THE SOLUTION OF DIFFERENTIAL EQUATIONS THROUGH INTEGRAL EQUATIONS by Charles Andrew Swanson A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the standard required from candidates for the degree of MASTER OF ARTS. Members of the Department of Mathematics THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1953-ABSTRACT A method of writing the solut i o n of a second order d i f f e r e n t i a l equation through a Volterra Integral Equation i s developed. The method i s applied to i n i t i a l value problems, to special functions, and to bounded Quantum Mechanical prob-lems. Some of the res u l t s obtained are o r i g i n a l , and other r e s u l t s agree e s s e n t i a l l y with the work done previously by others. ACKNOWLEDGEMENTS We are deeply indebted to Dr. T..E. Hull of the Department of Mathematics at the University of B r i t i s h Columbia f o r suggesting the topic , and f o r rendering invaluable assistance i n developing the ideas. Our further thanks are due to other members of the Depart-ment of Mathematics, e s p e c i a l l y to Dr. R.D. James and Dr. G.E. Latta, and to members of the Department of Physics at the University of B r i t i s h Columbia. We are also pleased to acknowledge the f i n a n c i a l support of the National Research Council of Canada. TABLE OF CONTENTS INTRODUCTION ( i ) CHAPTER ONE. THE GENERAL METHOD (1) Section 1.1. Derivation Of The Volterra Integral Equation (1) Section 1.2. Determination Of The Arbitrary-Constants (3) Section 1.3. The Vo l t e r r a Integral Equation For The General Cauchy Problem (5) Section 1.4. The Equation With The F i r s t Derivative Term Missing (10) Section 1.5. Solution Of The Integral Equation ... (11) Section 1.6. The General Solution Of A Related Integral Equation (16) CHAPTER TWO. APPLICATION TO INITIAL VALUE PROBLEMS ... (20) Section 2.1. Introduction (20) Section 2.2. A Problem With A Perturbation In The F i r s t Derivative Term (20) Section 2.3v Another Single Perturbation Problem . (22) Section 2.4. Multiple Perturbation Problems (23) Section 2.5. Another Treatment Of The Problem In Section 2.2 (26) CHAPTER THREE. APPLICATION TO SPECIAL FUNCTIONS (23) Section 3.1. Introduction (2$) Section 3.2. The Expansion Of The Solution Of The Confluent Hypergeometric Equation In Series of Bessel Functions (29) TABLE OF CONTENTS (Continued) Section 3.3. A Generalization Of The Problem (33) Section 3.4. A Further Generalization (35) Section 3.5. Solution Of Related D i f f e r e n t i a l Equations Expressed In Terms Of Bessel Functions (37) Section 3.6. Appendix To Chapter Three (39) CHAPTER FOUR. PHYSICAL APPLICATIONS (42) Section 4»1« Introduction (42) Section 4.2. The Integral Equation For Bounded Quantum Mechanical Problems (42) Section 4.3. The Bounded Hydrogen Atom Problem .(45) Section 4.4. The Bounded Rigid Rotator (49) Section 4.5. The Ground Level Of The Bounded Rigid Rotator (51) Section 4.6. Higher Levels Of The Bounded Rigid Rotator (54) BIBLIOGRAPHY (57) ( i ) INTRODUCTION The central theme of t h i s t h e s i s i s the use of a Yolterra Integral Equation to express the solutions of a second order d i f f e r e n t i a l equation i n terms of known functions. By the procedure which i s followed, i t i s then possible to derive c e r t a i n properties of these solutions systematically. The idea originated with Cauchy, Liou-v i l l e ( 7 ) , and contemporaries i n the early nineteenth cen-tury. In p a r t i c u l a r , L i o u v i l l e transformed the equation (a) u V ; +• f2-«&) = u&) into the i n t e g r a l equation of the second kind x (b) u&) = u&) coyf*.+ jruf(o)finf* + f-/ fr(t)s\*f(x<)u®tt. The c l a s s i c a l approach was to consider (a) as a non-homogeneous d i f f e r e n t i a l equation whose homogeneous part has known solutions cos j>* and si^f-x f and to apply the method of v a r i a t i o n of parameters or Laplace Trans-form theory to obtain the i n t e g r a l equation (b) i n terms of these known functions. The equation (b) was used to study the asymptotic behaviour of the eigenvalues and the eigenfunctions of (a) f o r large f • More recent investigators, notably Ikeda (4), Fubini (3), and Tricomi (15) have changed the viewpoint to that of comparing the unknown solutions of (a) with the known solutions of a d i s t i n c t d i f f e r e n t i a l equation (c) +• f^vC*) = O, or, i n general, of comparing the solutions of ( i i ) (d) P&) ufo-h uCr) = O with the supposed known solutions &) and i / ^ &? of (e) ir"&) -f R&) ir'fr) + SfrJ *rfr) _- o through a Volterra Integral Equation. This approach w i l l be used throughout the discussion. In the f i r s t chapter, the i n t e g r a l equation asso-ciated with the equation (d) w i l l be derived, and a procedure w i l l be given for the determination of the a r b i t r a r y cons-tants i n order that the i n i t i a l conditions be s a t i s f i e d . The whole idea w i l l be generalized to an n- -r*« order l i n e a r d i f f e r e n t i a l equation, g i v i n g a r e s u l t e n t i r e l y analogous to the second order case. The appropriate existence theorems needed i n l a t e r chapters w i l l be proved. The very nature of the method suggests that i t be used to get expansions of solutions of c e r t a i n d i f f e r e n t i a l equations i n terms of better known solutions of other equa-t i o n s . In the second chapter, we use t h i s idea to expand the solutions of the Confluent Hypergeometric Equation i n Bessel Functions of the f i r s t and second kind. The computational value of such an expansion has been discussed by K a r l i n (9). In the fourth chapter, we s h a l l show that boundary value problems as well as i n i t i a l value problems can be handled by adapting the method. In p a r t i c u l a r , the bounded Quantum Mechanical problems are discussed, and eigenvalues f o r the Hydrogen atom problem and for the r i g i d rotator problem are calculated. (1) CHAPTER ONE THE GENERAL METHOD 1.1 Derivation of the Volterra Integral Equation. The object i s to express the solutions of the second order d i f f e r e n t i a l equation (1.1.1) " V ? + P(*) u'?) -h