 2l 2 (BZU 52g) 523 R2 + 121622 6&2L 2 h2g 2 +A3(z) U=O a +n 5&2 5n 52 . § +n g n a +n We assume a solution U = gl(§)g2(n)e iuz to obtain 2 d g ' 2 1 2 6 - 6 + (kg-ue) a - T2: +-2-2-9- g = 0 , 2 2 1 d5 - 5 deg 2 and 22 + (k2-u2) n2 - T2n6 +.l:lgfl__ g2 = O . dn ' n Both of these equations can be reduced to Whittaker's differential equation, -2 -2 (2n) elm = g 2 Wm <'—; :2) or e. 2 “m g >, .2 - (25> gem) = n2 We“, ('1; n“) or n 2 MW ('2 n > 19 520 ago a2u 2 3 2 2 2 x2 —2‘ 2 2 Y“? ’"3" 5x By 52 - hx + + + w + I'D D; I + A 5(z) U a 0 we assume a solution of the form U kl(x)k2(y)eiuz to obtain d R1 2 2 2 2 x2_ 5 + k- -u -7 -T k = O , dx2 fix? 1 2 d k and 2 + 72 + 2dy - hreye k = O . dye 2 The first equation can be reduced to Whittaker’s differential equation 2 k2_'2 (26) k1(X)=\/31¢—wa-v,i[5 (1x2 ) 0?Ma-v’ 11‘5”): ).’ 6.1L, 2 . v =-£;—, d2 = 161362. The second equation can be reduced to the parabolic cylinder equation (27) k2(y) a D l(2w¢¥y - 26) or D 2 1 (-21J¥y + 26). +V-€2 '5 . V-€ '5 In parabolic cylinder coordinates the Schrodinger equation is: 21( 2 (52U ”Beg ago + k 2+d<§2_ n2)- 3 2_12é§6;n6)+ A5(z) U=O a +n 5:2 5n 522 ha n t +n we assume a solution of the form U = gl(g) g2(n) eiuz to obtain degi - 2 ‘2 2 h 2 6 3 —2-+ (k-u): +d§ -'r§ -—é—gl=0, d5 hé 2O 2 d s 2 2 2 2 u 2 6 3 2 + (k -u )n - dn - T n ----5- 82 = 0 - dn 1m and Both of these equations can be reduced to parabolic cylinder equations, (28) 31(5) \/__ Da+€2 (W/?- E + e); (29) 22(n)= ‘/__ Babe 62 (\f?- n + e) E. s(2,6) This case is somewhat similar to the S(l,h) case so we will consider two cases, which we will now indicate. VB1(O) + B3(z) = k2 + 2dz +-D§ - h12z2 - 1202 p SW) 95: <) <) 2 F E +F n 12 2 2 g k2+d(g2-n2)+ éb2 _ 12(52-n2) _ 12§2n2 \ § +n a n Case (1) d = O 1 2 o For cylindrical coordinates the Schrodinger equation is: p a + -—-+ ”ES? D2 672 :2 2 02 02 ’¢ we assume a solution of the form.U = kl(p)k2(z)e1H to Obtain d kl A_dkl 2 2 b _ 2 'd0 9 D . p 2 d Re 2 21 The first of these can be reduced to the form of Bessel's differential equation, (30) kl(o) s 20 (Va) , a = V11-43 , v = k2-72 For the second equation we have (31) k2(z) a sin 72 or cos 72. For paraboloidal coordinates the SchrBdinger equation is: b+B (¢) 1 1 a EU 1 a 6U 2 2 — (g—)+— (1] +-— + k + ——-— U = O §§+n§ g 5E 5g n Bfi' SE9 5.2 g2W2 -We assume a solution of the form U a gl(§)g2(fl)ejp¢ to obtain 2 d 8 dg 2 l l l 2 2 b-u d§2 g