Rigorous asymptotics for the Lamé, Mathieu and spheroidal wave equations with a large parameter

Transcription

1 This thesis has been subitted in fulfilent of the requireents for a postgraduate degree (e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following ters and conditions of use: This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated. A copy can be downloaded for personal non-coercial research or study, without prior perission or charge. This thesis cannot be reproduced or quoted extensively fro without first obtaining perission in writing fro the author. The content ust not be changed in any way or sold coercially in any forat or ediu without the foral perission of the author. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis ust be given.

2 Rigorous asyptotics for the Laé, Mathieu and spheroidal wave equations with a large paraeter Karen Ogilvie Supervisor: Dr. Adri B. Olde Daalhuis Doctor of Philosophy The University of Edinburgh 06

3 i Declaration I declare that this thesis was coposed by yself and that the work contained therein is y own, except where explicitly stated otherwise. This work has not been subitted for any other degree or professional qualification. Karen Ogilvie Septeber 6, 06

4 ii Publications [] K. Ogilvie and A. B. Olde Daalhuis, Rigorous asyptotics for the Laé and Mathieu functions and their respective eigenvalues with a large paraeter, SIGMA Syetry Integrability Geo. Methods Appl., vol., pp. Paper 095,, 05.

5 iii Lay Suary Real world phenoena are frequently described in science by di erential equations. Since their introduction by Newton and Leibniz in the latter half of the 7th century, the theory of such equations has been extensively studied by generations of atheaticians and physicists. Many di erential equations have a free paraeter. When this paraeter takes on special values, tered eigenvalues, thedi erential equation adits special solutions called eigenfunctions. These are the solutions which are typically of interest in physical applications. Often di erential equations cannot be solved explicitly, and thus soe approxiation theory is needed to describe solutions, and in turn the corresponding eigenvalues, if they exist. If the case of interest concerns soe variable or paraeter which tends to soe liit, we do this using asyptotic ethods. This thesis concerns three di erential equations, the so-called Laé, Mathieu and spheroidal wave equations. The case of interest in physical applications is a paraeter in these equations becoes large. We concentrate on approxiating the eigenfunctions and eigenvalues in each of these three cases, using unifor asyptotic ethods.

6 iv Abstract We are interested in rigorous asyptotic results pertaining to three di erent di erential equations which lie in the Heun class (see [] 3). The Heun class contains those ordinary linear second-order di erential equations with four regular singularities. We first investigate the Laé equation d w dz + h ( + )k sn (z,k) w =0, z [ K, K], where 0 <k<, sn(z,k) is a Jacobi elliptic function, and K = Z 0 dz p ( z )( k z ) is the coplete elliptic integral of the first kind. We obtain rigorous unifor asyptotic approxiations coplete with error bounds for the Laé functions Ec z,k and Es + z,k for z [0,K] and N 0, and rigorous approxiations for their respective eigenvalues a and b +, as!. Then we obtain asyptotic expansions for the Laé functions coplete with error bounds, which hold only in a shrinking neighbourhood of the origin as!. We also find corresponding expansions for the eigenvalues coplete with order estiates for the errors. Then finally we give rigorous result for the exponentially sall di erence between the eigenvalues b + and a as!. Second we investigate Mathieu s equation d w dz + h cos z w =0, z [0, ], and obtain analogous results for the Mathieu functions ce (h, z) and se + (h, z) and their corresponding eigenvalues a and b + for N 0 as h!, which are derived fro those of Laé s equation by considering a liiting case. Lastly we investigate the spheroidal wave equation d ( z ) dw + dz dz + ( z ) µ z w =0, z [, ],

7 v and consider separately the cases where > 0 and < 0. In the first case we give siilar results to those previously for the prolate spheroidal wave functions Ps(z, ) and their corresponding eigenvalues n for, n N 0 and n as!, and in the second we discuss the gap in theory which akes it di cult to obtain rigorous results as!, and how one would bridge this gap to obtain these.

8 vi Acknowledgeents For the last any years, I give y greatest thanks to y supervisor Adri. His support, encourageent, and atheatical guidance has led e to this point.