pGO u&) = O i n terms of the known solutions and of the equation (1.1.2) nru&) + R (x) *rV; + S&)*r&)^o by a Volterra Integral Equation. I t i s assumed that the U equation has the same s i n g u l a r i t i e s as the 'V equation. The r e s u l t w i l l be obtained by adapting the method of v a r i a t i o n of parameters f o r solving non-homogeneous d i f f e r e n t i a l equations: l e t (1 .1 .1) be rewritten i n the form (1.1.3) + + = CW-PCrlJ ur&? -4- Ls&l-Q&ijufr) with supposed s o l u t i o n (1 .1 .4) u&) = = LZ(zl-PQr)] - * + / where and fi} are constants of integration, b i s a con-stant , (1.1.12) and the Wronskian of ^ and i s given by AT, 'a? I I f (1.1.10) and (1.1.11) are put into (1.1.4), i t i s seen that U&) must s a t i s f y the i n t e g r a l equation (1.1.13) U&) - ci -f /?, *rj& + f NCttx)(^l- ffrl}(/e)Jx + f *M¥0(S-tp} u „ _ K * ; ^ . » | . u / ^ ; 941 foe? = f W^ ; n c e ^ ^ ^ l (1.2.3) 'd'iLift) = _ 7 where use i s made of (1.2.1) and (1.2.2). From (1.1.16), The r e s u l t now follows because of (1.2.1) and (1.2.3). From (1.1.15), (1.2.4) = t>}'+ fl^W and (1.2.5) u f e = f* ?£(9>*7Jx-f- K(**)u&). Upon use of the Lemma, Hence, f o r t = i> (1.2.6) uV ~ teQ,)-Pto}nQ = <«i') + /nr^O-) . The solut i o n of the l i n e a r algebraic equations (1.2.4) and (1.2.6) then gives and (3 i n terms of the i n i t i a l values u(i) and u'ii? : where (1.2.8) <%)= u ^ ; - { zto-wy uQ,) . This i s the r e s u l t that Ikeda (4) obtained by a di f f e r e n t method• (5) 1.3 The Vo l t e r r a Integral Equation For The.General Cauchy Problem. The solution of the n-th order l i n e a r d i f f e r e n t i a l equation (1.3.1) u PmJ5t? u assuming that i n i t i a l values U Q.) (4=/,Zj • - -i s to be expressed i n terms of the l i n e a r l y independent solutions ^(xl C^=/}i .--•*,_) of the equation (1.3.2) -v(Hh?-<- Z * W C " \ x ) = o through a V o l t e r r a Integral Equation. We suppose that the functions P and K &) are analytic for a l l required value of ^ . Let (1.3.1) be rewritten i n the form (1.3.3) "C"}(*)-h Z Z^)*"*)- Z f ^ K ^ 1 ^ ; with supposed s o l u t i o n (1.3.4) u(*) = Z Cr&)«rr*). >--=/ Following the method of V a r i a t i o n of Parameters, we have « g6r? = IE CrQr? 4/^V>, (1.3.5) fi-i? * c « V provided that the cr}s s a t i s f y (1.3.6) S, vJ&=o9 Putting (I.3.5) into (I.3.3), we get (1.3.7) %<^*>V""^ = Z { *~&>-rjfr)}u"'i)= $ • Let iV^/) be the Wronskian of the n functions ' ^ Ct) (*-l,Z ••• (6) (»->? ( 1 . 3 . 8 ) \AI& = Since \AJ^)4=-0 , the unique solu t i o n of the n algebraic equation ( 1 . 3 . 6 ) and ( 1 . 3 . 7 ) i s (1.3.9) Putting these values into ( 1 . 3 * 4 ) , we obtain where are the n constants of integration involved i n computing the cjs , and where t i s a f i x e d constant. Summing the determinants under the i n t e g r a l sign i n ( 1 . 3 * 1 0 ) , we have ( 1 . 3 * 1 0 ) " # = Z < ^ + I . ( 1 * 3 . 1 1 ) Writing ( 1 . 3 . 1 2 ) ,<•*-) V) V dx and replacing £? by i t s value from ( 1 . 3 . 7 ) , we get from ( 1 . 3 . 1 1 ) , ( 1 . 3 . 1 3 ) u « . i « i r ^ + / KUtfLi^V-Zr1*)} *l*T*tix. or upon interchanging i n t e g r a t i o n and summation, ( 1 . 3 . 1 4 ) «0 = Z*r*rti+LJ*NMl*&-Z& ( WvJft To transform t h i s into a Vol t e r r a Integral Equation, we need the Lemma. For r - / ^ . . - , and K-/,Z - • - (.»•*•-() , the following holds 3 W ( 1 . 3 . 1 5 ) [ \^ { % ) f ^ - ^ ) - t > J ] J = 0 * = 2r Proof. From the d e f i n i t i o n of ( 1 . 3 . 1 2 ) , (7) ye] WW 3* ^ CO v, en VK Or) Chi, 4? Each determinant involved in the partial derivatives contains a non-derived f i r s t row. Hence, for = ? , this f i r s t row becomes identical with the last row, and the determinant vanishes; therefore, (1.3.16) 3x 9 M ^ , * ? = 0 From (1.3.12) i t follows that each of the partial derivatives contains only terms with as factors, so by (1.3.16) (1.3.17) NC*.*) = = = Cvi-i") , For t-=./,zi . . * . , and z . . w-k-/ , we have Since each term in the summation has as a factor one of the (8) p a r t i a l derivatives l^NCzrf/a* (j=o i-. (yt-2)), equation (1.3.17) shows that the r i g h t side vanishes when ?c=z . and hence the Lemma i s proved. We now perform ( *~) p a r t i a l integrations upon the integra l s on the r i g h t side of (1.3.14), and use the Lemma at each stage to simplify the integrated part: b where /3e = £ K » A N D A 1 1 T H E ^JI.K a r e constants de-pending upon b • Putting these values into the summation i n (1.3.14), we get , . + f*NC [Ro (x)- f&Ju&Jx, where the f r are constants, and the kernel i s given by (1.3.19) K (i,x) - f ^ ^ " " £ c ^ { / V ^ ) [ ^ - ^ J ^ . (9) The a r b i t r a r y constants of integration )V are . determined from the i n i t i a l values U®0>) • From.(1.3.lg), = Z , y=.i v. Now, the expansion of 3/*' W&) . From t h i s and from the d e f i n i t i o n (1.3.12), i t i s seen that 3 N L ^ en Substitution of t h i s and the values (1.3.17) into (1.3.;l<9) gives (1.3.21) K(Z/ = « a) - . Hence, „ S i m i l a r l y , f o r the i-ru d e r i v a t i v e , we get an expression of the form (1.3.22) f£a) = z ) depends upon the i n i t i a l values U iS*\ ^„.J-^; and P {%) (f = ifi • • • m . j£=o, 1 z . - •• • The solut i o n of the l i n e a r equation (1.3.22) i s (1.3.23) V = -7-where i s the Wronskian of the n functions rffi) . In conclusion, we have reduced the general Cauchy (10) problem to the problem of solving the in t e g r a l equation (1.3.20), where a l l the constants tr are given by (I.3.23) i n terms of the Cauchy i n i t i a l values. 1.4 The Equation With The F i r s t Derivative Term Missing,. This section applies to the case w=A discussed i n section 1.1. From equation (1.1.16), we see that the kernel w i l l simplify to (1.4.1) { S&?-Q$)} i f Pfr?=0. We can arrange t h i s i n c e r t a i n problems by choosing the tr equation such that P= • however, i n other cases of in t e r e s t t h i s choice i s inconvenient, and instead we transform the variables so that the new functions P&9 and are zero i n (1.1.1) and (1.1.2). We now show that the l a t t e r e n t a i l s no loss of generality; that i s , s t a r t i n g with the second order d i f f e r e n t i a l equation (1.4.2) A"t?