An Asymptotic Study of a Glass of Ordinary Linear Differential Equations of the Second Order Citation Owens, Robert Hunter (1952) An Asymptotic Study of a Glass of Ordinary Linear Differential Equations of the Second Order. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/81ED-D440. https://resolver.caltech.edu/CaltechTHESIS:02082018-101404674 Abstract The asymptotic behavior, with respect to the large parameter µ, of two solutions of d 2 y/dx 2 + µ 2 [c 2 - x 2 + f(x, µ -1 )] y = 0 is given where f(x, µ -1 ) = 0(µ -1 ) is subjected to suitable hypotheses and all variables are real. These solutions are approximated to by the parabolic cylinder functions D v (±√2µ t) with v = µ a 2 /2 - 1/2 and t = Ø(x,µ -1 ). The function Ø(x,µ -1 ) and the quantity a 2 are constructed as a part of the analysis in which the parameter µ is restricted in such a way that v is bounded away from the positive integers. The relative error of the approximating functions is uniform for all x. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Erdélyi, Arthur Thesis Committee: Unknown, Unknown Defense Date: 1 January 1952 Record Number: CaltechTHESIS:02082018-101404674 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:02082018-101404674 DOI: 10.7907/81ED-D440 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10684 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 08 Feb 2018 18:42 Last Modified: 16 May 2023 21:17 Thesis Files Preview PDF - Final Version See Usage Policy. 11MB Repository Staff Only: item control page

AN ASYMPTOTIC STUDY OF A CLASS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER Thesis by Robert Hunter Owens In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1952 It is with pleasure that I acknowledge the assistance and guidance given me by Professor Arthur Erdélyie Our discussions of the difficulties I encountered are certainly appreciated. I welcome this opportunity to thank the California Institute of Technology, and the Mathematics Department who acted in their behalf, for their generous financial assistance which enabled me to continue my studies. In particular, I appreciate the teaching assistantships and the partial tuition scholarships which were awarded mee I wish also to thank my wife, Dorothy, for her help and en- couragement, and for her very excellent typing of this manuscript. ii Page Page Page Page Page Page Page 8 16 23 43 66 68 69 ERRATA 2 2 Eq. (2.8) d'v replaces dy * ax? ax? Eq. (2.37), A, replaces A Eq. (3.26), insert "=" after Dox) Eq. (6.15), ex replaces gi U t 2nd line from top,v*® replaces y7@ 2nd line from bottom, (7.18) replaces (7.17) 5th line from top, e "ox replaces qi ABSTRACT The asymptotic behavior, with respect to the large parameter hp, of two solutions of ae + |e - x? + toa")] y = 0 is given where f(x,y’) = O(ji') is subjected to suitable hypotheses and all variables are reales These solutions are approximated to by the para- bolic cylinder functions D,(+f2p t) with v = yp = - 5 and t = B(x,,'). The function P(x, 4i') and the quantity a* are constructed as a part of the analysis in which the parameter » is restricted in such a way that v is bounded away from the positive integers. The relative error of the approximating functions is uniform for all xe iii TABLE OF CONTENTS SECTION 1 INTRODUCTION 2 DESCRIPTION OF THE PROBLEM AND PRELIMINARY TRANSFORMATIONS 2el Description of the Problem 2e2 The Preliminary Transformation for Equation (2.4) 2e3 The Comparison Equation and Its Solutions HYPOTHESES FOR f(x,p ') AND CONSTRUCTION OF THE TRANSFOR MATION PROPERTIES OF THE TRANSFORMATION t = A(x, 1') 4el Behavior of the Bs k = less, mas x > +00 4e2 The Solution 6(x) of Equation (3.33) 4e3 Behavior of as x + +0 DEVELOPMENT OF THE INTEGRAL EQUATION ASYMPTOTIC FORMULAS AND BOUNDS FOR THE PARABOLIC CYLINDER FUNCTIONS 6.1 <A Summary of the Results of this Section 6-62 Bounds for D(x)» Die, (ix) in the Interval Vivt2 - V2 < x < VAve2 + V2 663 Asymptotic Formulas for D(x), ae (ix) in the Interval Vivt2 +V2 <x 6.4 Asymptotic Formulas and Bounds for D\(x), Di, (ix) in the Interval -/4v+2 +72 < x < VAve2 - 12 665 Asymptotic Formulas and Bounds for D(x) in the Interval x < -VAv+2 + V2 iv PAGE 15 18 26 26 29 32 43 50 52 54 SECTION PAGE 7 SOLUTION OF THE INTEGRAL EQUATION 55 7el Notation and Remarks 55 72 The Solution in the Interval a + _ <t 56 7e3 The Solution in the First Transition Interval 59 7e4 The Solution in the Interval -a + = st<a- = 62 7e5 The Solution in the Second Transition Interval 64 726 The Solution in the Interval t < <a - — 67 8 SUMMARY ¥ BIBLIOGRAPHY 74 SECTION 1 INTRODUCTION Investigations of the asymptotic solutions of ordinary dif- ferential equations containing a large parameter are not newe A very good account of these investigations is contained in R. E. Langer's Symposium Lecture of April 6, 1934 (a) * in which he dis- cusses the work that had been done up to that timee Moreover, he includes a very complete set of references. Of particular interest is the work that had been done on functions with transition points. A region in which the solutions of a differential equation change from monotonic to oscillating is called a transition region. Although the actual transition does not take place at a precisely defined point, it is often convenient to determine one point of this region and call it the transition pointe Different asymptotic formulas for the same function are obtained in intervals separated by the transition points, and it is important to know how these asymptotic formulas in the sep- arate intervals are related. ‘This knowledge is usually supplied by "connecting formulas" which are valid in the transition region. Where the intervals of validity overlap, the "connecting formulas” are asymptotically equivalent to the formulas previously obtained. In this regard the work of Langer (2] should be mentioned in which he treated * Numbers in brackets apply to references listed in the bibliography. mf a class of differential equations in the complex plane and containing a complex parameter under rather general hypotheses. His results include the case of one simple transition point. Also, Langer (3] considered certain differential equations in which the dominating term in the coefficient of the dependent variable has a double zero, and AP he considered the case of a simple transition point directly. ‘In this latter work the variable is real and the parameter complex. Finally, in 1950, functions with one simple transition point, in which all variables are complex, were investigated by T. Me Cherry [5] who treated the problem by a new method. Generally speaking, all the methods used, including Cherry's, are based upon the observation that "similar" differential equations are satisfied by “similar” functions. The use of the word “similar” will become clear in the contexte Consequently, the differential equation being investigated is compared with a similar differential equation, the solutions of which, along with their properties, are usually well knowne Then, the solutions of the two equations are compared in such a manner that the values of the known functions may be used as approx- imations to the values of the unknown functions. The novelty of Cherry's method is the way he uses a certain trans- formation by means of which the differential equation being studied may be compared with the "most simple" similar equatione The construction of the transformation is, naturally, an essential part of the problem. Although uniform approximation to the unknown functions by means of the known functions (with respect to the independent variable as the parameter tends to infinity) has been obtained by other mathematicians, the outstanding feature of Cherry's approach is the ease with which he obtains this desirable property which is enjoyed by his approximations. Various possibilities of Cherry's method were recognized by Professor Arthur Erdélyi who outlined how it could be applied to func- tions with two transition points in his seminar in Asymptotic Theory at the California Institute of Technology in the spring of 1951. The development of this idea will be carried out in this dissertation. Naturally, all results pertaining to a single transition point may be applied twice in the case of two transition points provided the coeffi- cients of the differential equation meet the required hypotheses. How= ever, the present investigation copes with the case of two transition points directly and gives very simple formulas for the asymptotic solu- tions of the differential equation. It is felt that Professor Cherry's paper is one of the more significant contributions to this type of researche These investigations are not to be confused with the asymptotic development of one special function, say the Bessel function, which is treated by methods especially suitable to the functione With respect to such developments it should be noted that Ne Schwid [6] applied the results of Langer (2) to Weber's equation (Section 2.3) and obtained asymptotic formulas for the parabolic cylinder functions. Schwid's results are subsequently used in the present investigation. Real functions of a real variable with two transition points, which satisfy a certain class of ordinary linear differential equations of the second order containing a real, positive, large parameter, will be studied. The asymptotic behavior of these functions with respect to the large parameter is desirede The class of differential equations has the form 2 = + p* [c? - x* + £(x,")| y=0 (1.1) where » is the large parameter, c is a positive constant, and f is subjected to suitable