9 Contents Introduction Theory of ordinary di erential equations Heun s di erential equation Introduction to asyptotics Unifor asyptotics for di erential equations with a large paraeter. 9.5 Preliinaries: Parabolic cylinder functions Rigorous asyptotics for the Laé equation with a large paraeter 3. Properties Previous Results Chapter outline Unifor asyptotic approxiations for the Laé functions Approxiations in ters of parabolic cylinder U functions Approxiations in ters of parabolic cylinder D functions A second ter in the approxiation in ters of parabolic cylinder D functions Nuerics Asyptotic expansions for the Laé functions and their eigenvalues Foral unifor asyptotic expansions of the Laé functions and eigenvalues General coe cients B s (t) and A s (t) Polynoial coe cients B s (t) and A s (t) Returning to the z-plane: odd solutions Nuerics Exponentially sall di erence between a and b Rigorous asyptotics for the Mathieu equation with a large paraeter Properties of the Mathieu functions Previous Results Unifor asyptotic approxiations for the Mathieu functions Asyptotic expansions for the Laé functions and their eigenvalues Exponentially sall di erence between a and b Rigorous asyptotics for the spheroidal wave equation for a paraeter being large and positive, and being large and negative Properties of the spheroidal wave functions Previous Results

10 Contents vi 4.3 Chapter outline Unifor asyptotic approxiations for the prolate spheroidal wave functions Approxiations in ters of parabolic cylinder U functions Approxiations in ters of parabolic cylinder D functions Unifor asyptotic expansions of the prolate spheroidal wave functions Foral unifor asyptotic expansions of the prolate spheroidal wave functions and eigenvalues Discussion outline on further work Unifor asyptotic approxiations of the oblate spheroidal wave functions Asyptotic expansions of the oblate spheroidal wave functions and their eigenvalues Conclusion Results obtained in this thesis Potential future work in the field Physical and atheatical applications References

11 Introduction This thesis is a study of second-order linear ordinary di erential equations which lie in the so-called Heun class of di erential equations, which are those ordinary linear second-order di erential equations with four regular singularities. This class appears to be a siple step up fro the ore widely known hypergeoetric class, which are those ordinary linear second-order di erential equations with three regular singularities, yet far fewer results are known for the Heun class. This is due to the analytical coplexity which is introduced. As such, rigorous results for special eigenfunctions and corresponding eigenvalues are lacking, a proble which we ai to resolve for several di erential equations. Rigorous and explicit error bounds for approxiations to solutions of di erential equations developed by Frank W. J. Olver [] provide the analytical tools to tackle these probles. This thesis is structured as follows. First in the introduction we give soe background theory for di erential equations, soe basic asyptotic results, and discuss unifor asyptotic approxiations for solutions of certain di erential equations. Chapter will discuss the Laé di erential equation, and we provide rigorous results for the Laé functions and their corresponding eigenvalues for a paraeter in the di erential equation becoing large. In chapter 3 we will realise analagous results for the Mathieu functions and their corresponding eigenvalues as a liiting case of Laé s equation. In chapter 4, the probles are twofold. First we consider the case where a paraeter becoes large and positive, and provide rigorous results for the prolate spheroidal wave functions and their corresponding eigenvalues, which are siilar to those given in the previous chapters. Secondly we give a discussion on an unfinished piece of work. In

12 .. Theory of ordinary di erential equations the case where a paraeter becoes large and negative, solutions are called oblate spheroidal wave functions. The analysis to provide rigorous and explicit error bounds for approxiations to these functions is yet to be provided, and as such we only provide an outline of how one would tackle these probles. Now we discuss soe introductory theory.. Theory of ordinary di erential equations Many special functions are known to satisfy linear ordinary di erential equations of the for d w dx + f(x)dw + g(x)w =0. (.) dx If f(x) and g(x) are continuous on a finite or infinite interval (a, b) then (.) has infinitely any solutions which are twice continuously di erentiable on (a, b). If the values of w and dw/dx are prescribed at any point then the solution is unique. Linearly independent solutions The di erential equation (.) has two linearly independent solutions w and w in (a, b), which satisfy the Wronskian relation W(w (x),w (x)) = w (x)w 0 (x) w 0 (x)w (x) 6= 0 (.) for all x (a, b). Every solution w of (.) on (a, b) can be expressed in the for w(x) =Aw (x)+bw (x), (.3) where A and B are constants, and such a pair of solutions is referred to as a fundaental pair. Classification of singularities An iportant topic of study is the behaviour of solutions to equations of type (.) in the neighbourhood of singularities. If