+ Ler)ji'e; + [nt) + 'hTei]JLe)=o with r e a l , continuous, d i f f e r e n t i a b l e c o e f f i c i e n t functions L£)t Mt) , and Tf? , and non-negative TP , i t i s possible to change the variables so that (1.4.2) becomes (1.4.3) [ Qfr? -+ 7iJ up) ^ O . F i r s t , introduction of the integrating f a c t o r enables (1.4.2) to be written i n the s e l f - a d j o i n t form (1.4.4) + ItV+TVJ^ti^O, where &T&. I f i n (1.4.4) we make the t r i a l s u b s t i t u t i o n (1.4.5) ' JL& = tf£lug) ; ^ = 9#7£x (11) then, i n order that the r e s u l t i n g U equation have the coef-f i c i e n t s of U11 and "}\U the same, and i n order that the U ' term vanish, the functions ^fr) a n c* &fr) must s a t i s f y the equations 2p9^4- [ pg'-h /&] fi-=^0 , and p9X = y? with solution & = (g/p) Vz, > = (pf)~,//4 Putting these into (1.4.5), we obtain the change of variables (1.4.6) Jit) - (Pf)-l/+u(?) ; ^= } which changes (1.4.2) into (1.4*3)• We now obtain special properties of the solutions of (1.4*3)* Let. two l i n e a r l y independent solutions be u{ andu^fr]: u/V; 4- [ Q#7 4 7v J - ufa&}is analytic i n the f i n i t e z-plane with zeros of order v*\j at 2 = % ) , where i s an entire function without zeros, p i s a non-negative number, and ba i s defined as b • Then the sequence {UH &} of successive approximations associated with (1.5.4) converges to the unique continuous solution of (1.5.4) f o r a l l z. (13) Proof. Let (1.5.4) be rewritten i n the form (1.5.6) U0 -- L ^ ; r ) u r ) J 3 C ; where By hypothesis, ^fr) may be expanded about any of the points i n a series of the form (i.5.«) *;« - 0(<*-yW], with i n f i n i t e r a d i i of convergence, where we define Successive approximations to the sol u t i o n of (1.5.6) are (1.5.9) «h& = -f f^&j where (1.5.10) Jt LC^z) ^ £c) ft^&Jx and U. (?) = , = 1 The proposed s o l u t i o n of (1.5.6) i s then (1.5.11) U.&) = *t Z. 'fr fe> . The nature of the functions ft^fc) w i l l now be examined; from (1.5.7) and (1.5.8), (1.5.12) L(hxl *0 - [^Ix-^+^te-^'XJfw^V. J (1.5.13) L ^ j ) . }[^C2-.p7lH]-Suppose now without l o s s of generality that the zeros are ordered according to increasing moduli Since the series on the right of (I.5.8) has an i n f i n i t e radius of convergence, the f i r s t series involving % i n (14) (1.5.13) i s uniformly convergent f o r a l l f i n i t e x , and can be integrated termwise. Now, the function/v ( ATJ$C) can be represented i n the c i r c l e . C -' 0 < I?- fyl < 6j , where for an ar b i t r a r y small number r , by the product of the Laurent Series given i n the second term on the right of equation (1.5.12). The product series represents either an analytic func-t i o n i n O , or a function with an i s o l a t e d pole at ^' . Since the expansion shows that the l a t t e r i s not the case, i t follows that ^C*) ^ fr) i s analytic i n C . Application of t h i s rea-soning for increasing ^ u n t i l a l l the points have been exhausted shows that ^P)^) i s analytic f o r a l l f i n i t e -x , and representable by any of the uniformly convergent series given i n the second term on the ri g h t of (1.5.12). From (1.5.10) we then have U.5.14) l ^ l ^ ^ ^ ^ ^ ^ ^ ^ - t By the same type of argument used above, the two series i n square brackets on the ri g h t side represent functions %&) and which are analytic i n the f i n i t e plane. Further, the series form-, ed from by putting £ ~o , dividing the f i r s t term by 2t^+ip + i , di v i d i n g a l l the terms by (g-t? 2- , and dropping superscripts i s 1=0 * with an i n f i n i t e radius of convergence, where (1.5.15) S' = ( 5 " , *=-°> Since the a n a l y t i c i t y of and n a s D e e n established, we (15) s h a l l hereafter use the series for j-o , and drop superscripts. Equation (1.5.14) then gives (1.5.16) 1^)1 < xri±iP_' A ( i»-t|* where i s a bound on the s e r i e s (1.5.17) S<'7 = Z S i " with Putting (1.5.16) in t o (li.5-.10). we get where A r i s a bound on the series with By induction, where A K i s a bound on the series (1.5.19) S . - ^ ' with . In (1.5.20) b r I ^ | / c 2 ^ ) ip+^v-n-^;^^ ,^ Since a comparison of (1.5.20)-with (1.5.15) shows that S V S 1 SS I f o r a l l ^ and * , i t follows that the sequence of nunbers £ A^} i s bounded above by some number f\ . Hence, (1.5.21) i f„&\ c <3-±*e=jr A - ,,_ t," (16) so that the series on the right of•(1.5.11) i s dominated by (1.5.22) a*f { (Zwf+ 2p_,; A H Iz-fcJ*} , and the convergence i s established. By the same reasoning as used i n the Liouville-NeumanmTheorem {&) ^Too U^CZ? s a t i s f i e s (i.5.6) To show the uniqueness of the bounded solution, sup-pose that u)fej i s another bounded so l u t i o n of (1.5.6), (1.5.23) toty = + V &) /* L(ixx) co , the general solu t i o n of ( 1 . 6 . 1 ) cannot be expected to exist f o r a l l 2- . However, i n the next theo-rem we s h a l l show that under c e r t a i n conditions the soluti o n does e x i s t s f o r a l l * excluded from a small c i r c l e P about -» . Theorem 2 . Suppose that the function 'Vl&)/{U-b)r i s analytic i n the f i n i t e z-plane3- with zeros of order one at by C^^O1/" %) * a n ^ with one zero of order f»o>/ 1><> = L> , where i s an entire function without zeros, and p i s a non-negative number. Further, l e t kfyx) = NOi,x) t>£t) , where Dftl - Ps (?-b)S f o r p o s i t i v e i n t e g r a l s . Then the sequence {. Kn&l} of successive approximations associated with ( 1 . 6 . 1 ) converges f o r a l l 4- f to the unique solution of ( 1 . 6 . 1 ) . Proof. Successive approximations to the solution of ( 1 . 6 . 1 ) are ( 1 . 6 . 2 ) where and ( 1 . 6 . 4 ) and where we define 1 The f i n i t e z-plane r e f e r s to a l l values of ? for which /*/ < I zj , where 3» i s f i x e d . (18) (1.6.