hypotheses among which is the requirement that f = O(,f'). However, the investigation will include the equation BE + P(xyt!) SE + [uP (x) + Qe") = 0 (1.2) where ho (x), like c* - x*, has, on the real axis, only two simple zeros and is positive between the zerose P,Q are also subjected to suitable hypotheses in particular P,Q are 0(1) for large p. Appropriate trans- formations will reduce (1.2) to (1.1). In equation (1.1), the outstanding characteristic, when y is 2 large, is the presence of the function c* = x*. The natural comparison equation is 2 4 + u2(a* = t?)u = 0 (1.3) The solutions of (1.1) are approximated to by the real valued parabolic cylinder functions D,(¥2u t) and D (-¥2u +t) which are solutions of (1.3), with t = A(x,;'), the previously mentioned transformation, and 2 yell 2 2” When v is not a positive integer or zero, D,(¥2p t) and D(-V2p t) form a fundamental set of solutions of the comparison equation (1.3) since their Wronskian does not vanish, but when v is a positive inte~ ger or zero, dD, (¥2p t) is a constant multiple of D,(-¥2p t), (Section 2-3) Comparison of the solutions of (1e1) and (1.3) requires asymp- totic formulas for D,(V2p t) and D,(-V2u t) when v is large. From these formulas (Section 6) it is seen that when v is not a positive integer or zero, Dy (+¥2u t+) is unbounded as t tends to + a. This behavior changes radically when v is a positive integer or zero for then D, (+¥2u t) tends to zero as It! tends to oo. The comparison of the solutions of (1.1) and (1-3) will be carried out with v bounded away from the positive integers so that this exceptional behavior of the parabolic cylinder functions will not occure Actually in connection with the differential equation (1.1) there are two distinct problems: (i) the general case, which is inves- tigeted in this thesis, and (ii) the special case, that is to say, the characteristic value problem where the differential equation has a solution which tends to zero as |x] tends to oo This occurs for certain values of p such that v = yp = - : becomes arbitrarily close to, or equals a positive integer. Work on problem (ii) will be continued, and it is hoped that satisfactory results can be given in the near futuree Also, applica- tions of the results, particularly to suitable forms of the one- dimensional Schroedinger equation, will be made in the future. It is believed that solutions of the Schroedinger equation may be obtained in more useful forms than those obtained by the presently used WKB method (the historical development of this is also described in Langer [1]) and possibly more quickly. A summary of the results of this paper is given in Section 8. The asymptotic formulas and conditions under which they are valid are precisely stated there. Notation. Capital letters A, aye K, K, refer to constants which j are independent of » and x or t, and the same letter appearing twice may represent a different constant each timee Arguments of the func- tional symbols are usually suppressed, or only one of the two variables may be shown when it is of primary interest. It is usually supposed that p is “sufficiently large." Attention is called to the "+y,o0,-" subscript notation which is explained in Section 22. SECTION 2 DESCRIPTION OF THE PROBLEM AND PRELIMINARY TRANSFORMATIONS 2el Description of the Problem. An investigation will be made of the asymptotic formulas for the solutions of the class of differential equations described by a ~1) ay , OE + pay!) + [pn glx) + wa(xri")] X= 0 (241) where p is a sufficiently large, positive parameter, and P,Q are analytic functions of x and - real and regular for all real x and o <p) Se The essential feature is that the analytic function h o(*)s which is real and regular for all real x, has, on the real axis, only two simple zeros b, < b, and is positive in the interval between the Zeros e Equation (2.1) may be converted by the transformation 1 Y = y exp(- 5 Pax) (2.2) into oy + y?[n (x) + ag! - +2 B) yy = 0 (2.3) ax” al Nast H 4 2 dx vo|Y ° and, since P,Q are regular functions of ji' for by < p, this last equa- tion may be put into the form 2 — + u*h(x,p Jy = 0 (24) where h(x,)') is an analytic function of x and i‘, real and regular for all real x and Hy < be It possesses the expansion sf = -n h(x,p') = 2. h(x) (2.5) n=o in which hy (x) is the function described above, and the h(x), o<n, are analytic functions of x, real and regular for all real xe When the transformation of the independent and dependent variables t = B(x), ult) = v(x) B(x), (2.6) in which 0 < #'(x) for all real x, is applied to 2 Pr] + H(t)u = 0 (2.7) the form of this equation is preserved, the result being Since (2-4) is of the same form as (2.7), a transformation can be constructed, independent of yp, by means of which the coefficient of the new dependent variable has the following property: in its expansion in powers of ji', the coefficient of u* may be any desired function which, like h(x), has two real, simple zeros and is positive between the zeros. The simplest function of this type is c* = z* where c is a positive constant. Consequently, the differential equation (2.4) may be con- verted, by a transformation which will be determined in Section 2.2, into the equation a*w ay ou *[ c* «ats (2, i")] w =0 (2.9) where f(z,j') is an analytic function of z and ji', real and regular for all real z and Hy < be It possesses the expansion f(2z,') = = f (Ze (2.10) n=l The present investigation will therefore apply to the three equiv- alent differential equations (2.1), (2.4) and (2.9). However, only the simplest of these three equations, equation (2.9), will be treated in detail. The desired results are approximations for the solutions of (2.9) whose relative error is O(n”), m being an arbitrary positive integers This error will be uniform for all real z with the exception of two arbitrarily small intervals including the points 2 = + c, respec- tively. The lengths of these intervals depend on pe In order to apply the results to equations (2.1) and (2.4), the preliminary transformations previously discussed must first be used in order to convert these equations into (2.9). It is necessary to develop such a transformation for equation (2-4). 2-2 The Preliminary Transformation for Equation (2-4). Introduce x= f(a), y(x) = w(x) At(2)* (2.12) into equation (2.4) which becomes, in view of (2.6,7,8), 2 mt te . oe * dO LOn + (3 oe " ‘ ren m ; w=O (2412) Now by (265) h[A(z)) = h[A(2)] + ra [A(2)Ju". It will be shown n=] “n that a positive constant c can be determined so that h [B(2)] B'(2)? = 7 = 2? (2.13) has an analytic solution g which is real and regular for all real z. Then the real zeros of the two functions c* = 2* and h,[0(2)) wi coincide so that $'(z) will vanish for no real z- Consequently, it is assumed that positive roots are taken, implying 0 < 6"(z) for all real 10 ze Then (2.12) may be put into the form (2.9) as desired. It is evi- dent that @(z) will be independent of p. Equation (2.13) is equivalent to dx)* _ 2. .2 h, (x) (G2? c Zz (2.14) and, since h, (x) is positive for b, < x < b,, equation (2.14) may be written in the two forms Vc? = 27 dz =Vh (x) dx, b, <x <b, z* = c* dz =V-h (x) dx, x <b, or b, < Xe 2 Since the zeros of h 9 (*) correspond to those of c* - 2*, integrals of these equations are m b | le- at -[ ‘Y-nE) a, x sb, Z x A oz - N A 4 Q A oe ‘ Q 1A iS) 1A Qe [ite - 7 at [4 ho(—) de, bs -c b, A La) ° 1A rs) [ve ~ ct at = [ 1%@ de, b, < ¢C 2 which give the results ¥-b,() at (2.15) 2-427 - o* + Cosh". =e Q Jno wy c? Cc where x $ b,, 2S “Cc, OS Cosh ““G,» x * c* - 2” + cos . = ef V4 (E) at (2.16) “1 where b, < XS by, “CS2ZSC, OF CoS -4 <u, and st x 5545? -c*%- Cosh +4 = Fi ¥ -h,() ak (2617) b, where b, SX» CZ2Z, 0 < Cosh /2. Since x = b, and z = c must correspond, these values may be sub~ stituted into (2.16), and the positive constant c is determined by 2 /° cs & [ Vb (E) ak (2.18) b, With this value for c, it will be shown that equations (2.15,16,17) determine an analytic function 2 = Y (x) which is real and regular for v; all real x with the property that 0 < ¥Y(x) for all real xe 2 2 Cc on It follows from (2.14) that the quotient “had is positive for all real x and z since the zeros of numerator and denominator have been made to correspond in such a way that the numerator and denominator dx always have the same signe Therefore, when x and z are real pi = dz ‘ and - = a do not vanish and are always positivee That is, the functions £ and Y will be monotonic increasing and inverse to one anothere Moreover, z = * c only when real x = b,,» Dae From (2.16) it follows that x 2c? - 2* + cos 12 ant Vhi(E) ak bz c* where 1 is introduced by using (2.18), and since cos = 2 =1- cos +2 with O< cos 1 < 1, this last equation becomes x te - 2* = cos tg = 5; ¥ h(E) ak (2.19) ba where b, SX < bg, “CL 2K Co This is equivalent to 12 [ve ~C ae [ 0) de, b, <x <b, (2.20) c be Since the analytic function ho (é) is regular for all real — and has simple zeros b, < b,, the following expansion is valid about b,: * Also ¥c* = C* may be expanded about © = c giving ¥c* = dag = V2 (c = r)*s da, (c = a eooe so that equation (2.20) gives 2V2e (o-n)** ad, ee 2¥-n' (b2) (b,x) + 2 o (byex) Steve (2621) It follows from (2.21) thet one may take arg —?