13 .. Theory of ordinary di erential equations 3 (i) f and g are both analytic at x = x 0,thenx 0 is called an ordinary point; (ii) x 0 is not an ordinary point but both (x x 0 )f(x) and (x x 0 ) g(x) are both analytic there, then x 0 is called a regular singularity; (iii) x 0 is neither an ordinary point nor a regular singularity, then it is called an irregular singularity. Irregular singularities can be further classed into ranks. Corresponding to each singularity is an exponent pair which corresponds to the behaviour of solutions in a neighbourhood of the singularity. The theory concerning solutions in the neighbourhood of singular points is rich and interesting, but is not relevant to this thesis so we will not discuss the details here. The reader is directed to [] for a coprehensive discussion. Nuerically satisfactory solutions Two solutions of a di erential equation of the for (.) are nuerically satisfactory if in a neighbourhood, or sectorial neighbourhood, of a singularity x 0, one solution is recessive, r(x) say, and one is doinant, d(x) say, eaning that r(x) d(x)! 0 (x! x 0). (.4) Any solution which is linearly independent of the recessive solution at a singularity is referred to as a doinant solution. A solution which is recessive at one singularity is not usually recessive at another, hence if there is ore than one singularity then several standard solutions usually need to be chosen to have nuerically satisfactory representations everywhere. Equations with a real paraeter Consider the di erential equation d w + f(u, x)dw + g(u, x)w =0, (.5) dx dx where u [u 0,u ], x [a, b], and f and g are functions which are continuous in both variables. If x 0 is a fixed point such that the values of w at x 0 are

14 .. Heun s di erential equation 4 prescribed as continuous functions of u, w/@x are continuous in both variables. The hypergeoetric di erential equation In this section we look at the Heun di erential equation, and discuss the coplexities associated with it. It is natural to first introduce a sipler, widely known di erential equation called the hypergeoetric equation, typically written in the for x( x) d w dx +(c dw (a + b + )x) dx abw =0, (.6) where a, b and c are real or coplex paraeters. This equation has regular singularities at 0, and with exponent pairs {0, c}, {0,c a b}, and {a, b} respectively. One way to see the regularity of the singularity at is to ap it to the origin. Any hoogeneous linear second-order di erential equation with at ost three regular singularities can be transfored into the hypergeoetric equation by a change of variables. If one replaces x by x/b and lets b!, and subsequently replaces c by b, we get the liiting for of the hypergeoetric equation called the confluent hypergeoetric equation in the for x d w dx +(b dw x) dx aw =0. (.7) This equation has a regular singularity at the origin with exponent {0, b}, and an irregular singularity at. The regular singularities of the hypergeoetric equation at b and are coalesced into an irregular singularity at. This special case of the hypergeoetric equation is of great interest in physical applications, and special cases of the confluent hypergeoetric equation include the Airy, Bessel, Herite, Laguerre, and parabolic cylinder di erential equations (see [] 3.6).. Heun s di erential equation Since any of the special functions of interest that arise in physical applications are special cases of the hypergeoetric di erential equation, a unified theory of this class of equations was of uch interest. It ight then see a siple step to consider a