d§ l g2 l deg2 1 d82 2 2 b-ue andF+WW+ “1*“ 712‘ 82‘0- Both of these equations can be reduced to the form of Whittaker's differential equation, (32)gl(§)--g—W€ (m?) orL 1% 6,2 In cylindrical coordinates the SchrBdinger equation is: B (¢) 16_(pét1) +1. fi+a_2u_+ k2 +1__ 1202 , {”222 1 2 p 2 2 2 2 2 p 3¢ 82 p p U = O 22 1u¢ We assume a solution of the form U a kl(p) k2(z) e to Obtain 2 d k dk 2 l l l 2 2 2 2 b-H '————'+‘—- -——-+ k - 7 - T p + k = O 2 d 2 ’ do 0 p p l d2k2 2 and + 7 - ht z k = O 2 2 dz The first of these equations can be reduced to the form of Whittaker's differential equation, 2 2 (3h) k1(p) =-l-W (Tog) or-l-M (Toe), €=g;-, V ='£;', p 26-V,O p 26- v, a a = Vb-u . The second equation can be reduced to the form of the parabolic cylinder equation, (35) k2(z) = D 1 (2‘\f;-z) or D l (-2'\f; z V--2' I"? In paraboloidal coordinates the SchrSdinger equation is: 1 1 62U (a §)+ (n— + .+me”0=° §2+n2 g a; néfi— an gene 5,? 2 B (¢) where F(§,fl,°) = R2 + 2b2 ’ T2<§2'n2) T 2g M“ + 2 2 g n 5 n ¢ We assume a solution of the form U a gl(g)g2(¢)ei “ to obtain d2g dg 2 21 +-%—-EEl-+ -9l%—-+ k2§2 - 12g6 g1 = o , d5 5 deg dg 2 2 _1_ 1 b-E 2 2 _ 2 6 and due + n dn + 2 + k I n 82 = o . 23 Both of these equations can be reduced to the form of Whittaker's differential equation, 1 T h l T h (36)gl(§)=—2-W a 35) 01' —2M 0 gfi); § 62E E, ('59): (37) 22 (n) =1—2 W a (gnu) or 17M 0 (gnu) Tl 5"); fl 6,? 30,2)4 TABLE 3 ‘ z z / <4) h.(x)=V—7W,,,(RX) "T's:- Mmgax) (5) hM='—W ( 2) Y J— 2 z w :3 a7 ° W M8,“: (3W) \ I (6) 9'(P)=va+1,e(af°‘) or 7;- Mv+Y’&(aF1) \ (7) 92(4): mm)"4 Niffify‘uosq») / (2) h|(’)=v—';’W«,t(/‘52) °Y w/I—f MmWi’z) (9’ W717- Wv’,(m‘) mr 3;? ngun‘) 5%“ (‘0) 3I(P)=T%Wx,e(ZVF‘-R‘P) °" me (aware) m) g,(¢>)= (I-sm‘q»)v" §:?_i:-&(sm¢) 3 (3,6) L (.2) 92“,): cos"2¢ (I+sund>'$t(I-smc§)¢zfiM-e-a-tfuec-Ql-2’t;cos‘(‘5’r¢4)) / (13) Nah “lg-Wm‘k't?) or _;_ Mv’cflats‘) (m tm= % mem‘) ‘7 MW (at) us) g.= 4, Wx+»,e(2k'"l Jr" AAM’Eum) ~ 1?: 7.6-"; K «6) 32(9):: an)”. (cos?) 21+ ‘1 TABLE ’4 SW?) 7 07) 2‘00: X)2 25-“. Va‘+x‘ X) case“) 08) Mb smw d ='c=o< ‘ b=V+-a‘ (19) 9.612,? W£)gi_(as‘) k (20) 32“‘=\;‘7W5g (W) = _L. , a. , (2|) hm WWWeflg (’tx) (12) ha“): 13,4, {-24% Y) CBSCQ) '1.th d=c.-0< (23) my): a .Efiz’ *lzfifi-i‘r‘l‘) b: MM" (24) g,($)=- 5-35 We) 6(‘1'; 5") V25) 32% YE" We), ($41”) I . z / (26) hum—x.— WM)? (m cascb) (27) k2“): Dv-a’thfiV'ze) “3-3 - _L b‘ 4 (28) 8"?)