$) U0&) = eL*/K&) -+J3^fr) • %afr)= I, -&&^<^&). The proposed solution of (1.6.1) i s then (1.6.6) uM\lM (1.6.9) 1 ^ - u>&\<. Vl^&iC, \%{&\ + vCzl£K&\ • The su b s t i t u t i o n of (1.6.9) into the ri g h t side of (1.6.8) gives Repetition of the process gives at the n-th stage —> O •* —? —* . The following generalization can be proved by sim i l a r reasoning : (19) Theorem 3. Suppose that the function ^ ^ / { ( e - i / j ^ J - i s analytic i n the f i n i t e z-plane with zeros of order at 2=b ' , where i s entire without zeros, and p i s non-negative. Further, l e t KC%)x)/'H(-^x) = 2- ^ C^-^O Cs^f^ ) Then the sequence., of successive approximations of (1.6.1) converges f o r a l l values of * ( i n the f i n i t e plane) ex-cluded from small c i r c l e s r y about by to the unique solution of (1.6.1). (20) CHAPTER TWO APPLICATION TO INITIAL VALUE PROBLEMS 2.1 Introduction. This chapter contains applications of the r e s u l t s of Chapter 1 to the type of i n i t i a l value problem i n which the d i f f e r e n t i a l equation to be solved d i f f e r s from a known equa-t i o n by terms containing small parameters. In p a r t i c u l a r , the i n t e g r a l equation (1.1.5) i s used to obtain solutions as power series expansions i n one or two of these parameters. Also, as a general r e s u l t , the formulation of the general Cauchy prob-lem as an i n t e g r a l equation, obtained i n section 1.3, i s used to give multiple power series expansions of higher order d i f -f e r e n t i a l equations i n several parameters. In the type of prob-lem considered, the m-th successive approximation to the solution of the i n t e g r a l equation y i e l d s a l l the terms of the multiple power serie s having the sum of the powers of the various para-meters l e s s than or equal to m. Since the d i f f e r e n t i a l equa-tions under consideration w i l l be assumed to have no singu-l a r points, so that the kernel tfte,*) and the second sol u t i o n are bounded, the Liouville-Neumann Theorem guarantees the convergence of the general sol u t i o n of the i n t e g r a l equation by successive approximations. 2.2 A Problem With A Perturbation In The F i r s t Derivative Term. Suppose that the solution of the d i f f e r e n t i a l equation (2.2.1) u"(*l + CLS-F&) uf(*) -f &&)ufr)=o having the i n i t i a l values tl(t>) and u'd?) i s required, when (21) ^k) and ^fr) are known to be solutions of (2.2.2) v-'fr)-/- QGr) «rfr) = O • Equation (2.2.1) d i f f e r s from (2.2.2) by a term containing a small parameter s . From (1.1.14),(1.1.16), and (1.4.8), - s LC^x) where Lfr,*) i s independent of s . From (1.1.15) and (1.2.7), (2.2.3) u&)~ ^ &)+- -f S J* L C^x) u&^JXj where °L and /? are l i n e a r i n s . With the f i r s t approximation to (2.2.3) i s The m-th approximation then has the form giving the complete power series up to the term i n 5 • As a simple i l l u s t r a t i o n , consider the problem, whose solutio n i s e a s i l y obtained by other methods, i n which F^Sr)-/ f Q&) = (for r e a l ^ ), are put into (2.2.1). In t h i s case, l< = — as- cor . I f the i n i t i a l conditions are u(o)=i u^o) = o then (1.2.7) y i e l d s oi=-\ , ^ — 3£ , and the i n t e g r a l equation i s U&) = Cafvt-t- -4- %f St*** - a s f*cosnC-x~*~)ufr)<}vC; First with A approximation (2.2.4) U.&O + Tc{/>* «* ~ ^ "* s'" rt* • (22) This checks with the f i r s t two terms i n the power series expansion of the known so l u t i o n (2.2.5) u © = cos (Mi 4- sin J where ^ = c ^ _ _ 2.3 Another Single Term Perturbation Problem. Consider the problem of fi n d i n g the solution of the equation (2.3.1) eQclu/&7+ [ Tfr)-(- sGCzll ucxp^o with the perturbation $G(x) i n the ufr? term, having the i n i t i a l values u(j?) and a %) , where ^ifr) and ^ & ) s a t i s f y (2.3.2) trig) + P&) v'C*? + 7-fr?«rfr?=-o • Again, use of ( 1 . 1 . 1 4 ) , ( 1 . 1 . 1 5 ) , and (1.1.16) shows that the i n t e g r a l equation f o r the problem i s (2.3.3) Ufr) - -h-ftf-ity-ir sj L£*tx) u&)Jx , where, from (1.2.7), oi and f) as well as Lfeix.') are indepen-dent of the parameter s # The m-th approximation to the so-l u t i o n of (2.3.3) then has the form (2.3.4) = J l ~&&) sX. £=0 * For example, suppose that the solution of (2.3.5) a" &) 4- { s(xl^ax)4- * } a&) = o with i n i t i a l conditions (2.3.6) u(o) = /; u/ and hence the f i r s t approximation to (2.3.7) i s o o where -£,(2-} = COS** 2.4 Multiple Perturbation Problems. Consider the d i f f e r e n t i a l equation (2.4.1) uH &) •+ ZS.Ffr) [ QCt) + t CCt)] u&) = 0 , containing two independent perturbing parameters s and 't , Suppose the solut i o n of (2.4.2) AT + QCz) yirfr?^ D are known to be *st(x) and • Then, from (1.1.15) and (1.1.16), ufr) s a t i s f i e s the Volterra Integral Equation (2.4.3) *^i&+/3*70r) + sj\&x)ufr)Jx+t(%htXJa&Uxi A> b where the functions are independent of 5 and £~ . I f the i n i t i a l values f o r are u(l>) and u ^ ? \ i t follows from (1.2.7) that the constants ) — / , a^ol-O , equations (2.4.4) give ^ / ; /»- i f . • Hence, the i n t e g r a l equation f o r the problem is (2.4.6) U£7 = ra$>tS 4- ;|-Jn-**- -f-/ *ni ^ — t f eUx-sfcofiCx-yfl+arfutUx . With Wo^3* f«»*-f|-/-i = ^ . + ( £ + - ^ V * M , * > The f i r s t approximation (2.4.7) gives a l l the l i n e a r terms i n the sol u t i o n (2.4.6). Consider now a general perturbation problem, i n which the solution of the n-th order l i n e a r d i f f e r e n t i a l equation (25) (2.4.8) cx% + £ 5 For)] uU'"h - O "•**t »-r —f having the i n i t i a l values U ^ . ( £~oJiJi/• - ••(-•>) ) i s required. Equation (2.4.8) i s changed by terms containing small parameters S^.y. from the equation (2.4.9) ^frl 4- £ frl «rc*"}*)= o whose solutions nrh ( x ) . (*"-/,? - - - "i ) are supposed to be known. From (1.3.18), the in t e g r a l equation for t h i s problem i s (2.4.10) U&) = 2. 2- S" H- J L fCr?«/xr where, from (1.3.19) L^C*.