_ x ~ 0 when x is sufficiently near b, (222) When x < b, take arg (bz - x) = O. Then, from (2.22), arg (c ~ 2) = 0 and z < ce Let x be continued into the complex x-plane from the left of b, around a semi-circular arc to the right of b,, either above or below b,- Then b, < x and arg (b, - x) = + ™ so that, from (2.22), arg(c - 2) = +41. It follows that c<z, 1 < Z, and cos +4 is pure imaginary. The following relations are then valids 1 = + Vc? - 22 = V2" = oc? ett? e < z,and h(E) = wh, (€) en tt b, < Ee These are introduced into (2.19) giving 7 = og = ay c* = iy cos “12s 2, f V ~hi(€) dé, ba c2 elz Fig 1g -lz and since cos “¢ is pure imaginary, e * cos c = Cosh “¢, so that 13 the last equation becomes x < 2* = c* = Cosh t2 = 3 f 1-@) a, b,<x, ec <2 2 which is identical with (2.17). Thus, by continuing x around b,, either above or below bz, equation (2.16) goes over into (2.17). When equation (2.16) is investigated at the points x = b, and 2 =-c, a similar discussion shows that, by continuing x into the complex x-plane from the right of b, around a semi-circular path to the left of b,, either above or below b,, equation (2.16) goes over into (2.15). In equations (2.15,16,17) let the left members be denoted by F_(z), F,(z) and F,(z) and the right members by W_, W. and W,, respectively, so that these equations become W=F(z), W,* F(a), W, = F,(z) (2-23) where the subscripts +, o or - are used in the following manner: Subscript notatione The subscripts +, o or = shall mean that the quantity bearing these subscripts is associated with the interval of the real axis to the right, between, or to the left, respectively, of the zeros of the function of intereste Sometimes the subscript o is not intended to convey the meaning of this subscript notation. This is evident in ho (x)- From (215,16,17) it follows that F'(z) = 2y z* - c*, Fi (z) = 27 c - 2°, Fi(z) = 4; z® = c* (2424) and that F_, an and F, are analytic functions of z, real and regular 14 for all real z except z= + ce It is seen from (2-24) that F', FP‘, and Fi vanish only for z = + c, ieee only when x = b,, b, by a previous remarke This means that equations (2.23) may be inverted in the neighborhood of any z # + ¢ giving z= H(W), 2=H(W), 2 = H,(W,) (2.25) where the H's are analytic functions of the W's, real and regular when real z # ¥ c, or whenx 7? b,, b,+ But since the W's are given by the right members of (2.15,16,17), they have expansions similar to the right member of (2.21) and are therefore analytic functions of x, real and regular for all real x except x = b, b,- Consequently, the H's are analytic functions of x, real and reguler for all real x except x = b,, b,, and they will be denoted by (x). Equations (2,25) then become 2= ¥ (x), 2* (x), z= ¥ (x) (2.26) where ¥_ is defined by (2.15), Y% by (2616), and ¥Y, by (2017). However, it has previously been shown that (2.16) may be continued into (2.15) and (2.17). This means that ‘, may be continued into WY and Y,° Therefore, these three functions are analytic continua- tions of each other and represent a single function Y (x). Moreover, the functions are defined and have first derivatives at the points x = b), b, so that these points must belong to the region of regularity of Y(x). Hence z = ¥(x) is an analytic function of x, real and regular for ell real x, with the property that O < ¥ (x) for all real x as was previously pointed oute This function is the solution of equation 15 (2.13). Its inverse, x = 6(z), is the desired transformation (2.11), and is an analytic function, real and regular for all real 2, with the property that 0 < #'(z) for all real 2 Referring to the state- ment following equation (2.13), it is easily seen that equation (2.12) may be reduced to the form 2 ay + 126? - 2° + #(2,6")) x = 0 (2.27) where f(z,,i') is an analytic function of z and ,', real and regular for all real z and by < pe This function possesses the expansion 1 2 en f(z") = 2 fi(z)p (2428) n=] and will be subjected to suitable hypotheses. The investigation will henceforth be confined to an equation of this form. 23 The Comparison Equation and Its Solutions. Equation (2.27) will be compared with the equation 2 7 + p2(a? = t?)u = 0 (229) which may be transformed into Weber's equation 2 2 fF + (ve5- Feo (2.30) 2 where v = - 5 » by the transformation ig ok. 2p » u(t) = w(z) (2.31) It should be noticed that the coefficients of the dependent variables in (2.27) and (2.29) have a common property. In their expan- sion in powers of ren the coefficients of y have only two simple zeros 16 on the real axis and are positive between the zeros. The parabolic cylinder functions D\(+ 2), D\_ (+ iz) are solutions of Weber's equation so that solutions of (2.29), using (2.31), are D(+V¥2p t), D (441 742p t). v =v! Some properties of the parabolic cylinder functions D (+ z) and Die (+ iz) are [7,8]: (i) They are integrel functions of z (14) nye) = A op (an) +e 2D (nts) (2432) a evtt D,(-2) + aos oF Mp (nts) (2.33) (444) D (iz) = atti ett Dy(-2) = e Di_,(-iz) (2034) (iv) They have the Wronskians D(a) & p(n) = yen) & v2) = ety (2.35) D(z) 2D (4z) - DL (iz) SD (2) = -6 > am (2.36) (v) If v is real, D(z) has [v 7 real zeros where [w + J v denotes the greatest positive integer less than v + 1 or zero. From (2.35) it follows that Di(a)s D,(-2) are linearly independent if v is not a positive integer or zero. It follows from (2.36) that D yl2s D (iz) and Dy (<2), D. (iz) are linearly independent for all =v! =v! values of Ve When v is not a positive integer or zero, the general solution of the comparison equation is u(t) = AD (¥2y t) + Ad (-¥2u t) (2.37) hy 4 2 where veu = - : (2-38) In addition to the foregoing properties of the parabolic cylinder functions, various asymptotic formulas and bounds for these functions are needede These will be discussed in Section 66 SECTION 3 HYPOTHESES FOR f(x,p') AND CONSTRUCTION OF THE TRANSFORMATION The differential equation to be investigated, (2.9) and (2.27), in which the variables are renamed, becomes a*y 2| 2 2 =t at tH [e - x* + f(x,f My = 0 (3-1) where the function f(x,)1') is subjected to the following Hypotheses: (4) £(x,j7') is an analytic function of x and i', real and regular for all real x and by < , and possesses an expansion of the form f(x,p') = 2D f(x) v (3-2) n=1 where the f(x) are analytic functions of x, real and regular for all real Xe (ii) When x is real andx++0, f(x,,i') may be put into the form f(x')=ax? +B xty +0(x"), O<n<1 (363) oO [e9) fe 2) “n -n - -n . where a, Z, abs B. 2, By} ’ Yu 2. Yy4 (3-4) the series for ai B. and Y, being convergent. From hypothesis (ii) it follows immediately that in (3.2) f(x) = a,x" +pxt+y, * O(x"”) as x + +00 (3-5) 18 19 In order to compare (3-1) with the comparison equation q2 Sz * He (a? = t?)u = 0 (36) this comparison equation will be expressed in terms of the variables x and v by putting % t = B(x), u(t) = v(x) 6'(x)* (367) and using equations (2.6,7,8). The result is ay + {atte ~ f) prt BOM 2G on (3.8) It is now required to construct the transformation t = (x) and to determine the quantity a* so that equations (3.1) and (3.8) will be approximately the samee The required transformation may be taken in the form t = B(x) = B(x!) =x + DP D(x) » (3-9) n=] The choice a* = ¢* would seem to be the natural one at first glance. However, it is necessary to choose a* so that @(x,,') has no real singularities. This requirement is met by taking fee) _ at =o%7 + 2 ah (3210) n=] in which the ay will be determined subsequently. Let the function g(x,t') be defined by - m9 "2 . e(x,f') = p?(a*-g*) p+ 2 , 7 : or ~ p2(c%=x*) = p?#(x,40') (3611) and note that the differential equations (3.1) and (3.8) would be identical if g(x,)') were required to be zero, and that g(x,,') meas- ures the discrepancy between these two equationse The equation g(x,)') = O is a non-linear differential equation for f(x,;i'). This 20 equation determines a system of equations for the b. of (3-9) and the a, of (3.10). Such formal series do not necessarily converge, but m -n their first m + 1 terms, x + = B(x) p and c* + = ant where m is arbitrary, will be computed and retained since they make the expansion of g(x,,i') in powers of jij’ begin with the term ym, In order to determine by a,» m +1 <n, the remaining system of equations for p, and a, will be modified so that (i) the series (3-9) and (3-10) converge, (ii) g(x,,i') = otc 4, and (iii) g tends to zero as x tends to to. Consequently, the differential equations (3.1) and (38) will approximate each other very closely for large » and all real xe Introducing (362,9,10) into (3-11) and using the Cauchy product for power series gives the result cenit) = |Z anh Bex Da .> F 6g.) ) a] + n=l] n=2 kel n=] n=] n=2 k=l] alt= 2 oa ak Dhlox> be -F & og _)s at [as Bb te oat i a n=] n=2 k=l 1 = tt) Pe = ' " . 2S goyma eS pe “-12 4 " ) | “(1 > pt ra ny - 2 _ rid (3-12) in which the coefficients of , n = -1,0,l,e«ee0e, m= 2 are set equal to zero, giving a system of differential equations which determine the 21 g and the an n= Lyccces Me n ys 2(c* = x*)f' = 2xf,= f, - a, (3-13) pos 2(c? = x*)f! = 2x82 =f, - 8, - 2a,f! - ~(c* = x*)pi* + g? + dxf pt (314) and in general, for Bi, ok+ ye 2, 2(c# = x* pt ~ 2x By. =f,+a, +P. ~p, (3.15) where k-1 RG) = 2 [Bing = (ot x) Oia 5 ~ ashy + aol) + k-2 > , fa r=1 [PO Pe jor ” ash k-j-r * aed ‘a Y L-jer] . ke2 j-1 k= j-1 +2 (2 6b. E BBY jan) (3416) je2 oral Coefficient of y ete in the expansion py, (x) = 1g 3 gv (3.17) in powers of ,1' of se %, gre It is most easily seen from (3.13) that P (x)= 0 (3-18) and from (3.16) that PL (x) is a polynomial with constant coefficients in pb. pi, a, and x, n= 1l,eses, k-le From the terms in (3.12) contain- ing second and third derivatives it follows that p, (x) = pa(x) = 0 (3.19) and also from these same terms in (3.12) that p,, (x) hes the form 22 Polynomial with constant coefficients in P(x) = pi, ne ns n= Lyeces, k - 26 (320) It is quadratic in pr and linear in om. Equations (3613,14,15) may be used to successively determine Dy guwowy o, and a,,ee60, @ when appropriate boundary conditions are imposed. These conditions will be chosen so that the 6, will have no real singularities. It is sufficient to discuss the general equation (3-15) which may be put in the two forms bie P. - ie xD, : £,(x)+ i (*) Py, (x) celle er (3.21) K Vo?