‘43— D5+£1 (1&5 +E) K (29) 3,01%“ V431: D642 (fin +2) 25 or COSBY I 1. °' 72%: (3’) \ 2. or W Meg: (1‘1) °r ‘T/Lx— Mv-ze,J——“;13 (T X‘) 0V .Dwyz (ilk-7C Y) or émy .F. @4332; 1‘3”) or 5‘35 Me fit“ ) or V135 M2,: G '1?) \ ' z or "xi—x- Ms-va/E (tx) or Dv_£z_yz {-zfiwze) °V V—_é-_Dc+£z(:\ff$-E) file-Et (V? 'l“ E) TABLE 5 3(2,6) / (50) k.(pI=Z¢(v.o) “sew (an) Irez(z)= smarz ’t=cI=0 ‘ <32) g.(s>=?WE),Z-_(ns‘) \ (as) 9:"1I=%‘W,¢(LkVI‘I I? / (3+) Mp) =;TW2£_),FWPZ) (35) RZIZI‘; 0,42 (HE 2) casem d=q=o (as) %.('§I= “—éswg’g (it: 5") I (31) 92W:- le—Wffijqévt") 26 or cssYz I w E'Mg)%(tks‘) I CY 71- ME)? (Unit) L 2 W P MZE-vfiu'P) or Dwzéz 'V‘E z) I or 37 M6543; 3*) I 0v viz-Magma IV. A METHOD OF GENERATING INTEGRAL REPRESENTATIONS In this section we shall introduce the theorem which enables us to generate integral representations between the various special functions, appearing in Tables 3 and h. First we will introduce appropriate notation. Let x1, x2, x3 represent mutually' orthogonal coordinates (any system appearing in Table l). The Schrodinger equa- tion, V2U + ¢ U a 0, for the various forms of ¢ in.Table 2, can be solved by the method of separation of variables. Thus U(xl, x2, x3) = kl (x1) k2 (x2) k3 (x3). The associated ordinary operators will all be selfadjoint, e.g. of the form a 5 L1 3&1 Pi (xi) $1 + qi (xi) 1 g l) 2) 3' When we write U(xl, x2, x3) 2 K(xl, x2) k3 (x3) we obtain a partial differential equation of two variables of the form (38) Fl(x1) Ll K(xl, x2) - F2(x2) L2 K(x1, x2) The theorem is: Let (a) c be a path in the x2 plane with endpoints a, b. (b) B be a domain in the x1 plane. (c) fé(x2) be a separated solution of L2f2(x2) = 0. (d) K(xr32) be a regular analytic solution of (58) for x1 (59) when m1) =fc m1. x2) F2'1) W,,,,,E(.o*) d sm‘icp Rho-v) o eflgiggig P26 P Let z a p2 and examine this expression for small cos ¢. zc+i cos 4; R-16) \ 6’5- z t(¢)3 smyz¢ r0936.» 3 oz WK‘IJZ) WV+X,E( ) dz 4- (”85-2% R") '- 1'”?- W (2) W (2) a ran/15¢ ram-» Z Z 8;: v”): 2 O These integrals can be evaluated (Bateman-l, Vol. 2, Page th (h2)). ‘4‘) t(¢) $1". Coé‘t’ét> R163 (R%+£+I+S)E3-€+t+s) R-Z’t) 3é+£+t+67re+t+ sfi-XVC ‘] 25"",14’ WT“) Ryz‘Z-t) fiz-v-n‘bs) 3E[ Hit , ‘z—v-xruc 3 + Rim-1&6) Rafi-em s (at) [%+e—’t+sfi-E-t+6’ v1-3-1. ] R’é’X-t)r(z-v-‘l-'t+6) 32 “741,2-v-x-t + 6 ; | + cook-25¢ [_(26) What-5) [(31% +t4)‘Ezz) [3&+£+t-6,32-6+t-6,’5-3+2 ‘ ] 25sz¢ Ryzi’sfi) WE'X-t) f-(z-v-n’t-s) 