15 .. Heun s di erential equation 5 general linear second-order di erential equation with four regular singularities instead of three, with the property that any linear second-order di erential equation with at ost four regular singularities can be transfored into it. For this reason Heun in 889 introduced the following equation known as Heun s di erential equation d w dx + x + x + x a dw dx + x q w =0, (.8) x(x )(x a) where,,,,, a and q are paraeters, and where a 6= 0,. Any linear secondorder di erential equation with at ost four regular singularities can be transfored into the for (.8). This equation has regular singularities at 0,, a,, with exponent pairs {0, }, {0, }, {0, }, and {, } respectively. The su of these eight exponents ust equal two (theory of Fuchsian di erential equations (see [3]) state that if a di erential equation of second-order has finite singularities and a singularity at, then the su of its exponents ust equal ) hence there is a condition on the paraeters that + += + +. (.9) The paraeters,,,, and are regarded as exponent paraeters, a as a singularity paraeter, and q an accessory paraeter which plays the role of an eigenvalue paraeter in soe applications. This class of equations is one step up fro the hypergeoetric class of di erential equations, but the coplexity introduced in studying this equation is significant. We give two iportant exaples of the di culties faced. When one tries to obtain power series expansions for solutions, the recurrence relations which deterine the coe cients are now ore coplicated than for the hypergeoetric case and it is generally ipossible to write down explicit series representations. Another analytic drawback is the general lacking of integral representations for solutions in ters of sipler functions (in general sipler eans that the di erential equations they satisfy are less coplicated), which are available to those in the hypergeoetric class. Just as the hypergeoetric equation has the confluent hypergeoetric equation as a significant special case of interest, the Heun equation has four special cases of

16 .. Heun s di erential equation 6 interest which occur when regular singularities erge to for irregular singularities: the confluent Heun equation, the doubly-confluent Heun equation, the biconfluent Heun equation, and the triconfluent Heun equation. These result by the confluence of a finite singularity with the singularity at, the confluence of two separate pairs of singularities, the confluence of two finite singularities with the singularity at and when the three finite singularities confluence with the singularity at respectively. The special cases of Heun s equation discussed in this thesis are as follows. Laé s equation In (.8) setting = = = eans necessarily fro (.9) that + =, and writing =, = ( + ), q = ah, (.0) 4 (.8) becoes Laé s equation of order d w dx + x + x + dw ah ( + )x + w =0. (.) x a dx 4x(x )(x a) Without loss of generality it is assued that is replaced by. Setting since (.) is unchanged when a = k, x =sn (z,k ), (.) where sn (z,k ) is the Jacobi elliptic sine function with odulus k (0 <k<), (.) becoes d w dz + h ( + )k sn (z,k ) w =0. (.3) We consider in this thesis solutions in the interval [ K, K], and where h, k and are real paraeters such that 0 <k<and, and K = K(k) is Legendre s coplete elliptic integral of the first kind (see [] 9.(ii)). When h assues the special values a or b + for N 0, Laé s equation adits even or odd periodic solutions denoted Ec z,k or Es + z,k respectively. Laé s equation first appeared in a paper by Gabriel Laé in 837 [4]. It appears

17 .. Heun s di erential equation 7 in the ethod of separation of variables applied to the Laplace equation in elliptic coordinates. Laé functions have applications in antenna research, occur when studying bifurcations in chaotic Hailtonian systes, and in the theory of Bose-Einstein condensates, to nae but a few (see [] 9.9). Mathieu s equation Mathieu s equation is a special case of the confluenced Heun equation. It can be recognised as a liiting for of Laé s equation, where k! 0 +. Mathieu s equation in its ost recognisable for is d w dz + h cos z w =0. (.4) In this thesis we consider solutions in the interval [0, ], and take paraeters. When and h to be real assues the special values a or b + for N 0, Mathieu s equation adits even or odd periodic solutions denoted ce (h, z) or se + (h, z) respectively. These functions first arose fro physical applications in 868 in Éile Mathieu s study of vibrations in an elliptic dru [5]. Since they have appeared in probles pertaining to vibrational systes, electrical and theral di usion, electroagnetic wave guides, elliptical cylinders in viscous fluids, and di raction of sound and electroagnetic waves, to nae but a few. In general, they appear when studying solutions of di erential equations that are separable in elliptic cylindrical coordinates. For an insight as to how Mathieu functions appear in physical applications see [6]. The spheroidal wave equation The spheroidal wave equation is another special case of the confluenced Heun equation. It is ost coonly given as d ( z ) dw + dz dz + ( z ) µ z w =0, (.5) and in this thesis we consider solutions in the interval [, ], where, and µ are real paraeters. If assues the special values n for, n N 0 and n, and