=x? 2 V c*=x* 2 Vcr—=x? p £, (x)+#P, (x)-p, (x) (FE Bi, ten «celine pms, gt fall CNR x7=¢7 2x? 0? 2Vx*-c* Integrating (3.21) gives the result f (5) *P, (E )=p, (E) a, V o* * —X a- = cos” t= (3-23) x f= | ie 2 lame -E 2 c el where -c < x < cy, O< cos =e <7 from which a, can be determined by putting x = c giving c £,(E) + P(E) = p(é) a = 4 ms _ x ae (3-24) -c V c* = hg With this value for ay and the subscript notation of Section 2e2, where the function of interest is c* = x*, it follows from (3.22) that (6) +P, ( )ep, (E) =[- eee ae ag - —K cosh™-¥ (3.25) Cc £2 =o? 2V x*=c? By _ (x) . aS where x < =c, O< Cosh" = % 23 from (323) that x £,(E)+P_(E)=p, (€) a, - B, (x) — = x x dt - ——— cos 1x (3-26) a Co l. 2V cr? 2¥ tax? ° -1 where -c < x <c, O < cos -% <7 and again from (3.22) that _ 1. f* £,.(E +P, (E)-p, (8) — oiling B,.,(x) aes [ end dk > = Cosh G = (3627) where c < x, O< Cosh 1% The functions By» B49 b. assume indeterminate forms at x = +c, but they may be defined there by their limiting values, obtained after applying L'Hospital's rule. These definitions are f,.(-c) = f, (-c) = + ——+ . (3428) fi (c) + P(c) = p(e) = A, (c) = 6,,(c) = - ———_e Fe (3.29) It is easily shown that Bos 69 and Bh. are analytic continua- tions of each other and are therefore elements of an analytic function B(x)» real and regular for all real x. This assertion is first established for B, (x). All the functions involved in the definitions of Bios Bios and Bis (3-25,26,27) are analytic functions of x, real and regular for all real x, except x = + c, and if x is continued into the complex x=plane along a semi-circular path around + c, whether above or below +c, Bro goes over into Biss respectively. These functions are defined at x = t c by (3-29) and (3-28) so that + c must belong to 24 their region of regularity. This implies that f,_, 6 ro? and B14 are elements of an analytic function £,(x), real and regular for all real xe The assertion follows for B(x) from (3+25,26,27) by induction. Having determined the Bs ap» k = lyesee, m, the system of differ- ential equations (3.15) is modified, as mentioned on page 20, in order to determine the remaining Bs ay Thus, when m + 1 < k, the quantity P(x) in (3-15) is deleted so that By» a, are given by (3-24,25,26,27) after crossing out the p,.(E) in these equations. This modification is equivalent to deleting all terms in (3.12) which contain second and third derivatives, when the differential equations for the f,, m+i1<k, are formed. As can be seen from (3-17) this is also equiv- alent to using only the first m - 1 terms in the expansion of 4 or ~ : in powers of »' for the determination of all the Dy» aye Since a convergence discussion of the series for i) and a* is difficult, this required result will be obtained from properties of the functions themselves. A differential equation which determines f and a*, and hence all the 4, a, is easily constructed. For this purpose, define F(x) = f(x) - Py, (x), k 2 lees, m (3-30) F(x) = f(x)» m+ls<k (3.31) first m-1 terms in (os) th the expansion of F(x") = 2. FA(x)w = f(xei') = 2 om 3 ore (3232) n=] z UF 4 2 in powers of ji’ and introduce (3.32) into (3.11) which gives, because of the above modification, the following differential equation for pf: 25 u*(a? = f*) Br* = y2(c? = x*) = pF(x,i') = 0 (3-33) Moreover, once the ff, yee, Bb, and a,,e+«, a have been determined, F(x,))') is a known analytic function of x and ;/', real and regular for all real x and Hy S He It will be shown that (i) the solution A(x‘) of (3.33) exists and is an analytic function of x and »', real and regular for all real x and Ky < py, and its expansion in powers of re has as coefficients the B, previously determined for k = 1,++e, m, (ii) a* can be deter- mined from (3633) and is an analytic function of ,', real and regular for », S$ 4» having as coefficients in its expansion in powers of {i the a, already computed, k = 1,++«, m, and (444) O < B' for all real x and <1 so that t = O(x,n') is a reversible transformation with respect to Xe From (3-33) and (3.11) it follows that "2 1 gm n2 = -n “2p i gpet 2 (Rf) & and using (3.30,31) this last equation becomes 1" " m - g(x,y) tee -w > pw” (3-34) n=1 SECTION 4 PROPERTIES OF THE TRANSFORMATION t = £(x,11') 4e] Behavior of the By, For this purpose Hypothesis (ii) will be used. A detailed dis- » kK = lheeoy MAS X + tO. cussion will be given for #,(x). From equations (3+27,18,19) f, { )a 1 - at =o hy glx) = bf” AS cost, e<x (4el) c E? = o* and, making use of (35), this equation may be put into the form a + y a 2 Ver - co 2 1 eit) ated tagee 0(¢77) c Er? = ¢? x VeEF=c* giving, after integration, the result -1 a, B, Cosh % S45 1 ak vampenewesmnitions i -nemmmpinioes 1-1 > . 6, ,(x) = = + @, Vxtao® i 2a + O(x )y as x 00 (43) 1 /@ £, (f) = a, £7 ~ BE- y, where 6,, = > ai ag (L404) é Et? = 2 a, acy, and ©, = 3 = 5 7 > (405) Differentiating (4.3) gives “lx Cosh w &. 4% t (x) == % = c lt “2-1 Bal) = = Pm Om area) a Ga ape OT) ne) 26 27 It follows from (3.16), but is most easily seen in the right member of (3-14), that P,(x) is given by P(x) = -2a,f! + (x* = c*)p!7 + B? + dxf, BD! (4-7) In order to compute P, up to terms of order o(x 7), f'* must be computed up to O(x 2"”), 2? up to O(x ”), and 6,8! up to o(x 277), Making these calculations with (4.3) and (4.6), and introducing the results into (4.7) gives a, P2,(x) = 3 a?x* + 2 a, B,x - 7 a,+ 3 a7c? + : a,y, + + Be + 0(x 7) (48) in which w, has been eliminated by using (4.5). From equations (3.25), (3018) and (3.19) 1x*=c? 4 (x) = ; i oe - =" Cosh"+-% » x<-0 (409) into which (3.5) is introduced, the result, after integration, being a, 8. Cosh 1-3 6 B(x) =- 7 x- > o, Vx? = ¢? "Te -o* + o(x72"") as x > =o (4.10) where ®, is given by (4.5) and -c f = CA iby, af" (E) = a8? = BE = y — § == i 2 a Ve? = co Substituting # _, B'_ into (4.7), and following the procedure stated below (4.7), gives the result 2 P2_(x) = 3 a?x? + - a, B, x a a,+ 73 a*o* + : a,y, + Ce + O(x 7) (4012) where, again, w, has been eliminated by (4-5). Equations (4.8) and 28 (4-12) show that P,, and P\_ are identical, that is, P,(x) has the form P(x) = A,x* + Bx +C, +0(x”), asx++to (4013) which is of the same form as the f,(x) given by (3-5). Therefore, the 6, given by (3-25) and (3.27), using (3-19), will have the same form as ~,~ Because of the form of A, in (4-3), the assumption is made that a 6 <meta lier Nile + (xt) (4014) + J f= Vx* = ¢? as xX * OW, j s lyeveeyk -1 and the form of B.,(x) will be established for all k by induction. The functions Drs and their first derivatives are substituted into (3.16), the computations being similar to the procedure outlined below (4e7)- After a straightforward, but long calculation the result has the form Pyy(x) = Ax? + Bx +0, + (x ”) (4015) Differentiating (4.14) gives the results Bs +(x) = d; + o(x"), 56x) = o(x 27), Bi (x) = O(x 3-1) (4416) which are introduced into (3-20) giving Pry (x) = O(x ren (4017) If (4017), (4015) and (3.5) are substituted into (3.27), the result has the same form as the right member of (4.2) so that Cosh “lx Pp(x) = x + e+ 4 “= Ve ——- +000"), asx+ oo (418) -c* -c? 29 Equation (4.18) establishes by induction the form of 6, 4(*)» when x * oo, for all k in view of (4-3), (4014), and (4018). By noting the form of f,_ in (4210) it can be established in a similar fashion that the form of B(x), when x + -00, for all k is Cosh “= 6 = ay, = & ke + O(x ls as x > -00 (4.19) Vx? = co? Yx*=c? The derivatives of 6, from (4.18) and (4019) have the forms given by B(x) = bx + o (4-16). When these derivatives are introduced into (3-20) the result is Pp, (x) = o(x7""), as x + +00 (420) 4e2 The Solution f(x) of Equation (3.33). Equation (3.33) may be written (a? = f*)ft* = (c? = x*) = F(x,)i') = 0 (4021) It is necessary to discuss the behavior of the function o* = x* + F(x,p') (4222) which occurs in (4021). It follows from (32,32) and (3.30,31) that (4-22) is equivalent to m cn xt + Fi ct xt +f + (Fe fl) ect xt +f - > PH (4023) n=1 and from (4.20) that m 2. PH = o(x"? ) as x* tO, 4H, sufficiently large (4-24) n=1 30 Equations (3.3) and (424) are substituted into (4.23) giving the result c® - x7 +P = c* = x* + a x* + Bx *¥, + O(x”) as x ++ @ (4.25) For », sufficiently large it follows from (3-4) that la <i, BI < 1 which, with (4.25), give the inequality c= = x* +F <0, c << [x], psu (4.26) ° Since the analytic function F(x,,') is real and regular for all real fo) x and p_ < p, and possesses the expansion F(x,\i') = =. F (x)p” ° n=l “n by (3-32), it follows that pF is bounded for by S bs X, S$ x < XQ where the numbers X,, X, are arbitrary, but in the present discussion are chosen such that X, < “2c, 2c < X,- Consequently, for Hy large enough by Sell (427) [Fl <c*, X,sxsX» u Then, it follows from (4.26) and (4.27) that the function c* - x* +F is negative at least when 2c* < x*, and from (4.27) that this function is positive for |x] small enoughe Moreover, 0 < ec? + F < 2c* by (4027) so that the function c* = x* + F of (4.22) has precisely two real zeros which evidently depend on y» and tend to +c as yp tends to co. The equation c* = x* + F(x,j') = O implies x = x(ji') is an analytic function of ji', real and regular for y, < », and consequently, the zeros may be represented by the expressions 0 ee) x, = <c + ~~ x, nH 