*) = c-o"'~'^'> { A / r w P » C-^,-Here, N i s given i n terms of : ( "-^z, ' ' ) by (1.3.12). The constants KV , given by (I.3.23), w i l l i n general have the form where the exponent on each of the parameters 5; i s either one (y) or zero, and the c o e f f i c i e n t s • v „ depend upon the i n i t i a l values i4% , i r ^ V t / , and F{\) , (/^>/ •••(*>-(/ ) The m-th approximation of (2.4.10) i s (2.4.11) f u ^ - z z X-/*/ ; Tisdc-f- y 2 n /v~-i(*') — st«i* y the r e s u l t i n g i n t e g r a l equation i s (2.5.7) Ufr)= cos«* 4- &^*-{*^x-tO[^F\^^{f/&)]u97Jx With Uofc}- cos**-4-r<^ n , the f i r s t approximation to thi s o l u t i o n of (2.5.7) i s (2.5.8) + **{^frtfiU - J L f ^ n ^ + ^(f^^rrux (27) The sequence of successive approximations associated with (2.5.7) actually does converge to the unique continuous solut i o n of (2.5.7) provided that the function together with i t s f i r s t derivative are bounded f o r a l l values of the argument. The m-th approximations of the form where the powers of S higher than s , represented by the second summation, w i l l i n general receive contributions from l a t e r approximations. As an example, suppose that f — C i n &.5.I); then (2.5.8) gives f o r the approximation , (2.5.9) U,&) = A* f*tc_y#r -Ji« «*t-r In t h i s p a r t i c u l a r example, the f i r s t approximation (2.5.9) gives a l l terms up to those containing & , and likewise the m-th approximation gives a l l terms up to those containing s , since there i s no overlapping of terms at the successive stages of approximation. The r e s u l t (2.5.9) checks with the expansion of the known s o l u t i o n (2.2.5) up to the s 3 term. (28) CHAPTER THREE APPLICATION TO SPECIAL FUNCTIONS 3.1 Introduction. The object in this chapter i s to use the result of Section 1.1 to obtain expansions of special functions i n series of better known functions. Ikeda (4) f i r s t used this method to expand X,@x) and Y„ 6*7 i n terms of J"Mc*? and Y*,fr/ res-pectively, where 7K<*7 and V*.(x) are the Bessel Functions of f i r s t and second kinds of order n . In addition to rederi-ving Ikeda's formal results, we have examined the convergence of the series; in particular, we have found that the IT, &*) series converges for a l l x and a l l d. , but that a restriction must be imposed upon in order that the Y„'(_**) series converge, (for a l l x excluded from a neighbourhood of the origin.) For details, see M.A. Thesis of D.A. Trumpler (16). More recently, F. Tricomi (15) has obtained expansions of the Confluent Hypergeometric Function in series of Bessel Functions. Using Laplace Transform methods, he arrived at an expansion for the well-behaved solution of the Confluent Hyper-geometric Equation, and gave a four-term recurrence for the coefficients in the series. Also, by setting up an integral equation similar to that which we have derived in Section 1.1, he obtained asymptotic formulae, but no general expansions. In this chapter, we use the result of Section 1.1 to obtain the general solution of the Confluent Hypergeometric Equation as series in T^q and Y^(x) , and as a special case, the well-behaved solution of this equation as a series in Xx(*') • (29) Further, we arr i v e at a sim i l a r series of Bessel Functions f o r the solution of a generalized Confluent Hypergeometric Equation. T h e o r e t i c a l l y , the procedure could be generalized to obtain expansions of various other functions i n terms of known functions except f o r the computational d i f f i c u l t i e s i n evaluating c e r t a i n i n t e g r a l s involving the l a t t e r . 3.2 The Expansion Of The Solution Of The Confluent Hypergeo-metric Equation In Series Of Bessel Functions. The object i s to express the so l u t i o n W(a)c} t) of the Confluent Hypergeometric Equation (3.2.1) irW11^ -h (c-6) W ' - «UJ<£) « O i n terms of the solutions a n ( * VI, <£) of Bessel's Equation (3.2.2) trA"6r) + ? -h ( 4- •£) Xit) = O -We now proceed to set up an integral equation l i n k i n g the solu-tions of (3.2.1) and (3.2.2). In order to obtain the simple expression (1.4.1) f o r the kernel, we use (1.4.6) to get the transformat ion (3.2.3) t ir&l; 17=*> which changes (3.2.2) into (3.2.4) AT" Cf? + [ 4 ^ •+ * ]ir#? = 0 • Likewise, we can remove the f i r s t d erivative term from (3.2.1) by the change of variable (3.2.5) <*lfc)= tf* which changes (3.2.1) into (3.2.6) + J£ - 1L^> } U t f ~ o^ (30) where (3.2.7) 1^= f - / ; \4= s= . The further transformation changes (3.2.6) into (3.2.9) aV;+ { % r - 7* + <} o} where (3.2.10) £ = n= U!+\ = c-i 7V-^H -We now use the r e s u l t of Section 1.1 to write the solutions of (3*2.9) i n terms of the known solutions cx^- X*fr? and Y^Cx) of (3.2.4) by a Volterra Integral Equation. From (1.1.14, (1.1.16), and (3.2.3), we obtain upon taking £x& , JL*,6r) = , and using the i d e n t i t y (See Watson (17) ) (3.2.11) Yji]{f)**J*)Jx (fj'x^ + 0 ^ ; + . ^ ; •t-fjk M"! Hf» + I 1 See the footnote on page 2 (3D where > i s a p o s i t i v e i n t e g e r , and cfy (L-I,z,i,^) a r e c o n s t a n t s o f i n t e g r a t i o n . The r e c u r r e n c e r e l a t i o n s ( 3 . 2.14) and (3 . 2.15) are well-known ( 1 6 ) , and t h e r e s u l t s ( 3 « 2 . l 6 ) and (3 . 2.17) w i l l be e s t a b l i s h e d i n S e c t i o n 3«6» I f we t a k e i/t@j = * * l^TQi) + /3 Y„&} and c a l c u l a t e t h e f i r s t few a p p r o x i m a t i o n s o f ( 3 . 2 . 1 3 ) , i t becomes apparent t h a t t h e s o l u t i o n o f ( 3 . 2 . I 3 ) w i l l have t h e f o r m (3.2.18) « » - .* f [AXt) 3>>+ eM\*-)]> upon rearrangement of t h e terms i n t h e s e r i e s . We t h e n s u b s t i -t u t e (3 . 2.18) i n t o ( 3 . 2 . I 3 ) and d e t e r m i n e t h e n e c e s s a r y r e c u r -r e n c e f o r m u l a e f o r t h e c o e f f i c i e n t s and S K so t h a t (3 . 2.13) i s s a t i s f i e d . The r e s u l t o f the s u b s t i t u t i o n i s A p p l y i n g ( 3 . 2.14) and ( 3 . 2.15) t o t h e B e s s e l F u n c t i o n s i n t h e summation under t h e i n t e g r a l s i g n , and t h e n u s i n g (3 . 2.16) and (3 . 