18 .3. Introduction to asyptotics 8 µ =, then the corresponding special solutions are split into two classes; if > 0 then solutions are called the prolate spheroidal wave functions, and if < 0 then solutions are called the oblate spheroidal wave functions. These are the non-trivial solutions which are bounded at the singularities of the di erential equation at z = ± and are denoted by Ps n (z, ). This equation coes fro separation of a special partial di erential equation in spheroidal coordinates where is real, but negative or positive depending on whether the spheroids are oblate or prolate (see [] 30.3 & 3.4). It is a generalisation of Mathieu s equation, and appears ost proinently in applications related to signal analysis..3 Introduction to asyptotics In physical applications, situations of interest often include when soe paraeter in the syste becoes very sall or large, and to study these one typically uses asyptotic analysis. In particular in this thesis, we will study the behaviour of a function when soe paraeter in the function becoes large. Sybols To describe the behaviour of a function as its variable tends to we use the following notation: (i) If f(x) g(x) (ii) If f(x) g(x) (iii) If f(x) g(x)! as x!then in this liit, f(x) g(x),! 0 as x!then in this liit, f(x) =o(g(x)), is bounded as x!then in this liit, f(x) =O(g(x)). Two notable cases include f(x) =O() as x!which eans f(x) is bounded in this liit, and f(x) =o() as x!which eans f(x) vanishes in this liit. This notation can also be used when the liit is soe other point. Asyptotic expansions The following results about asyptotic expansions can be found in of [].

19 .4. Unifor asyptotics for di erential equations with a large paraeter 9 Theore.. If P s=0 a sz s converges for z <r, then for each non-negative integer n X a s z s = O(z n ). (.6) s=n However any expansions are often not convergent and are called divergent series. Suppose we have the function f(z), where z is real or coplex, and the foral power series expansion (not necessarily convergent) P s a sz s such that nx f(z) = a s z s + R n (z). (.7) s=0 If R n (z) iso(z n ) as z!in soe unbounded region Z in R or C for each fixed n then we say that P s a sz s is an asyptotic (Poincaré) expansion of f(z) as z! and that f(z) a 0 + a z + a +... (z!in Z). (.8) z This definition can be altered for soe other finite liit point in Z. All convergent series are asyptotic series by definition, but the use of the ter usually refers to divergent series. Divergent series are, in a sense, often ore powerful than convergent series, as typically when divergent series are truncated at an optial point the errors are beyond all orders in the expansion paraeter..4 Unifor asyptotics for di erential equations with a large paraeter Many special functions satisfy an equation of the for d w dz = u f(u, z)+g(u, z) w, (.9) where u is a real or coplex paraeter, and solutions of interest involve u becoing large. In this case, asyptotic solutions which are unifor with respect to u are needed for z in soe interval or region Z in R or C. Here unifor eans that the error bound

20 .4. Unifor asyptotics for di erential equations with a large paraeter 0 corresponding to the asyptotic approxiations holds uniforly in Z. The for of the asyptotic approxiations for solutions depends on the nature of the transition points of the di erential equation. Transition points are zeros and poles of f(u, z), with the zeros of f(u, z) being ore coonly referred to as turning points. This leads to a discussion of three ain cases. We first discuss the ethodology that applies to all three cases. Introducing new variables {,W}, related to {z,w} respectively by W =ż / w, where ż = dz d, (.0) (.9) takes the for d W d = u ż f(u, z)+ (u, ) W, (.) where (u, ) =ż / d g(u, z)+ż d ż /. (.) The transforation is specialised by specifying the relationship between z and. This is done by defining f such that ż f(u, z) = f(u, ), (.3) where z and are analytic functions of one another, and such that d W = u f(u, z)w (.4) d has solutions which are functions of a single variable. In order for this transforation to ake sense, e f(u, z) ust have the sae nuber and order of poles and zeros as f(u, z) in the region considered, and (u, ) should be negligible as u!. This transforation is coonly referred to as the Liouville transforation. One chooses f(u, ) conveniently such that it satisfies the above conditions and so that approxiants are in the ost natural for. These approxiants exhibit siilar behaviour as the solutions you want to approxiate. We now discuss the siplest case, where there are