2 X, = ct > x nt _ (4-28) n=1 n=1 * These results may be summed up in the statement that when He is sufficiently large and », < 4, the function ce? = x? + F(x, p') of (4022), which occurs in the differential equation (421), has two real zeros 31 given by (4-28) and has the property that c* - x*+F <0, x< x, or x, <x and 0<c*=-x*+F, x,<x< x,° For a fixed value of pp, a discussion of the solution A(x, ') of the differential equation (4.21) is identical with the discussion of the preliminary transformation in Section 2.2. For, c? - x* + F(x,,)') is then a function of x alone with two real zeros and is positive between them so that (4.21) is of the same form as the differential equation (2.13). Therefore, the analysis of Section 2.2 applies to the solution f(x,}i') of (4621), and it follows that for fixed » the transformation t = f(x,,') is an analytic function of x, real and regular for all real x. However, all quantities in (4.21) and the limits of integration (4.28) depend on » and are analytic functions of ji', real and regular for p, < He Consequently, t = O(x,p') is an analytic function of x and ji', real and regular for all real x and by <M» with the property that 0 < O'(x,y') for all real x and by S be From equations (2.15,16,17,18), in which the necessary changes have been made, it follows that the elements f#_, Bb O, of the function O(x,\7') and the quantity a* are defined, respectively, by the equa- tions EVP = oF + cosh =~ Ze fer - oF - resi] (429) where x < X,» O<ra, OF cosh"+.8 x “a & Va? = 67 + cos 1-8 = 2 f [c? ~ £2 + P(E, ) ir (430) *, a § where x, SxX<X, was Bsa, O< cos ben 32 a x ru £ (@ =o - conf = 2, f [e* - o* - F(E,a')] “ae (4431) x where x, <x, asf, O< cosh*8 a? a 2 [ [c? - Ee? + F(E,i")] ak (4032) where Xj» x, in these equations are given by (4.28). Finally, if 0, gt, and a* are expanded, uniquely, in powers of u' and substituted into the differential equation (4-21) or (3-33), the coefficients of y must vanish so that the quantities By» ays k = lyeoey my, originally determined, are recoverede Moreover, the series J = x + z=, oe a* = c* + =, a converge. This completely establishes the three assertions on page 25. 4e3 Behavior of $6 as x > £0. Introduce (4.25) into the integrands of (429,31) which become aj ¥yte* ‘ (1-a,,)E* z B niga Bb since [a <1 for », sufficiently large. Because J&| is large, this equation gives [ez-c?-F(E, i") *= Vi-a, & - Bu - ¥ote = 2Vi-a 2 E B" ale = hte + O(E mM) (4-33) (1-a,)"6 so that (4.29,31), after integration, become, respectively, 33 Bx 4(1=a )(y +07) +B Vina, 4(1-a, )*# log x + gp V 2-27 + a®Cosh +. 8 = Vi-a. x* - + e.. + 0(x"”) as x >-c (4.34) where ie = ao [oe + a* cosh” 2-8 - Vi-a,, ¥ + Bux 4(1-a,,)(y,+e7)+BF la nth E los x| (4235) 3% 1-a, 4(1-a,) Bx ; 4(1=a,,)(y +e? )+B5 3 = - 1-a,, 4(1-a,,) . log x + DY fr-27 - a?Cosh 18 =Yl-a x* - m > ae * 0(x"”) as x~* oo (4036) a where @.4 = ae E V f2-a? - a*Cosn™28 - Vi-a, x* + Bux = A(1-a,,)(yyte*) +87, + + log x (4-37) Vi-a, 4(1-a,)* The left members in equations (4.34,36) are large if and only if } | is largee But this takes place only when |x| is large, and then these left members may be written ¢* (2 + Oo(f *n)| e Using this result and taking the square root of (434,36) gives phos?) = (a, (2 - Te “em + to (4.38) ay) x 2(i-a,,)x x as x > +00 (4. It follows from this equation that f and x are of the same order of magnitude, that is 34 £7 (1) es x + te (1439) Consequently, (438) gives the result f=(l-a js - Pu +0(x”) as x ++ (4-40) - 2(1-a,,)°# The derivatives of %, from (4.40), are B= (1-a,)* + (x7), p= o(x*), gu = (x27) as x >to (4.41) Returning now to the expression for g(x,,)') given by (3.34), it is evident that g is an analytic function of x and ji’, real and regular for all real x and y, < we This follows from the fact that O and its derivatives are analytic functions of x and » ', real and regular for all real x and up, < pH, and that the derivatives of the Bb.» which occur in the Py? are analytic functions of x, real and regular for all real x, with p, = pe = O by (3-19). Moreover, the first m terms in the expansion of g in powers of ,' cancel since # was constructed to insure that this would occur. That is, roma? is an analytic function of x and y', real and regular for all real x and Hy < pe When real x is bounded, all the functions involved in g of (3-34) are bounded, since O < f#' for all real xe Consequently, [e(xi)| s4u™ » XSxeX, wee (4042) where the X,, x of (4-27) are usede When [|x| is large, it follows from (3234) and (3019), (441,24) that 35 le(xsir)| < Ay px!" where A. is a function of up, and since pele is regular for Ko SH [er e(x,it')| < eta pt < A IxI"7"" as x + +00 (4043) A suitable adjustment of the constants in (4.42,43) gives the results “mt | le(xi | cae, -c-Ssxse+8, wosu (4044) Jeli) | saw x1, xg-e- Sore +S ex, wosu (4045) where the A may depend on m, and 6 is an arbitrary positive numbere SECTION 5 DEVELOPMENT OF THE INTEGRAL EQUATION Equation (3-1) will now be compared with (3.6), that is, with the transformed form of (3.6) given by (3-8). Introducing (3-11) into (3.1), the result is i + {¥ (a* - p*)pr* +5 : et = g(x,{1' )y (5-1) in which the left member is identical with (3.8). Equation (3.8) has fundamental sets of solutions given by (a), valx) = 2, [18n owl] 6x)? v, (x) = Dy [Y2u Bx] BY(x)*, ve(x) =D, [4¥2u Blx)] or(x) (5.2) and st + ¥, (x) = D,[-W2u A(x} B(x)", F(x) = DL, [-4v2u B(x) Br(x)* (5.3) which follow from (3-7) and Section 2.3. These pairs of solutions are fundamental systems for the homogeneous equation belonging to (5-1). Now let v(x) be any solution of (3-8). Then, formally, the general solution of (5.1), and hence of (3.1), is given by x y(x) = v(x) + x | [v.(x)v, (E) - v, (x)v, (E)] ele dy (E Jae (504) le] where X, is an arbitrary constant and A = v, (x)v}(x) - v, (x)v} (x) (545) A similar equation is obtained by barring all the v's and the A in 36 37 conformity with (503). From (5.5) and (52) A = v,[Yan o(x)] ar(x)* Sp, [stan A(x] a ei U ~ Dyn, [V2u M(x] Ox)? SS [Yau B(x] Brx)* (5.6) so that ed a gst ed d gy2] Asm [aes Bm” Dyay 2 of, | ax Py Oy so ge D D dD dD evgd DD -svi a Ds 7 fp sei =D v Bax W' Bt ax * “ai f) a2 #) The last expression in brackets is the Wronskian of the parabolic cylinder functions of argument V2 f- Therefore, by (2.36), A= {2h {-s ein} (567) In exactly the same way, from (5.5) and (5.3) A = -\2h {-s om} (548) Introducing (562) and (57) into (54) gives the result 2 ins y(x) = v(x) + - ° f ° {ae [sven B(x} o(V2n A(e)] - Le) ~ dy [V2n B(x)] Dyat [1v2u ac] NED () Z ak (5-9) B'(x)* P(E) with a similar equation resulting from the use of the barred quantities of (53) and (58). After the existence of solutions of these integral equations has 38 been established, estimates of the integrals will be required. However, the analysis will be facilitated if the solutions of (5.9) and (5.9), hence of (3+1), can be expressed in terms of variables which make the functions involved as simple as possiblee This can be accomplished by reversing the transformation t = f(x) of (369) which is allowable since 0 < O'(x) for all real x. For this purpose the following definitions are made: xe V(t) HXt), Plt) = Ry (5-10) c= ¥%(s)2 6s), ¥(s) = BE) where x = ¥(t) is the inverse of t = 6(x).e Equation (5.9) then becomes y{¥(t)] (ell , ing rt — a = ; D_ (i¥2p t)D. (V2p s) - wee By I, [Paes (AFR +10, (02 4.2 y[¥(s)) - Dy (2p t yD (aly sf eH] ¥(s) Vat ds where, for convenience, further definitions are made: G(s) = e((s)] ¥(e)?= 2S, (5.12) or(E) [¥ (+)] z t) =D Seyret] oly ct) (5-12) " v(t)? i pF u(t) - rlX(a)] = v[(¥(t)) or [y(t)] * (5.23) (+t)? and, with these definitions, the integral equation becomes 39 in} -t wie) = (a) + ROU [b,_, (Re o,f 8) - 2p » - D(¥2u t)D,_, (a¥2u s)| G(s)w(s)ds (514) Equations (5010), (5.13) together are simply the inverse of the transformation (3.7), by means of which (3.6) was converted into (3-8). Consequently, w(t) satisfies the equation iy + Etcs - 7) = a(t)] w= 0 (5415) and u(t) is a solution of (3.6), which was discussed in Section 2.3. Since (5.14) and (569) are equivalent equations, the existence of a solution of the simpler equation (5.14) will be established, and then an estimate of the integral will be made. Remarks Because the analysis will be carried out in terms of the variable t, it might seem more reasonable to construct originally the transformation x = Y(t) of (5.10) and hence, convert the differential equation (3.1) into (515) at the very beginninge However, the function Y(t) is not nearly as easily determined as (x), the required manip- ulations being much more complicated. The behavior of G(s) in (5.14) is required, and for this, the results given by (444,45) will be usede It follows from equations (4+29,30,31) that the points x = x,, x, correspond to the points t =f =a, ae Then from (5.11) and (4.44) it follows that {c(s)| < awn, ww he <s<at 4. (5.16) vp =l 40 since o[Y(s)] is bounded and positive for s in the above interval, and from (5011) and (445) that |a(s)| < \z\ eT ca yal ie (73 prc)? ssna- ee, atess (5-17) since, by aids x and # = t, or & and 8, are of the same order and g(c)=(Q-a,) + oe) = (1 = a,)* + O(s 2"), It will be seen in Section 6 that D,(¥2u t) tends exponentially to zero as t tends to infinity. Consequently, t will be taken as +oo in equation (5-14) and u(t) = Dy (V2u t). In the corresponding "barred" equation, t, will be taken as -oo and u(t) = D,(-12u t)e Making these changes, (5.14) becomes in co w(t) = D)(¥2u t) + io rf [D,(/2p t)D,_, (412n 8) - - Dy, (4¥2n +), (V2p s)| G(s)w,(s)ds (5.18) It is easily shown that the corresponding "berred" equation is