32. \+Z’t)Z-v-X+’t—6', Halve—ts) R’ré—‘r—S) Rn) 2+e—1— qX-E-t-cfi-X-‘t ] I—(K'MfiRZ-v-I-t—s) 3E \-7.'t)z-v-34_--5 3‘ 33 2. "',.2't16 When we examine the various forms of (l-Cos4>)y+ ¢z€_y1:’&(cos¢) for small cos ¢ and compare them with the previous expression for t(¢) we find that we have a linear combination of the functions -Z’t 26'- ’2 y ~~zt, '26-"; (I- -Cos 1'ch) Y.“ 3y: (c.0540 and (l-Cos 2W 2‘- ya (cosdr) when. m s+1=~6 J5) ~ -2t,25-Vz1[ e ( 9",”; m3‘*’,,, [T+6+£+‘& {he-ed 2(CO )7- . 1] 1E ‘9' sq) 2“"I—I+zc)l_e— t-6+‘2)R€+‘t-s+’2) Z ' ”’25 ,C051> So if we define #él— 26)R|+z$) Eut- 6'4-‘0 REM-y t") T (v,1,o',’t,£) 21r 26.6015 6+‘t Jr.) fiyz-c-v) (2 fi-Zt)r(35+e+t+6)r(3i-e +t+c) [c+‘t+e+7i,c+‘t—£+§a.,y2."+t ] l rm’V't) I—(Z-v-X+’t+s') SFZ I+2t,2-v-X+‘t+s 3 [_(Z't) r(5i+£-’t+6) [-051 -t+6) Em Fruu‘ig-r-e‘kfi-x-z ] [-(‘i'hfi [_(z—v—r-tm) az \-1t,2.v—z-'r+¢- 3' we can write (42) ]; Whig) WmW') Wy+1,e(f") dp J}. .‘L - T6» {1,6, t,e) 0-1x'coso) Y3? _Z: (cow) + Th ,1,- at, e) (\- cage) Y ~21:- ,z’i («,qu provided |Reol + |Re 1'] + IBeel ( é. 31+ We shall now check this result against known cases by assigning special values to the parameters. Let v: 64-"2 , and Y= 23-5 . The integral of (12) becomes 00 26% 22+”: \ 6+1 “5- cos 4) sm ¢ ? Z 82' WON-2356(2) dz when [0122. O This integral can be evaluated (Bateman-l, Vol. 1, Page 337, (8)), and yields x, y £052“ 4> smuucb R6+t+a+3é)r(6+t-e+°/z) I (43) — 2 we now must show that the right hand side of (#2) reduces to (MS). First we note that T(s#z,’t+’i,—c,’t,e)-?O and also the last half of‘Rm-Vzgn ’t+¥;,,6,‘ca) vanishes due to the factor [_(‘i- 3+t) We have 27'E V; I—wc) r’t- (+8 +5) fi-t-s +£4JQRX+ER+JR "*‘t*‘) 2:“. 8““ 8-6 +1: Vt.) T(6f’i;t+kl., 6) t)€')-2 -' t, hence. (I C(54))” Y2:_:6-yz (506$) TG+5,t+vz,6,’t,£) : £05265 4. a nu”: 4; — LRMt-wfi’z) Row-t- 03$) VI. INTEGRAL REPRESENTATIONS USING s(1,h) Referring to Section III we recall the case S(l,h) was subdivided into three special cases. In this sectionwe will Obtain integral representations for cases (1) and (2). A. For case (1) we have: 1 2 . 2 . '--s d 2 2 2 h F1(‘)L1 - age + c1 + (k -u )z + :2 Flu) - 1 P1“) - 1 2 2 s -d 2 2 2 I? F1001? - —3+ cl - (k -u )n - 2 F201) - 1 p201) . 