21 .4. Unifor asyptotics for di erential equations with a large paraeter no transition points. (I) If f(u, z) has no zeros or singularities, specifying that Z ż f(u, z) =, giving = f / (z)dz, (.5) gives approxiations for solutions in ters of exponential functions. In this case, foral solutions of (.9) can be given in the for Z w (z) f /4 (u, z)exp w (z) f /4 (u, z)exp f / (u, z)dz, Z f / (u, z)dz. (.6) These approxiations are widely known as the WKB approxiations, naed so after Wentzel (96), Kraers (96) and Brillouin (96) for their independent contributions to the theory. Their contributions however weren t the for of the approxiations theselves, but rather deterining the arbitrary constants in the forulae which connect the approxiate solutions on either side of a turning point. Even this work was unknowingly predated by Je reys (94) who solved this connection proble, and as such the ter WKB is soeties aended to WKBJ, or soe perutation thereof. Je reys also later noted that he hiself had overlooked an even earlier derivation by Gans. The approxiations theselves date back to Liouville (837) and Green (837) who published the independently, with siilar techniques dating back as far back to the work of Carlini (8). As such, the solutions are also soeties referred to, especially in the work of Olver, as the Liouville-Green approxiations. The proof that these approxiations are asyptotically equivalent to particular solutions of the differential equation can be shown by constructing a so-called Volterra integral equation for the di erence between the solution and approxiation. This ethod is now a part of a larger general theory which dates back to the work of Langer, and is not how these results were first derived. Langer, in a series of papers starting fro 93, first developed a transforation for this siple case in a siilar anner to the above. The singularity of the functions in the approxiations at a turning point, i.e. at a zero of f(u, z), is introduced by the transforations given to solve the proble. Thus he developed a siilar ethod for the case where there is a zero

22 .4. Unifor asyptotics for di erential equations with a large paraeter of f(u, z), and provided unifor asyptotic approxiations for solutions in this case, which we will call case (II). Previously only local approxiations in the neighbourhood of a turning point were available. (II) If f(u, z) contains one zero point, specifying that ż f(u, z) =, giving 3 3/ = Z z z 0 f / (t)dt, (.7) where z 0 is the zero of f(u, z) inthez-plane, gives approxiations for solutions in ters of Airy functions. In this anner, the zero of f(u, z) in the z-plane is apped to = 0 in the - plane. Langer provided the odified version of Liouville s transforation for this case, and gave the leading ter of the asyptotic approxiation together with a Liouville- Neuann series for the corresponding error. This theory was further developed later by Cherry (950) and Olver [], the latter who derived explicit bounds for the error ters. The earliest investigation of a transition point being a siple pole appears to be in Langer s 935 paper. We call this case (III). (III) If f(u, z) has a siple pole, specifying that ż f(u, z) =, giving / = Z z z 0 f / (t)dt, (.8) where z 0 is the siple pole of f(u, z) inthez-plane, gives approxiations for solutions in ters of Bessel functions. In this anner, the siple pole of f(u, z) inthez-plane is apped to the siple pole at = 0inthe -plane. For large values of u, Langer derived asyptotic approxiations for solutions in ters of Bessel functions, which were valid in a shrinking neighbourhood of the transition point, which vanished as u!. Further situations are treated in a siilar anner. To suarise this ethod using the Liouville transforation, the di erential equation is essentially transfored into a for such that approxiants for the transfored equation behave siilarly and in