walt) = D(Vau t) + 8 = D,(-1u t)D,_, (-412h 8) - ~ Du, (al Zu t)D\(-12y s)] G(s)w,(s)ds (5419) If in (5.19) the transformation s = -o0, t = -T is introduced, the result is 41 cl al A ine co WOT) = DURE T) + 88 f [bien MD, (AEH) - au Tr - DV, GRE DWT) ] Ge dwg (- eae (5-20) Therefore, by associating w,(-7) with w,(t), it is sufficient to treat equation (5.18) only. This will be done subsequently, but it is first necessary to discuss the parabolic cylinder functions that occur in the above integral equations. SECTION 6 ASYMPTOTIC FORMULAS AND BOUNDS FOR THE PARABOLIC CYLINDER FUNCTIONS 6.1 A Summary of the Results of this Section. The formulas which will be derived in this section are expressed in terms of the quantity 2vtl J_ x yf _x* x x" Jintd n, * - 1 = log +] - 1})VAvt2 <x = (601) x 2 Yive2 Avt2 Vive2 Av+2 ’ Notation. The symbol [a] shall be used with the meaning [Je a+e ge n=1 in which the Bh are functions of v and xe They are bounded for all real x and o < vy Sve The expression a~f is used in the sense that Bo le [lve @ Dy (x) ~ i Vave2 +12 <x (6.2) Le ces i ) wa of 2.2. Avt2 vti _¥ -Ny | D,(x)| <Avieze V4vt2 < x < V4ve2 +12 (64) otas % Ne [p,., (ix)| s AV?" Fe? « Vived sx < Yanez +¥2 (6.5) 43 = : D_, (ix)| ¢ Ae? v273 {4ve2-¥2<x<Thvt2 (606) —-o wee |D, (x)| <Ae*? yee Y4v+2 - 12 < x < VAvt2 (6-7) as © D(x) = (2) 2G Jeos p + o(w4)} ~f4vt2 +V2 < x < Vhve2 -¥2 (6.8) aia aN 3 xv ae t where M) = 28 e2 yz 8 (6.9) pt = é Avex® + (v - 2) sin ae - % (6210) ati Las [D(x)] < Ae? v2"8 ~fivt2 +12 < x <Vive2- V2 (6611) 4 3.2 |DV., (ix)| s Ae? v?FvS -V4v+2 +¥2 < x <¥4ve2 -¥2 (6.12) o¥ Yoo [D(x)| <she*tve @ -Vave2 < x < = V4vt2 + V2 (6.13) wm wee “TL |D, (x)J se? y2 "2 five * + A, 93|sin vt] o*} ~Viv+2- V2 <x < -Yivt2 (6614) vy o> F vn =-7) x D. (x)~ ——*—_ +8 fi] e “* -¥2 sin wnfije x < - Vave2 - 12 (6415) 6.2 Bounds for D(x), Diy (ix) in the Interval Y4v+2 -12 < x <Mvt2 +12. The results of Schwid [6] will be usede These results are stated for functions w,(z) and w2(z) which satisfy the equation 44 2 rr + (2v+1-2*)w=0 (6.16) The parabolic cylinder functions satisfy Weber's equation (2.30), that is, 2 2 see te +3- y= 0 (6.17) which is transformed into (6.16) by the relation (6.18) nih Consequently, the perabolic cylinder functions can be expressed as a linear combination of w, and w,- The relations (39) in [6] give the results h aati D_, (ix) = Beary {regs w(K) - arg +1) we} (6.19) vo! D,., (ix) = Bey {r rH) w (Hh) + arg +2) w(Z) (6.20) Introducing these expressions into (2.32) gives N D(x) = % Tr \r r (4) cos = W, (35) + arg +1) sin > tv, (Z)} (6621) Before Schwid's formulas for w, and w, are substituted into (6.19,21), the gamma functions which are present will be replaced by their asymptotic representations. The following formula can be deduced from [7] Chapter XII, page 263, example 44: "(2 +a) =fan e228" 3 fa + Oo(z' )| » jarg al <1 (6622) 45 Making appropriate choices of z and a and introducing the resulting expressions into (619,21) gives D._, (ix) = 5B? Sirk {ta w, (5 - i {a) fav “Go (6-23) D (x) - ot oF wa {ta cos = W, (5) + fn) ¥2v sin = xe) (6024) The interval under consideration is Viv +2-%2sx<Vavt+2+V2 (6.25) and because of (6.18), formulas for w, (2), w, (2) are required for Yav+1-lsez<Vavelt+l (6226) Schwid's results in Theorem V are stated for bounded E, but Theorem 6 in Langer [2] contains these results as a special case. This latter theorem states that they are valid for general values of EF. However, when the error term is large in comparison with the other terms, these formulas are of little use heree Schwid states that in $4 w, (z) - {2 ef (F) Mave { 08 Zs(E) +c Jy (E) + Ov | (6.27) in ‘ w, (2) - (22 e & (Sa rf: Js(8) +O Jy (8) + O(v' (6.28) where Es ss 4 (sin 20 - 26), . anh et 4 8 (6.29) 9 = cos) (6.30) ¥2v + ob. 46 and the Ook are bounded functions of v alone which are listed in (6) Table Le The numbers z occurring in (6-26) may be obtained by putting z2=V2v+1+8, -lsBP<sl (6.31) From (6.31) and (630), it follows that ai B -28 3 a @ = 1 )= y" {1 + o(v®) 6632 coo” (1 +)» I" + os] iad and from (6-32) and (6-29) that ie 3 ' g = Hav + 1)* (28)? [2 + o(v8)] (6433) ($7 -/# (2v +1)* fi + 0(98)] (6.34) Asymptotic formulas for the Bessel functions [7] are i(EFE -$) -i(EFE -§) J, (&) = 1+0(é')| + -& 1+0(f' ) > (6.35) of) BE Poe + Fe Bee] se > and J, (&) = 0(€*3) as E+ 0 (6.36) wey Remarkse It follows easily from (6.33) that & is a bounded function of v in part of the interval -1 < B < 1 and unbounded in the remainder. When B is negative, — is real by (6.33). Then, for unbounded £, Jy (8) = o(é2) by (6.35). At worst, when B = -l, J,4(6) = o(v 8) and the error term in (6.27,28) remains small in comparison with Jy (6) When B is positive, — is pure imaginary by (6.33). Then, when & is unbounded, the function e +5 which occurs in (6.35) is largee That is, the error in (627,28) is again small in comparison 47 with J, fb) Consequently, equations (6.27,28) give useful results in the interval of (6.26,31). Equations (6.27,28) may be put into the form w, (2) -/2 eit yt nH S, (zyv) T, (zyv) (6.37) w (2) - (28 ott Gte% 5. (2,v) 7, (2,0) (6-38) where S,(z,v) = ( a ta 1 » S2(z,v) = BF v (6.39) T (zv) = £2 et fo? gu(e) + G2 a (E) + 06") (6.40) ’ W “¢ 12 3 ty(ayv) =e AE IE aCe) +O 3, () + ov} (6442) in which = ¥av+1 Ck . [6] Table 4, is a bounded function of v alone. From (633,34) it follows that S, and S, are bounded and from (6.35,36) that T, and T, are bounded. Consequently, bounds for w, (z) and w,(z) are [w,(z)| < ave fen*8], Iwy(z)| < av? [e748], Vav*+l-1le2zsV¥Yav+le*l (6442) From (6633), & is real when -1 < B < 0 so that |e 6] =1. ‘Then, from (6.42) and (6.31) (w,(z)] < avé, |wi(2)| < av’, Yaved-1s2sYavel (6243) Introducing these results into (6.23,24) and using (6.18) gives (6.6,7). 48 When 0< B <1, it follows from (6.32,33) that 6 and & are pure jmaginarye The transformation (6.18) is inserted into (6.30) in which the relation © = da,» where a. is real and positive, is used. The result is =] x a. * Cosh = — (6-44) * Viv +2 Then (6-29), becomes r= i(2v +1) (Sink 2a, - 2a.) = in, (6.45) vhere n= 2¥+-+4 (sinn 20, - 20.) (646) x 4 x x and (661) follows from (6+44,46)+ Equations (6.42) become, in terms of NN. » 1. 7) oe n \w, rot < Av*e = [v,(=)| < Avie x five2 < x < VAve2 +92 (6-47) 2 2 which, when introduced into (6-23), give (6-5). A satisfactory bound for D(x) cannot be obtained from (6427,28) in the interval of (6-47) so that the required bound will now be derived from Whittaker's integral [7] which is x2 ¢(o") . D(x) == Ped ss | ° exp(-xt - ty(-t)” "at, ree) farg(-t)] <1 (648) 2 The function f(t) = -xt - $- (v + 4) 1og(-t) has saddlepoints given by £1(t) = - S(t? + xt +t 2) = 0 which are t = - 3 + SV x*=(4v+2)- The saddlepoint of interest is t = - 4+ 51x*-(4ve2). Put 49 so that t, is non-negativee The integration in (6.48) is carried out f (6.50) 2in t e , “t= te (en <9<n), as follows: ae ih . ial oO foe) be in In these integrals substitute -t = se and -t = “a, respectively. Then t foe) 2 ue | os ~ovet)an f exp(-xs - = s ds fe 9) oo _ er) 2. and / ste = * (v#l)in I exp(-xs - —) s ‘as te t ° _ re @) gh. niga so that i i 2 = -2i sin(v + 1)q0 f exp(-xs - > )8 ds fee) t o®” in ty which gives the result fo fat The second integral on the right of (6.50) is +? < ol me) (6251) in t_e* n @ [ ° = - f exp[t(-t,6*°)] 1%, e? ide = - o TT The real part of the function £(-t,6°°) has a maximum at the saddlepoint, that is, when © = 0. Then this integrel gives the result 50 ein 42 te 2 < 2n t_” exp(xt_ - +2) (6.52) | f | fo) fe) 2 ° Since (6-52) is larger than (6.51), the bound for D(x) of (6.48) is +* |p (x)[ < AM + 1)t2% exp(- + xt, - 2), Vv FB sx (6.53) where the constant A is less than 3 for large v- From (6.49) - 20 22+ be =(iv*2) \nich vith t, is introduced into (6453) The result may be put in the form v (y+ + : + ¥x*=(Avt2 | 2vtl _ x x* D A Pv) -= w [Py tI] s (wept RE) PT 2 ayaa! Av*? and finally [D, (x)| < ap eet) rs le ay 1 ise [= + 1x? meal? Vinee < x (6.54) where 7, is given by (6.1). Replacing the gamma function by its asymptotic form (6.22) and omitting the second term in the exponent gives (604) ° 6.3 Asymptotic Formulas for D,(x), D_, (ix) in the Interval ¥4v42 +V2 < x. The results of Schwid (6) Theorem I may be used, since asymptotic forms for w,(z), w, (2) are required in the interval fave+lti<z (6.55) because of (6618). In this interval — is unbounded as can be seen from (6.33). These results are 51 1 ezin et w,(2)~ = eI? fi) e%% - 2fsin 2 om (6.56) vtl ~ Meh 1 e2 4E ~it Wo(z)~ : [1] e + cos v (6.57) ‘ aby - 1) | vavei im I where the coefficients Be from [e] Table 2 have been used, since by (6.30,29) arg £ = 5 so that h = o from Schwid's relation (14). However, the coefficients of . in (6.56,57), a are listed in this Table 2, contain the ambiguous symbols (- 1)? and (1)? They must mean erit erin » respectively, as in (6.56,57). This interpre- tation is deduced from (6.27,28) since these equations are valid for unbounded —, as previously mentioned, and are therefore valid in the interval (6.55). When the asymptotic formulas for the Bessel functions (6.35) are inserted in (627,28) these equations are asymptotically equivalent to (656,57) only if the above meaning is attached to the ambiguous symbolse Using the relation z = e from (6-18) and the definition of 7, from equations (6.46) and (6.1), equations (6.56,57) become 1 2s “1 n Ww, Q~s za Dt e"fijJe *- 2[sin v | e * (6-58) 4vt2 Ww, (m~ oF “ (2) e — [cos v | e* (6.59) a y% | Yaver Yavel . These functions are substituted into (6.23) giving the equation 52 Vv 2 ¥ <7), D,., (x~ — Fs fet") - be * - 2v2 lla om - 1) = 24( [eos a - sein wa) o* and since the first term in the braces is asymptotically negligible equation (6.3) is obtained. Now put (6.58,59) into equation (6.24) from which it follows that vy _y = 2 6 2 Sin vt vn, "x D, (x) 5 jez" ([1) cos 2-4 (2) sin >) e + , 12( 792 -1)* ° ° n + 2(sin - [cos st - [sin “| cos >) s” and since D(x) +0Oas x > 0, the coefficient of e * must vanish, not only asymptotically but exactly. This leaves the result (662). 