2.17) , we o b t a i n i n t u r n where W, and /?, are c o n s t a n t s depending upon d. , (3 , and ('='^,3, * ; * = '/V'. ' ). R e p l a c i n g r b y > + / i n t h e f i r s t and t h i r d terms under t h e summation on th e r i g h t s i d e , we may r e w r i t e t h i s i n t h e f o r m «> (32) Equating separately the c o e f f i c i e n t s of (±y+3j (*) and V2-' I fc) on both sides, we obtain the ~ recurrence r e l a -t>r | we can rewrite r+i J Putting 4 , and ^ (3.2.18) i n the form (3.2.20) u&- ** f W, £ ^ ^ J ^ ) + A f b r (ff Y ^ J ? where now (3-2.21) jl} ' "' )• As i n (3.2.6), y\ and J? are constants of the problem. The case discussed e a r l i e r (f=/) is r e l a t e d to the quantum mechanical problem f o r an harmonic o s c i l l a t o r i n space. The more general form here (and the generalization considered i n Section 3«4) could therefore be interpreted as an anharmonic o s c i l l a t o r i n space. The change of variable (3.2.8) transforms (3.3.1) into (3.3.2) u > ) + { - ^ ! _ +{ y u^r) - Oj where now (3.3.3) * - ItkfJ? . As i n Section 3*2, we use the r e s u l t s of Section 1.1 to com-pare (3.3.2) with (3.2.4), and obtain the i n t e g r a l equation (3.3.4) Ufr) = ^TS^i + /3 X® + ^ ^ f / f t ^ X ^ ) - W t o J ^ « W c / > r In order to obtain the solut i o n of (3.3.2) which is f i n i t e at (34) the o r i g i n , we take a-o ; a simple modification would give the general solution, (as i n Section 3 . 2 ) ' . Following the method of Section 3 . 2 , we look for the solution of ( 3 . 3 . 4 ) i n the form ( 3 . 3 . 5 ) u&>=*>* 2 where f i s a po s i t i v e integer. This w i l l be proved i n Section 3 . 6 . Substituting ( 3 . 3 . 7 ) into ( 3 - 3 . 6 ) and using ( 3 . 2 . 1 5 ) , we get, upon interchanging summation and integration, ( 3 . 3 . 8 ) / The change of dummy s=r-p-f leads to i- •• - 4*? . From ( 3 . 3 . 8 ) ( 3 . 3 . 1 0 ) 40 - ' ; 4, = •• • - l) -•••{> . i n summary, ( 3 . 3 . 1 0 ) and ( 3 . 3 . 1 1 ) give the c o e f f i c i e n t s 46 a • •- dtp , and ( 3 . 3 . 9 ) then gives a l l subsequent *»«<)••• ^ ft The successive changes of dummy s = r-f+^ , and t = S-2.p-f2.g_ transform (3.4*5) into (3*4*6)^ ~° . where and' * * f ^ " - < j c V T i T ^ ^ ^ +c^LL%z-^ & ^ [(^- c^^j x+t+s® •+ Equating the c o e f f i c i e n t s of y r ^ i n (3.4.6) gives "4- t^> —1 the recurrence r e l a t i o n (3.^.7) ^ ' i l ^ M ^ ^ ^ ^ , where ^ ^ --- 4^ are obtained from (3.4.6) i n any given example. However, the complicated nature of the functions (37) and l-Jv,Cip makes i t i n c o n v e n i e n t t o g e t g e n e r a l e x p r e s s i o n s f o r t h e s e c o e f f i c i e n t s . F o r f i n i t e numbers P , Theorem 3 o f S e c t i o n 1.6 shows t h a t t h e sequence o f s u c c e s s i v e a p p r o x i m a t i o n s a s s o -s i a t e d w i t h t h e i n t e g r a l e q u a t i o n (3.4.4) a c t u a l l y converges t o t h e unique c o n t i n u o u s s o l u t i o n o f (3.4.4). As i n S e c t i o n 3.2, however, we have r e a r r a n g e d t e r m s i n o b t a i n i n g (3.3.5), so t h a t f u r t h e r a t t e n t i o n i s r e q u i r e d i n o r d e r t o e s t a b l i s h t h e con v e r g e n c e . 3.5 S o l u t i o n s o f R e l a t e d D i f f e r e n t i a l E q u a t i o n s Expanded I n Terms Of B e s s e l F u n c t i o n s . I n t h i s s e c t i o n , i t w i l l be shown t h a t the s o l u t i o n s of a number o f i m p o r t a n t d i f f e r e n t i a l e q u a t i o n s a re r e l a t e d t o t h e C o n f l u e n t H y p e r g e o m e t r i c F u n c t i o n t h r o u g h v a r i o u s changes o f v a r i a b l e . Hence, t h e r e s u l t s of S e c t i o n 3.2 can be u s e d t o e x p r e s s t h e s e s o l u t i o n s as s e r i e s o f B e s s e l F u n c t i o n s . N u m e r i c a l v a l u e s f o r t h e s e s o l u t i o n s c o u l d t h e n be computed a c c u r a t e l y by making use o f th e e x t e n s i v e t a b u -l a t i o n of the B e s s e l F u n c t i o n (17), and i n f a c t , f o r A « 1 , o n l y a few terms o f t h e r a p i d l y convergent s e r i e s (3.2.19) would be needed t o guar a n t e e a c c u r a t e r e s u l t s ( 9 ) . (a) The W h i t t a k e r F u n c t i o n ( 1 8 ) . P u t t i n g i*i=vt?-A£ i n t o (3.2.6) g i v e s (3.5.1) v"Gr)+ i - i _ + %+'^irlvC^=6' w h i c h , by (3.2.5), has as i t s s o l u t i o n the W h i t t a k e r Func-t i o n (3.5.2) »„.. ft - H / < V ; t) (3d) Since (3.2.7) gives f , H = € - c t , equation (3.5.2) can be written (3.5.3) A l ^ ^ , e ^ £ * U/(*»*~i-hs ^ which i s now i n a form to which (3.2.21) can be applied. (b) The Laguerre Function Density; By the substitutions (3.5.4) ± - ^ ; equation (3.2.6) becomes (3.5.5) L."tf+ i -4 - UQ> --fr] L&?^6; with sol u t i o n 0.5.6) Lnj(,)= n,,^& = n „ i t H . (c) The "Associated Hermite Equation". The so-called Associated Hermite Equation (3.5.7) r ' W f L-(2*HKtj+i?_ _ t f + M j r 6 ; = o , obtained from (3.2.6) by the substitutions (3.5.8) t= -f ; v&= gfrbi has the solution (3.5.9) n>) - k r ^ r ^ H ( £ i or (d) Hermite Ts Equation (18) Hermite Ts Equation (or Weber's Equation) is (3.5.10) t>£(*)-r E(3+i) - P# . which i s related to (3.2.9) by the transformations (3.5.11) * = x = «(*)=pd&>; «=i Hence, the solution of (3.5.10) may be written where i n the expansion (3.2.19) for U 'f and i n the (39) recurrence r e l a t i o n (3.2.20), ; ij + i • (e) The Equation For The Harmonic O s c i l l a t o r In Space This equation i s (3.5.12) K/'fc?* i ^ C-A f+vJ ^b)=oJ which i s related, to (3.2.9) by the transformation (3.5.13) "CJ= K ^ f ; U*. to) . The solu t i o n of '(3.5.12) i s then (3'.5.14) K,: ?• ** - f*[*~*HZ X, h„ «x^Jx - E-7M 7^ **»] * + / * ^ T _ x**l«/x using the r e l a t i o n (17) (3.6.3) = * X ^ ^ J T ' W -A p a r t i a l i n t e g r a t i o n of the l a s t i n t e g r a l on the r i g h t gives (3.6.4) , * / AH . /r* Upon use of the i d e n t i t y (17) (3.6.4) becomes x + z A + 2. ^ * T ^ T T v- = f T j x - J j % / (40) Upon transposing terms i n t h i s equation, we get f i n a l l y (3.6.5) f'lnlfU™*-^[XuZfoF-JjoZj*)?*]* Since the functions Ynfr) s a t i s f y the same recurrence r e l a -t i o n s as T*(z) , the following r e s u l t i s obtained i n the same way as (3*6.5) (3.6.6) S\*>T„e) - ^ lWT^c*>^\0 T ^ ? n * . Upon use of (3.6.5) and (3.6.6), we get Now, from (3.6.3) and (3.2.11), (3.6.7) Y H H (tF) Z&) - & Y..G-) = a / r r * , and the r e s u l t (3.2.16) follows. Again, following the same procedure with X.+. (x) replaced byY , we get, instead of (3.6.5) and (3.6.6) (3.6.8) J^XVX+M ^ [ l ^ l ^ - r c ^ l ^ x ^ ] (3.6.9) f\lnln*)x'*,J*- ^ £ Y A W ^ - Y « U V from which the r e s u l t (3.2.17) follows. (b) Proof Of (3.3.7) We need the following Lemma. For J?