1 _ an n - 1 2 2 --s Fi(x)Li . d 2 + k2-n2-72 + h 2 Fine) - 1 pi(x) - 1 dx x - 2 I 'd _ 2 t I The separated ordinary differential equations corresponding to these Operators are solved in Section III. we assumed that iuz iuz U . kl(x)k2(y)e , so we simply list the or U - sl(£)se(n)e solutions we will use in this section, which are detennined by our choice of path and the vanishing of the bilinear concomitant. For our kernel we choose the product of (19) and (20) l 2 l 2 W, 2 2 K(£.n)a—W a (6§)- W 0(an),a= u-ls , V; €,— V1] '6’;- ’ 2 2 and formula (18) for {2(12) - sin7y. If we choose a suitable path and use K(§,n) sinyy as our integrand, the integral will then represent Bessel functions of order to and argument VkE-uE-yz x , except for a factor ii'. It is convenient to use the notation a2 a uz-ke introduced earlier. 35 56 The integral t“) :[V—w WEAR?) _\/-"L-W£%(a|12)sm3ydy V will be some linear combination of )(2 Itch/d117- x) , where I is a modified Bessel function. Our choice of path C is the real axis from -00 to +00 , since this causes the bilinear con- comitant to vanish, e.g. W5.z(35 )W.s (a'l‘) W (35W 5":(a'l) + $\MY§-y[ W {- ]- 6”": W 5—;[5\MY]] =0. —00 provided [Real > 0. 00 Q) The integral can now be written (44) “>0 =[Vr—W Ea" (EON-am WE, New WY )SMXY d)’ [V’W EC (aY an]; N) WE d.(- ay MVX‘ar I) sad, (1’. To determine which solution of the modified Bessel equation t(x) actually is we consider only the first part, namely 00 , i p' --:—and when x a 0 this is valid for all real values of y provided |Re ha, + [Re V166 -3|<2. By the theorem of Section IV we may assume t(x) has the following form: -1 izfcy —L’tY7' t(x)= We o')v,’t) e |Fl (:52; 'E' ZL’tY )+B(€,6,\3,’C) Q ‘/ $0222} 2 ,ZlTY) It remains only to determine A(e,a,v,‘r) and B(e,a,v,'r). To determine B(€,a,v,'r) we consider the “mo 25%)") +00 hm 3:533; 8_'12W(-3 1 -t AX 8(2)‘>”>2)= Y-vo V\/£)6(2X)VV€)‘(-z-x’) WU_2£[__\L:-a (Tx‘) 72 -00 A ) easy“ 3"1 Since __ + —Y— _ —8y , the integrand tends to zero hence B(E,S,v,’c)s o . To determine A(e,a,v,1') we let y approach zero. 00 9aw§6 t; x9]Zw-v.z£m (sz) ix; )7 we notice that Kummer's function does not depend on a, and s so we consider the special case where a = e --%—. Our integral for A(e,e-g,v,1) becomes m v )-\/ :S‘ues1 W ( ‘) 9‘;- E)E-z,v)’t — 22’ o -y—Lg)m ZS 32. where 3=V§ X. Let zs‘=u to obtain co 9’ 3/2; 22-? 