6-4 Asymptotic Formulas and Bounds for D(x), Ben (ix) in the Interval -Viv+2 +¥2 xs Vavt2 -V2. The results of Schwid's Theorem 3 hold in the right half plane and within the circle |x] = ¥4v + 2e Watson [9] derived corresponding results which hold within the circle |x| = 2%v, and consequently yield a single pair of formulas which cover the interval under consideration. Watson's formula for D(x) is ¥ D(x) = (wen) or = {cos p +0 fares? (6.60) Wav2(1- x)" Vv (4v-x*) 53 where p is given by (6.10). As x approaches + 2W the error becomes large, and it is necessary to check the order term in (6.60) at the end points of the interval being considered.- At these points the vt+l- ¥2v+1_ G(Vaval - 1)"% valid throughout the interval. order term is 0 ( = 0O(v4), and therefore (6.60) is From Watson's result ((9] p- 140) a bound for D., (ix) is easily obtained. It should be noted in this reference that when -2%v < x < 2¥¥V, y = if, so that Watson's asymptotic formula for D., (4x) yields & vt [2] sa, -Viv+2+¥2<x<fly+2-%¥2 (6.61) 4 a- > Av The error term in Watson's formula for Di. (4x) is identical with that in (6.60), and consequently, the bound (6.61) is also valid throughout the interval under consideration. Also, only a bound for D\_ (ix) is needed in this investigation so that the asymptotic formula is not statede The asymptotic formula (6.60) shows that D(x) oscillates in the interval under consideration. The coefficient of the braces of (6.60) takes its greatest value at the endpoints of the interval. This value is bd u 3 1 t Coa) of vF_ 8 98 oF Fah + 058) iv? (ava -1)* which, essentially, is used in the following 54 Definition: ' nje vie + | ~ (6.62) M) will be called the modulus of D(x) when - Vive2 +72 < x < Vavt2 -V2. Using M,, equation (6.60) gives (6.8). Finally, (1 - i is smallest when x is an endpoint of the interval in (6.62), taking the v value ( Introducing this value into (6.8) gives (6.11) and into (6-61) gives (6.12). 6-5 Asymptotic Formulas and Bounds for D(x) in the Interval x < -Vav +2 +V2. These relations may be obtained by expressing D(x) in terms of D (-x) and Dy, (ix), in which y4v + 2 - 12 < =x, and using the for- mulas already developed in the interval Viv +2 -Y¥2< -x-e For this purpose, equation (2.33) is put into the form bal D(x) = ie D (+x) {2 sin vnf"(v +1) e2 " D., (nix) (6-63) Then, equation (6.13) follows from (6.63) into which (6.6,7) are inserted. Similarly, (6.14) follows from (6.63) and (64,5). Finally, equation (6.15) follows from (6.63) and (62,3). SECTION 7 SOLUTION OF THE INTEGRAL EQUATION 7.1 Notation and Remarks. Equation (5.18) may be put into the form ind w(t) = D (V2p %) i * i K(p,t,8)G(s)w(s)ds (7.1) where (1) K(ustys) =D (V2p t)D,_(4¥2u s)-D(V2u s)D. (a¥2u t) (762) 2 (44) ven = ~ ; from (2.38) is bounded away from the positive integers (444) from (5216,17) it follows that co G(s sas “mt! | ds <K SK (723) [og las s x,5 eo qa ty TH (Gs)las ck SK yt! (764) “es 8 h los sxeke™', (slsate (7-5) Remarks 3 (4) The arbitrary integer m designates the number of approximat- ing functions computed for equation (3.9) which may be written Plx,i') =x + Bp! + coos + oe” + O(p ”'). However, when the number of functions used is 0 or 1, it follows from (3.34) and 55 56 (5016,17) that in (7.3,4,5) the right members may be replaced by ca (44) For brevity, the asymptotic formulas and bounds for the parabolic cylinder functions will be expressed in terms of x and ye Therefore, the following notation will be freely used: x=¥2pt, y=V¥2us (76) in which x is not the x occurring in the transformation f(x,{)'). (iii) With this notation, the transition points t = + a of equation (515) correspond to the transition points x = +Thv +2 of Weber's equation, (2430). (iv) The asymptotic behavior of w(t) will be described in the three intervals t < -a- i » cat de < <tsa- - » and a+ == 7 < ty but the asymptotic behavior in the transition region will not be obtained. (v) In the notation of (7.6), substitute (233,34) into (7.2). The result is K( wy —,2) = [p, (=x)D,_(ay) - Dy(-y)By_(- ix)} (igh OE ae aor od) Von Yu —— (77) 7e2 The Solution in the Interval tes or Y4v +2+V2<x (7.8) Since (662) indicates D(x) never vanishes in the interval (7.8), the integral equation (7.1) may be treated in the form 57 int D (y2u s) wt) yy a (* DENN) oe ajGtsy <n ty 109) D (f2u t) Yan Yt Di (¥2u t) Dy (2p s) For brevity, the following definitions are made: via) « =e 7.10) Dy (¥2u t) D(¥2u s) K( gty )=—— nee Dp Rp +t) K(p,t,s) (7.11) Using these definitions, equation (7.9) becomes int W(t) =1 + —. K(p,t,s)G(s)W(s )ds (7.12) mM In order to carry out the usual iteration process, the functions W(t) are defined by the relations in® foe) : ta*" 2 - W(t) =l1+ i I K(p,t,s) G(s )W,_, (s)ds, W(t)= O, kk @1,2,ce0ce (7213) From these relations it follows that W,(t)= 1 (7.14) and that ine = Wey, (t) = W(t) = is [ K(uytys)G(s)[W,(s) = ¥,_,(s)] ds (7-25) From (7.11,2,6) it follows that D_, (ix) 2) = Dy) DY (ay) = Dy(y)? K(p, —&, (7.16) "au V2 D (x Vv 58 and from (6.2,3) that = vt ‘ -27 +2) Ks Be |S ag ee ] (Fa2 From (661) it is easily seen that m,, is an increasing function of x so that the above bound is simply increased by replacing Ty by 7,° For t < s implies x < y, and therefore Tas ys The result is Ku, &, | < Av (7.17) The quantity in parentheses in the denominator of (7.17) is smallest when y = Yavt2 + V2. Introducing this value gives the result er: y. Av* er = t-)| — (7418) : By om y which becomes, using (7.6) x A K(ust,s)] < Ae < (7.19) JEvstse)| us” ys since », v are of the same order by (2.38). The iteration in (7.15) may now be performed. The steps are 3 |wo(t) - w(t) < AK (720) which follows from (7-14,19,3). Now assume yr [w,. (t) - Wy, _, (t)| < (AK ri (721) and then (7.15) gives the result 3 [me44(t) ~ We(t)] < (aK (7622) 59 In virtue of (7+20,21,22), equation (7.22) is established by induction for all ke Hence, the sequence W(t) converges uniformly in the 3 interval of (7.8) provided AK uF < 1 which is certainly true when Ho Sh is sufficiently large. Then W(t) = W(t) (7.23) — is the solution of (7-12), since the uniform convergence permits the appropriate limits to be taken on both sides of (7.13). It also follows from (7.22,23) that w(t) = dm w(t) = w(t) + > [Was (t) - W(t) n?o n but W(t) = 0 by (7.13) so that 1l- ai al [2.2] [w(t)| < 2 Jw, (t) = w,(t)] Introducing this bound for W(t) into (7.12) gives the desired result 3 © aie wit) = af x aaat fo (S22) as < arm (7.25) t From this equation and (7.10) it follows that —w(t) 1+ ont), > <t (7.26) D, (12 t) =r 7.3 The Solution in the First Transition Interval. This interval will be treated in the two parts (a) agtsat#, (b) a- 7st <a 60 Part (a): a<st<at + or Y4av+2<x< Vv +2412 (7227) Let the bound for D(x), given by (64), be denoted by By (x). Then equation (7.1) may be put into the form 1 w(t). (vay t) + rd a ree 2 — x(u,t,8)6()—2—as + By(¥2n +) BY(W2y t) Qn “+t By(¥2p t) BY(12p s) . D, (V2ya" +72) wa + i) -1 (728) BY (V2u t) D (Ya + ¥2) The existence of a continuous solution w(t) is assured by the classical theory of the Volterra integral equation, since all the functions involved are continuous, and the interval is finite. From (7.2) and (64455) [D,_, (4x)] Bi(y) K(p, —< ) By x Bia) Se rt Fe - By(y) (D,., (4y)| * By(y)* ‘ “27 +2, ' <ve(A,+aApe % *) < av (7.29) where the last inequality is obtained from the same argument as the one leading to equation (7017). It follows from (6.4) that D, (¥2ua + B, (Tay +2 +¥2) Vv < so that with (6.4) and (7+29,5,30,26), equation (7.28) gives the result w(t) <1+A wie K poy) (a +z - a) + aut BY(¥2u t) BY (¥2p t) e 61 £ Since p, v are of the same order, A,veyr' K = A ys Ky and this quantity is less than one for Ke sufficiently largee The last equa- tion then gives the bound -m+ 1 + Asp mt u | w(t) | < a \ (7631) B(¥2u t) 1- aK it Introducing (7.31) and the foregoing bounds into (728) gives the result < Ab K + a, pecan’, w(t) . D (Ven t) By(¥2n t) BY(¥2u t) actsatye (7232) Part (b): acs:<ct<a or fly +2-V2<x<V4v +2 (7.33) hr Using the M, of (6.9), equation (7.1) may be put into the form D. (¥2p t) in® a (s) we =— 7 + 42 f[ K(p,t,s)G(s) — ds + v v V2p t v By(W2ua") | via) 2a | + - (7234) M, B(V2pa®) By (V2ua* ) a t) a From (67,9) 77 < Av (7235) Vv From (7.2) and (6.6,7) | K(u,t,s)| < Av @ (7236) V be Zz From (664,1) B (Yay +2) = Ave Fe 2 By (¥2pa" ) BL (VAv + 2) 2 so that 7 = 7 <Av (7-37) Vv Vv 62 Introducing these bounds, together with (725,32), into (7.34) gives Mats ave A, ve iPr, euP ll -(a- a) + A; ye ym s ol Since y,v are of the same order, A, v 27 2 vu K =A u*K, which is less than one for KM, large enough, so that ’ -m+ Ape ays ‘ sup | ¥{t2| < ; ” v az S Ap (7238) 1-AKypF 2u Using this inequality, (7.34) gives the result D (v2u t) ' 23 ‘ £ w(t) Vv ~- - -mta -m+ _———e < AvP K, pet + a vy teku” '; (7.39) v v -s as ‘i= t<a This completes the treatment of the first transition interval. 