-!^, --- f> , the following r e l a t i o n holds (3.6.10) = CF? • We now prove the r e s u l t (3.3.7) by f i n i t e induction upon P For p-/ , (3.3.7) gives (3.6.11) £ &) - &G}*Z+lCX? - ( i ) l ^ l (41) where f o r convenience, we put n-tr ( y--o,/jz) - - - - ) which i s correct by (3.2.14). Assuming the r e s u l t (3»3»7) i s true f o r P=fr ^ o b t a i n , with the help of (3.2.14), (3.6.12) Z & - {T'f-H-Vj-L Tfi, where ( X- 0,1,1, • •--Hence, upon app l i c a t i o n of the Lemma. Putting t h i s into (3.6.12), we get f which completes the proof by induction. (42) CHAPTER FOUR PHYSICAL APPLICATIONS 4.1 Introduction. Although the method of Section 1.1 was: o r i g i n a l l y designed f o r i n i t i a l value>• problems, i t can be adapted to solve boundary value problems. In t h i s chapter, we s h a l l discuss a type of boundary value problem which a r i s e s i n Quantum Mechanics. Now, i n the usual problems treated i n Quantum Mechanics, i t i s required to f i n d the solutions of the Schrodinger Wave Equation which s a t i s f i e s a set of "natural boundary conditions", f o r which the p o s i t i o n of the mass p a r t i c l e i s u n r e s t r i c t e d . The p r o b a b i l i t y i n t e r -p r e t a t i o n of the wave function then leads to the boundary conditions of f i n i t e n e s s at the singular points of the wave equation. I f , however, the system under consideration i s en-closed, then these conditions are replaced by the " a r t i f i c i a l boundary conditions" that the wave function vanish at c e r t a i n ordinary points of the d i f f e r e n t i a l equation. In f a c t , f o r these so-called bounded Quantum Mechanical problems, the boundary conditions require that the wave function vanish on some surface i n f i n i t e three-space, such as a sphere or a cone. The corresponding physical condition i s that there be an i n f i n i t e l y high and i n f i n i t e l y steep potential wall on t h i s surface. 4.2 The Integral Equation For The Bounded Quantum Mechanical Problem. Let generalized c u r v i l i n e a r coordinates x, , , and x3 (43) i n three dimensional Euclidean space be chosen so that the surface on which the wave function vanishes i s ^ - c , where C i s a constant. We assume that the surface i s of s u f f i c i e n t l y simple nature that the Schrodinger Wave Equation i s separable (12) i n the chosen coordinates xx , "*z_ , ^ . equation The space dependent wave Ais, f o r a p a r t i c l e of mass M , (4.2.1) ~ ^ V > = Ce-Vlifs, where i s Planck's constant divided by , £T i s the i. energy constant, V i s the potential energy, and V i s the Laplacian operator i n the coordinate system , ^ , ^ c3 . The s u b s t i t u t i o n permits the separation of (4.2.1) into three ordinary d i f f e r e n -t i a l equations for the functions Xfa) • The equations for Y^Cxv) and y3faj) have the same solutions as i n the unbounded problem, ahd the l a t t e r are supposed known. The Xt(^) equa-t i o n has the form < ^ 2 - 2 ' 4 [ & ) a ; ] -t- i•%(•*,)+ YA)'0 where JP i s the quantum number a r i s i n g from the X^lXJ equation. From Section 1.4, (4.2.2) can be transformed into (4.2.3) u'Oj where ^Ofl s a t i s f i e s the boundary conditions (4.2.6) srQ,] F,«,+e ; = o We suppose that a so l u t i o n of (4.2.5) which i s analytic i n the f i n i t e plane i s known to be (4.2.7) t/7 = Mt<&, and that the eigenvalue i s known. From Section 1.4, the second solu t i o n of (4.2.5) and the Wronskian of the two solutions are given by (4.2.8) ^ 7 = C j f f g f o f , and (4.2.9) - C -Hence, from (1.1.14), (1.1.15), and (1.1.16), the i n t e g r a l equation connecting the solutions of (4.2.3) and (4.2.5) i s (4.2.10) The convergence of the solution of (4.2.10) by successive approximations i s established by Theorem 1 of Section 1.5, since i t has been assumed that -#\h fr) i s analytic i n the f i n i t e plane. The f i r s t of conditions (4.2.4) requires that /3~o . I f we take u0&) A*&, the f i r s t approxi-mations to the solution of (4.2.10) i s (4.2.11) u, & - A* m [<- A ) ( * f i«?T? and parts of the wave equation c l e a r l y have solutions which are i d e n t i c a l with the solutions corresponding to the natural boundary condition It remains to solve the r a d i a l part of the Hydrogen atom wave equation (4.3.1) W&+ [ f ~ + 1] « & ~ O, under the a r t i f i c i a l boundary conditions (4.3.2) F^rte ; i4(*J=o, instead of the natural boundary conditions ( 4 - 3 . 3 ) "(0? F.Vrfe ; u(o°7=o . The conditions (4.3.3.) give solutions of ( 4*3 « 1 ) e a s i l y by the Frobenius method; the eigenvalues are 7i,(w? = =L , f o r p o s i t i v e integers , and the eigenfunctions are the Laiguerre Function Densities. However, (4.3.2) require that 'A s a t i s f y the equation u C ; ) = o , where c<£>j ^ denotes the Laguerre Function Density (corresponding to the eigenvalue " A , ) which i s related to the Confluent Hypergeometric Function U/(a,c> *) ( c f . equation (3.5.6)). Michels et a l (10) have found approximations for the eigenvalues of the ground l e v e l , and de Groot and ten Seldam (2) have extended t h e i r method to the 2s and the 2p l e v e l s , g i v i n g graphs and tables for the s h i f t i n X . Soon afterward, Sommerfeld and Welker ( 1 4 ) applied the formulae of Michels et a l f o r values of 2» equal t o three and four times the Bohr radius. Also, Sommerfeld and Welker stressed the importance of a general investigation of the be-haviour of the Confluent Hypergeometric Function near =2- = <=*> Sommerfeld and Welker ( 1 4 ) have also discussed a graphical method for obtaining the eigenvalues, which gives accurately the curve " X - A ^ 7 f© r small values of ^ . By t h i s method, the known standard solutions u C ? ; are plotted f o r various pos i t i v e i n t e g r a l n , and the f i r s t zeros of these solutions are located. These functions are then solu-tions of the problem f o r the p a r t i c u l a r values 2 ^ of ^ 8 . A graph of against * i s drawn, and