2 Q(€)E-‘é)v)’t)—2—zi u e quj-W (u) d“ o This integral is known (Bateman-l, Vol. 2, Page hO6, (25)). A 33 a Vz)‘)VZ-v hm _Aa \ff G34 2£+Y%3%+K,2€-figtqg%f)§,l\ H(E,£-’é)v,‘t)= ""° n522€r(’z+V+ZE+464M)l—(Vz+v+1E-V4e’—+E+‘4) Let R = 1¢§62-he +-%- and use the definition of the Meier G- Function (Bateman-l) to obtain 1&7 1:." Q(£,£-’i,v,’t)= V—a . R5+v+ze+R) “if—(Yzi-vHe—R) I W R—ZR)R:A-2€-F§) R5+ZE+R)l—(;—t+R) R5 *WZE +R) )5: Ze+R,‘2+V+2£+R,Ze oR ] “0 aze'R [_(HZe-MR) 3E3 "ZRx‘flemflx‘nHiR 3 6 Km) [Ex-25m) R1§+ZE-R)R25-R) Wanna—R) |&+22-R,ze-R ”,1“, +224? item [—(H-ZE-R-A) 3F3 I-ZR)I+2.£+X'R,I+Zg-ka 3-3 [-H) W?“ l—(Vzw-A) RA+2£-R)RM2&+R) ['Q-x ,-A ,ie-Aw _ D 3 ax [—U-ZA) I-Z\)|-A-ZE+R,l-A-Z£ -R; Since R-x) = 1%— in the neighborhood of x = 0, we obtain \f’g I—(zuR) [-(za-R) I—(w 1g) 2.25 [-(VzwnuR) I—(Vzwne—R) . Q(€)E-Vz)v)t) = For the special case a = e - 5 our integral becomes 1,41- x2. (SI) WEE_,£(’§‘ 5)W #262 Vt‘) W u- -ze git—’5 m“ E Rzom)r(2s-W)RV*’Z\ 3“” F (realism?) 222 R’2+v+2nfie‘-4eo§)r(’i*‘”u 445—4234) I ‘ VII. EQUIVALENCE OF FORM FOR THE INTEGRAL REPRESENTATIONS USING s(3,6) AND s(1,2) To show the equivalence of form we will begin by considering S(l,5). Referring to Section III we have 2 __l..§_ .9. _ 2 2 .9:P;H . - Fl(§)L1 - g dg (: dg + 2a c0 + k g + £2 F1(§)-1 P1(§)=§ F ( )L -l.d ( d + - - k2 2 - d+b‘“2 F ( )-1 ( ) 2 n 2 3 n Efi' “'Efi co n “;2" 2 n ‘ p2 n =n ' '39— 2d 22.. ' ' 2 Fl(r)L1 dr (r '5?) + ar + k r cl Fl(r)=l p1(r)=r F'(9)L' . i9. (sin 9-9—9 -c sine - d‘”2+b cose F'(e)=1 '(6)=sin9 2 2 d6 d6 1 sin6 2 p2 The separated ordinary differential equation corresponding to these operators are solved in Section III. we assumed that U a k1(§)k2(n)eiu¢ or U a g1(r)g2(9)e1“¢ so we simply list the solutions we will use in this section, which are determined by our choice of path and the vanishing of the bilinear concomitant. For our kernel we choose the product of (13) and (1h), K(e,n) =-%—wv,a (kig2)-%- ”7,1 (kin2) and (15) for F2(x2) =-%- w (2kir) 1+9 By the integral theorem of Section IV we have Me) = é—WVFUQ'LSI) 41. me'f) L, wwnux) .1. C’ where t(9) represents one or possibly a linear combination of the various functions denoted by Muffin (cos‘é) "’21'26 z ‘ If we choose a suitable path and use K(§,n) €2(x2) as our integrand, the integral will represent a generalized hypergeometrfl: equation (Leitner—h). For our path we choose the real axis from O to d3 since this causes the bilinear concomitant to vanish e.-. ) 2. 3 i - 1 ' 1W ' ° r 5-; [5i WWII“? ) WmQRW) 7 vafihmr) '- i I J :— WVJAELS-L) Mfimmz) ‘3? [_L— vaahma] O 5’! provided lReO‘l + IRC'EI +‘Rcil