23 .4. Unifor asyptotics for di erential equations with a large paraeter 3 a sense sipler, again eaning generally that the di erential equation corresponding to the approxiant is sipler. The theory of such transforations and explicit error bounds corresponding to the approxiations can be found in []. It is these developents by Olver in constructing explicit error bounds corresponding to unifor asyptotic approxiations for these types of equations which allow us to copose the work contained in this thesis. Suarising Olver s technique for the error analysis, once the di erential equation is in the for d W d = u f(u, z)+ (u, ) W, (.9) a pair of solutions can be written down as W (u, ) =V (u, )+ (u, ), W (u, ) =V (u, )+ (u, ), (.30) where V and V are solutions of (.4). To obtain rigorous bounds for,substituting the first of (.30) into (.9) gives the di erential equation for, d (u, ) d = u f(u, e z)+ (u, ) (u, )+ (u, )V (u, ). (.3) Then using variation of paraeters, we get the Volterra-type integral equation (u, ) = Z K(,t) (u, t) (u, t)+v (u, t) dt, (.3) where K(,t)=V (u, )V (u, t) V (u, )V (u, ), (.33) and the liits of the integral are chosen to identify the approxiation with the correct solutions. The sae type of integral equation can be written down for (u, ). In the rest of this section we give the general theory for how to bound errors of this type, which can be found in 6 of []. Take the standard for of the integral equation

24 .4. Unifor asyptotics for di erential equations with a large paraeter 4 to be h( ) = Z K(, ) ( )J( )+ 0 ( )h( )+ ( )h 0 ( ) d. (.34) Assuptions are as follows: (i) The path of integration lies along a given path P coprising a finite chain of R arcs in the coplex plane. Either, or both, of the endpoints and be at infinity.,say,ay (ii) The real or coplex functions J( ), ( ), 0 ( ) and ( ) are continuous when (, ) P, save for a finite nuber of discontinuities and infinities. (iii) The real or coplex kernel K(, ) and its first two partial derivatives are continuous functions of both variables when, (, ) P, including the arc junctions. Here, and in what follows, all di erentiations with respect to are perfored along P. (iv) K(, ) = 0. (v) When (, ) P and (, ] P K(, ) apple P 0 ( )Q( apple P ( )Q( K(, apple P ( )Q( ), (.35) where the P j ( ) and Q( ) are continuous real functions, the P j ( ) being positive. (vi) When (, ) P, the following integrals converge ( ) = Z ( )d, 0( ) = Z and the following suprea are finite 0 ( )d, ( ) = Z ( )d, (.36) apple sup (Q( ) J( ) ), apple 0 sup (P 0 ( )Q( )), apple sup (P ( )Q( )), (.37) except that apple need not exist when ( ) = 0.

25 .4. Unifor asyptotics for di erential equations with a large paraeter 5 Theore.. With the foregoing conditions, equation (.34) has a unique solution h( ) which is continuously di erential in (, ) P and satisfies h( )/P 0 ( )! 0, h 0 ( )/P ( )! 0 (! along P). (.38) Furtherore, h( ) P 0 ( ), h0 ( ) P ( ) apple apple ( )exp(apple 0 0( )+apple ( )), (.39) and h 00 ( ) is continuous except at the discontinuities (if any) of ( ). ( )J( ), 0 ( ), and The bounds for h( ) and h 0 ( ) can be sharpened in the following coon case: Theore.3. Assue the conditions above, and also that ( ) = 0 ( ), ( ) =0. Then the solution h( ) satisfies h( ) P 0 ( ), h0 ( ) P ( ) apple apple exp (apple 0 ( )). (.40) apple 0 Olver derived these bounds by constructing uniforly convergent series for the errors, with bounds derived by ajorizing these expansions (see [] 6.0 for details). These bounds are only eaningful if (u, ) is in soe sense sall copared to u f(u, z) as u becoes large. This technique of bounding the errors is prevalent in this thesis. In the next three chapters of this thesis, we consider di erential equations on the real line which have two turning points which coalesce as a paraeter in the equation becoes large. In this case, specifying that ż f(u, z) =, (.4) gives approxiations for solutions in ters of parabolic cylinder functions. In this anner, the coalescing zeros of f(u, z) we have in our equations in the z-plane are apped to = ± in the -plane, and these will coalesce into the origin there in this case. The theory involving this case and explicit error bounds are provided by Olver in [7], which is an iportant paper in relation to this thesis. We suarise the ain result as follows.