by i n t e r p o l a t i o n , (47) the value of n(it) corresponding to a given value Z6 i s taken from the graph. Then the ground l e v e l eigenvalue i s ^ L%) ~-i/n • The eigenvalues for higher l e v e l s are obtained by a similar procedure. The preceeding gives an h i s t o r i c sketch of work done on the problem up to the present. We now proceed to give our own treatment, using the method of Section 4.2. The mathema-t i c a l problem amounts to solving equation (4*3.1) under the boundary conditions (4.3*2), when we know that the solutions of the equation (4*3*4) i r " & + [ f - £Ug) + * J = o, s a t i s f y i n g the boundary conditions (4*3*5) •vfc) Fi'«i-fe • v<**>l — o are (4*3*6) vx@) = A\&) , with Since the problem thus presented i s of the same type consi-dered i n Section 4*2, the Vol t e r r a Integral Equation cor-responding to equation (4*3*1) is (4*2.10), with f i r s t approxi-mation (4.2.11). For the ground l e v e l , Ai& - -and (4*2.11) gives (4*3*7) ut®= *ec~*[ i-(* + 0 f**e-** J* r^-^r] With U - / * , and dtf= x l c - w < / r a p a r t i a l integration gives (48) or - i = f + f *t)[ - f [ ± + £ + i b } * < } . Now, Hence, _ * M_, 4s c-1 IL± I 1 1 * ^ z 2 -*wy - _ x S" _&*2— and (4.3.7) gives (4.3.8) 2 " - ^ Z ^ f k ~ ] -Application of the f i r s t of the conditions (4*3«2) gives (4.3.9) * f f (^ , -I +~±%i,r The second approximation to the i n t e g r a l equation i s ob-tained by putting (4.3.8) back into (4.2.11): e—JtcU 7 Again integrating by parts and l e t t i n g ' 6 0 , we get (4.3.10) u «, -*«.-'«/• l - f i & L + < ^ Z A Z M J Upon making the approximation fa+l) = C V •+ I,) , where A 0 J i s given by (4.3*9), and applying (4-3«2), we get for the second approximation to the eigenvalue, (4.3.11) \ , 0.) - X^o) + ,~ W l * The r e s u l t (4.3.9) i s i n essential agreement with that ob-tained by de Groot and ten Seldam (2), and i n f a c t the c a l -(49) culated eigenvalues f i t the curve of Sommerfeld and Welker (cf. the bottom of page 46) better than the r e s u l t s of de Groot and t e n Seldam. The values of are within one percent of the correct value when zg i s at lea s t f i v e times the Bohr radius. Sums of the type appearing i n (4.3*9) and (4.3*11) are most e a s i l y handled by using a method of de Groot and ten Seldam, which depends upon the properties of the exponential i n t e g r a l /•x — **» which i s tabulated (6). 4.4 The Bounded Rigid Rotator. For the general rotator problem i n three-space, a mass p a r t i c l e f i s r e s t r i c t e d to rotate at a constant distance * , c t o € , or ~l- J (4.5.4) ^ J V t ^ [/+ j\ A i j ± ^ T ] . Application of the second of conditions (4«5*4) gives f o r the f i r s t approximation to the ground l e v e l eigenvalue, (52) (4-5.5) y « l % ) . = ± > where (4.5«6) -») = 14- c°s • The second approximation to the solution of (4«5«D i s obtained by putting (4.5.4) back into the right side of (4.5-1) : Again making the sub s t i t u t i o n (4.5*3), we obtain or X (4*5*7) ^ In order to evaluate the expression (4.5.8) - T - ^ ^ l / ! * ( , + 5 ) ^ on the r i g h t side of (4.5*7), we need the following r e s u l t s : (4.5.9) ( \ ( i+>))) i s the Riemann Zeta-Function. Putting (4.5.12) in t o (4-5.7), and us-ing the value = ^/^> , we get (53) or (4.5.13 ) « ^si'n^g / )+ A i , l ± ^ J t ~ A / U ' ± ~ £ + / . l 4 S - ^ J f j By applying the second of conditions (4.4.4) to equations >jOJ L 0.0 (4.5.13) and using the approximation 7*= f/\('tf'. where 6; 0,0 i s given by (4.5.5), we obtain for the second approxi-mation to the eigenvalue (4.5.14) ^3) where ^ i s given by (4.5.6). Again, the t h i r d approximation to the eigenvalue i s found to be (4.5.15) where III the l i m i t ^ -» o , equation (4.5.14) reduces to (4.5.16) - + f i T l l ' which i s the re s u l t obtained by Sommerfeld and Hartmann (13) by a d i f f e r e n t method. The following table gives, f o r small values of ^ , a comparison of the eigenvalues obtained from (4.5.16) and those from the f i r s t three approximations (4.5.5), (4.5.14) and (4.5.15). TABLE I rr-ze *) CSH) A 0,0 o°4S' \0"* O. tow OAOHS O. \0^\ 0. IU2 10" O.I3I5 01450 OA443 OJ4SS I0'2 o.iSU O.ZIOZ 0.1.20-3 il'zf zw~l 0.9.111 0.XH-3Q 18° ll' 5 W0"z o.illi 0-3/22 0.30*2 0.344C Z5°5\y 10" 0.333S 0.3100 0.3UO 0.44+1 (54) 4.6 Higher Energy Levels Of The Bounded Rigid Rotator. For the 1-1 l e v e l , Putting these values into (4.4.8), and following the same procedure that was used i n Section 4.5, we get for the f i r s t and second approximations t o the eigenvalue (4.6.1) 7.- ^ (^=i+coizo)y a. -*t 2. * W + or f o r small values of ->\ , ^ (4-6-2) * [ W + £ £ i i > * * The rapid convergence of the successive approximations for small values of ->) i s i l l u s t r a t e d i n the following t a b l e : ( A \f' = ^ ~ 2) ' , 1 ' hi TABLE 2 ^ 7 A *?, %°34J \o-3 1.0011 2 0021 own r G' I D " 2 1.01S5 10211 OOZ<\\ I f " 2 7 ' ix lo"1' 20541 2.0S&S O-0S(>5 18° ll' 5XlcTt 2.12(>(p 2-1343 0.i3>42> 2S°s\ 10'' 2- 230 2-253 o.xss 2.-40{ 0.4C5 (55) The following graph compares the r e s u l t s of TABLES 1 and 2: GRAPH OF EIGENVALUES FOR THE (0,0) and (1,1) ENERGY LEVELS * *o C<*cSr«e*) *0 IS tO S o The graphs i l l u s t r a t e that, as ->j-^o , the increments i n the eigenvalues of the bounded rotator approach those of the free rotator for the (0,0) and (1,1) energy l e v e l s . Further, there are v e r t i c a l and horizontal tangents res-p e c t i v e l y to the (0,0) and (1,1) curves at the o r i g i n . For higher energy l e v e l s , we have calculated only the f i r s t approximation to the eigenvalues. The r e s u l t s are tabulated below. TABLE 3 Pj (a*) ( 2 , 2 ) sin** (3,3) s i n 3 * /2 • H S i n at 20 -l/C^f- its Cr}r) Si'n'ac (o}0 cos x 2 (0,1) -5/ C-in 2 ~ V") OA) SlU X cos* C 0-,3) S i ' h * x cos* l-Z -7 / (JU 2 " Jos V-) (56*) In TABLE 3, the extreme right hand column involves only the highest power of °\ which appears i n the exact formu-lae for ^ l l ^ , ' • Also, for the (r,r) l e v e l , KV i s a > o constant which can be determined f o r each p a r t i c u l a r value of r ; for example, - /j <3 = 2;