7e4 The Solution in the Interval ratectsa-% or -Y4v +2+V2<x<V4v+2-V2 (7.40) It follows from the differential equation (5.15) that w(t) is an oscillating function in this interval. The zeros of w(t) cannot be expected to coincide with those of Dy (124 t), (668), and consequently (t) the quotient, —— 4s unbounded in this interval. For this D (V¥2u t) reason, w(t) and the error will be compared with the "modulus" M . Equation (7.1) becomes 63 p J ~ w(t) Dy (12h t) iet"3 on th w(s) hi K(p,t,s)G(s) “a8 + v v V2 t v wa = = Dy (v2pa* = {2) v Vv From (611), | Pal Fe t) M v < A, and from (6.11,12), [K(u,t,s)/ < Av * Inserting these bounds together with (7+5,39) into (7.41) gives the relation [82 <a, +a, v4 pex, sup Se (2a) + A, vate =i wid me and because ,v are of the same order, A, v* HFK = A, ut Ki which is less than one for Me large enoughe This gives the result an + A, + As p iil. sup | #2] < +z <A (7242) M, 1-A K Ge 2X, # : and again using (7.41) gives D, (V2u +) wail emt £ aka _ « < ax p™tt + A, m+ 8 (7.43) v v which may be written D. (¥2p t) 5 w(t) _ _v -mt § . i =n ike M M + B ), a "Es tsa Th (744) v v 64 725 The Solution in the Second Transition Interval. This interval will also be treated in the two parts (a) -asts-ats% (b) -a-Festsre Part (a): masts-aty or “Van +2<x<-Whv+2+2 (745) Equation (7.1) is put into the form lie K(p,t,s)G(s) w(s) ds + M M M v v t v » ¢- aud Dy(¥2p t) . sei"? [ a W2u w(-a + +.) ; D,(- 2ua* +12) + m, v7 (746) From (6.9,13) | Det" t) <A a (747) M, K(p, ee age 1. | (u ‘oP a < Av (748) These bounds, and (75,43), with (7-46) give the relation ' ‘ ' 5 w(t) 24 mG w(t) eS -n+ 6 ey <a, v tA, rm K, sup | ml Ls * iE (-a)\ + Ayu ‘ r - Since u,v are of the same order, A, vou! K = A, a K which is less than one for Hy sufficiently largee Therefore, this last equation gives the result < — © A p* (749) sup | w(t) M, 65 Inserting this last result into (7.46) gives D (¥2p t) 4 a 5 5 Vv Vv “a<st<scat ra (7.50) It is necessary to distinguish between integral and non-integral v in the remaining discussion of the integral equatione The behavior of D(x) changes radically when x < -Vhy + 2 according as one or the other subsists, as is easily seen from equation (6.63). Consequently, the restriction that v be bounded away from the positive integers will be used in the subsequent analysis. Part (b): -a - HS t<ea or -VY4v+2-V¥2<x<-fav +2 (7-51) Let the bound for D)(x), (614), be denoted by C(x). Since sin vn is bounded away from zero, it then follows from (6.14) that n “!, = (7.52) nile rj [Dx] scx) Zac?y Making use of the quantity C(x), equation (7.1) may be put into the form D. (12 in¢pea © (12u wt), Pv ee HS) k(ystya)(s) —MED— ag Cyn t) CW2pt) Wn ~t ci(f2p t) C(¥2p s) pe ope. (7.53) M M .< Cy (v2p t) ) Vv 66 From (7+7), (64,5) and (7.52) C ee om K(p, s v® [A,exp(-2n,, + 2m) + A, a Tee It follows from t < s that x < y, -y £ “x so that ty a There= fore, replacing Nay by a merely increases the bound. Consequently, C y(¥) L G(x) Kes re Zi) <a! (7254) 2u 2p It follows from (69) and (7-52) that M M ¥ < Vv Cyan t) C,(-V2pa”) since -J2ua* corresponds to -/4v+2 and is * O at this point by (6.1). Substituting these bounds and (7+5,50) into (7.53) gives the result ~2 omet wit <1+ave mr tk 8 up|——witd | fa - (-a - &)]+ avs ome c(¥2u t) cy(i2p t) . Los vi and since A,v° p Ki = An Ky <1 for 4, sufficiently large -m+z (t) 1 + Aap sup | wit | Cas sO (7-56) C(V2p t) 1- Aue K Putting (7-56) into (7.53) gives the desired inequality that D. (¥2p +t) 7 £ 7 yi), cayeK y™ eases cay ’ Cy(W2p t) oy (¥2p t) =i = Hs t<-a (7.57) This completes the treatment of the second transition interval. 67 7e6 The Solution in the Interval tgra- ze or x<-YV4v +t - ¥2 (7.58) 1 From (6.15), bounds for D(x) and D, (x) may be obtained. Since sin vm is bounded from zero vevyMn ,u“8 = lo(x)[ <¢ 2e 2 (7.59) ( sh ag Av +2 and a air | ( Fi A(T 2 1) 1 D(x) = vy n vai “1. = e 2 y2 [72 sin ve (2) 0 *[- [#2 ta] o = 2 z A, (745 1) “Ns ab e = - ut 2 ~2n_. aE so that the bound for is Dy (x) , = 2 "x ee 1h Dr tert < Aqua oY ane) e 2 y2 since A, eax <1 for He sufficiently largee This assertion is valid because .™ is greatest when x = -Vin + 2 - 42, and for this point Nx is an unbounded function of v in virtue of (6-45) and the remark following equation (6.36). Since D, (x) never vanishes in the interval (7.58),as can be seen from its asymptotic formula (6.15), equation (7.1) is put into the form 68 and 2° yd (12 8) w(t) =l+ ie (d i a: nae Seba K(p,t,s)G(s) __w(s)__ ds + D,(v2u +t) V2p Dy (2p +t) Dy (V2p s) 1 . C(- 2pa* = ¥2) w(=a - tf ; Dy (-¥ 2ua* - 42) ia) Dyan t) | C(-V2pa? -¥2) 6 (-V2ya? - 12) From (FT) and (662,3) (a3 7 «fa, exp(=7_, + ty? + A, exp(-7_y 7.) (7262) and this equation combines with (7-59,60) to give Dy (y) Dulas) Kus er roy rr [4 mae, * 2n_y) . A,] From t < s, it follows that x < y, -y < =x so that Ty i a Hence, the right member of the above relation is simply increased if Nay is replaced by "x° Consequently, D (y) = ¥ Bit Mm 2) s As trae) 2p V2p _y* (45 Zvea 7 2) in which the last quantity on the right is obtained from the same argument as the one leading to equation (7.17). Therefore, Dy(¥2p s) Dy (v2u t K(p,t,s) (7-64) ~ (8) 69 C, (-V¥2pa* - 12) A bound for is needed which, in terms of x, is D (¥2p t) Cc (-Y4v+2 - 2) Ci(- Yav+2 - ¥2) Zz v ai (-V4v+2 - va] * This follows from (752,60) provided (760) increases as (=x) decreasese But this is easily seen when the notation of (6.46) is used, since ‘ x 7 x 2v+1 (4 * -1) e ex [- 3 (Sinh 2a_,- 2a_,) + L + i log Sinh “| and d 2v+l 1 . at=2) - i (Sinh 2a, 7 2a_,) oe log Sinh “| 7 1 Cosh a. da_. = {-(2v+l) Sinh’ iad 4 Sinh a. a(-x) da 1 1 - os "5 (x? ~(4v+ 2)] + ie ee ry i(=x) The braces have the value -(1 +2129 aq) +3 Weven +) “ (1 + 2V¥2v m1)? da when =x = VA4v+2 +72, and O < Itax) o1Wayss so that the above derivative is negative at -x = Viv+2 +V2 and decreases as x decreases, remaining “x negative. Hence, ( -l1) e increases when -x decreases as : Av+2 required. 70 Introducing (7+64,4,65,57) into (7.61) gives the relation +1 w(t) =e Zz amt D (W2u t) Dy(W2p t)t me and since A,y* K.* 1 for p, sufficiently large 3 (4) 1+ Awa™ sup | WAti—— | < — < k (7.66) D,(V2p t) 1-4, Kt Inserting this bound into (7.61) gives eS -a=- 3 Vp enti. -mt & —eed— a] 5 8 |S) Jas +a, vPymMFs yom (7.67) Dy (¥2u +) t and finally wt) 1+ o(umt 8), t<-a- * (7.68) Dy (Wu +t) i. SECTION 8 SUMMARY All the results and the conditions under which they are valid will be summarized here. In this investigation uniform asymptotic formulas have been obtained for two solutions of the differential equation a*y 2 2 2 -! ae rte |e mx + f(x,0 )| y = 0 (3-1) in which p, a large positive parameter, is restricted as in (2.38) below, c is an arbitrary positive constant, and f(x,p') is subjected to the following Hypotheses: (i) £(x,') is an analytic function of x and ji’, real and regular for all real x and o < He < py, where Ke is sufficiently large. This function possesses the expansion = “n f(x,i') = 2 £,(x) p (3-2) n=1 where the fn are analytic functions of x, real and regular for all real Xe (41) When x is real andx+ +00, f(x,,') may be put into the form f(xf') sax +B x+y, + Ox”), O<n<1l (3.3) 7. 72 oe) e) oe) where a = pa ap”, B= Z. B i 2. YH (3-4) Pon 7 oni ® ene ® the series for Gi B, and %y being convergente Asymptotic formulas for two solutions of (3-1) which may be deduced from the analysis of Sections 5 and 7 are y, (x) 8" (x)® = >, [Van A(x)] (a + o(u™*)] » e+5<ex (8.1) z +35 y, (x)6'(x)” = > [van A(x)| + Mow ™ 8), -c+6<x<0-6 (82) in which M = 3 ct ta iy-33 y, (x6 (x)* = v, [iE Axi} [2 + ou 4], xg- 0-8 (8.4) and y, (x)p (x)* = D, [Va A(x)| [1+0(u"™*8)] » e+6<x (8.5) U § Ya(x)p' (x)* = D, [Wa B(x)| + Mou), -oc+6<x<ec-8 (8.6) in which M) is given by (8.3) Ya(x)B'(x)* = D,[-Vou A(x] [2+0(u™*#)], x<- 0-6 (8.7) where 6 = = * O(ji') (8.8) vepo--3 (2.38) in which the values of » must be such that v is bounded away from the positive integers. 73 “m=! ) (oe) a® = ¢* + > aye c* +a t' teceet anu + O(n (3-10) n=1 in which the a,, n=l,++., m are given by (3-24). ie.9) B(x) = B(x,f') = x + 26, (x)H™ = xt iter Aue ou ') (349) n= in which the o n=1,.e+e., m are given by (325,26,27) and termwise differentiation is valid. Remarks The arbitrary integer m designates the degree of the approx- imation made to the comparison equation (3.6) by the functions p. and the numbers an? In the case of the Oth and lst approximations, equa- tions (8+1,2,4,5,6,7) may be used with m_= 2.in the order symbol. As x > +00, O(x) and £'(x) have the forms A(x) = (l-a * - %,* O(x”) (4040) B 2(1 - a.) B(x) = (1 - a + o(x 27) (4042) 1. 26 Be he 5 66 Te 8. 9 BIBLIOGRAPHY Re Ee Langer, The Asymptotic Solutions of Ordinary Linear Differential Equations of the Second Order, with Special Reference to the Stokes Phenomenon, Bull. Amer. Mathe Soce 40 (1934) ppe 545-582- Re Ee Langer, On the Asymptotic Solutions of Ordinary Differential Equations, with an Application to Bessel Functions of Large Order, Trans. Amere Mathe Soce 34 (1932) ppe 447-480. Re Es Langer, The Asymptotic Solutions of Certain Linear Ordinary Differential Equations of the Second Order, Trans. Amer. Maths Soc. 36 (1934) ppe 90-106. Re Ee Langer, The Asymptotic Solutions of Ordinary Linear Differ- ential Equations of the Second Order, with Special Reference to a T. Me Cherry, Uniform Asymptotic Formulae for Functions with Transition Points, Trans. Amer. Math. Soce 68 (1950) ppe 224-2576 Ne Schwid, The Asymptotic Forms of the Hermite and Weber Functions, Trans. Amere Mathes Soce 37 (1935) ppe 339-362. Whittaker and Watson, A Course of Modern Analysis, Chapters XII, XVI, XVII, American Edition, The MacMillan Co. (1948). Magnus and Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics, American Edition, Chelsea Publishing Co. (1949). Ge Ne Watson, The Harmonic Functions Associated with the Parabolic Cylinder, Proce London Math. Soc. 17, series 2 (1918-19) pp. 116-148. 7h