26 .4. Unifor asyptotics for di erential equations with a large paraeter 6 Consider the di erential equation d w d = u ( )+ (u,, ) w. (.4) Denote and to be the endpoints of an interval containing, with negative and positive, where both ay depend on or be infinite. Additionally, (z) denotes a conveniently chosen continuous function of the real variable z that is positive, except possibly when z = 0, and satisfies (z) =O(z) (z! ±). (.43) With this, the error-control function is introduced as Z (u,, ) F (u,, ) = ( p d, (.44) u ) the choice of integration constant being iaterial. Theore.4. Assue that for each value of u, the function (u,, ) is continuous in the region [0,A], [0, ),and V 0, (F ) Z 0 (u,,t) ( p dt (.45) ut) converges uniforly with respect to. Then in this region equation (.4) has solutions w (u,, ) and w (u,, ) which are continuous, have continuous first and second partial -derivatives, and are given by w (u,, ) = U u, p u + (u,, ), w (u,, ) =U u, p u + (u,, ), (.46) where (u,, ) M( u, p u (u,, )/@ p un( u, p u ) apple E u, p u exp r u l u V, (F ), (.47)

27 .4. Unifor asyptotics for di erential equations with a large paraeter 7 (u,, ) M( u, p u (u,, )/@ p un( u, p u ) apple E u, p u exp r u l u V 0, (F ), (.48) and l (b) = sup (z)m (b, z)/ z(0,) b (b apple 0). (.49) The functions U and U are parabolic cylinder functions, and the functions E, M and N functions are defined in ters of parabolic cylinder functions, and will be defined explicitly in the following section. At the end of chapter 4, we will discuss briefly another case, where a turning point coalesces with a siple pole when a paraeter becoes large. If f(u, z) has a siple pole and a turning point, if we specify that ż f(u, z) =, (.50) then approxiations for solutions are given in ters of Whittaker functions. In this anner, the siple pole of f(u, z) is apped to the origin of the -plane, and the turning point will be apped to =. In our equation the tuning point will coalesce with the siple pole, thus we will have! 0 as u!. The theory for this is as of yet incoplete, and as such we only provide a short discussion of the proble. For all of these cases one ight consider whether asyptotic expansions can be derived for solutions of the di erential equations, instead of just a one ter approxiations. Indeed for case (I), one could try for solutions of the for W (u, ) e u X s=0 A s ( ) u s, W (u, ) e u X s=0 ( ) s A s( ) u s, (.5) as u!. In case (II), one would need to add a second ter in the solutions involving the derivative of the Airy functions, W (u, ) Ai(u /3 ) X s=0 A s ( ) u s + Ai0 (u /3 ) u 4/3 X s=0 B s ( ), (.5) us

28 .5. Preliinaries: Parabolic cylinder functions 8 W (u, ) Bi(u /3 ) X s=0 A s ( ) u s + Bi0 (u /3 ) u 4/3 since, if this second ter wasn t there, the recurrence relations the coe X s=0 B s ( ), (.53) us cients A s ust satisfy can only hold when (u, ) = 0. The other cases are treated in a siilar anner, which we shall discuss in this thesis. The error analysis for these expansions follows siilarly fro the theory detailed above..5 Preliinaries: Parabolic cylinder functions The parabolic cylinder functions satisfy the di erential equation d w dz = 4 z + b w. (.54) When b<0 this equation has real turning points at z = ± p b, and solutions oscillate in the interval defined by these endpoints. When b>0 there are no turning points and no oscillations. The two standard solutions of (.54) we shall use are U(b, z) and U(b, z), and the case b<0 will be of interest in this thesis. U (,z) U (,z) U (,z) U (,z) U ( 3,z) U ( 3,z) U ( 4,z) U ( 4,z) Figure.: Plots for assorted parabolic cylinder U and U functions.

29 .5. Preliinaries: Parabolic cylinder functions 9 Properties of U and U Here we will suarise the relevant properties of the parabolic cylinder functions, which can be found in ( []) and [7]. These functions satisfy the Wronskian relation W U(b, z), U(b, z) = r b, (.55) take the following values at z =0 U(b, 0) = U(b, 0) = p a+ 4 3 p a b, U0 (b, 0) = p a+ 4 b 3 4 a 4 + b, U 0 (b, 0) = 4 + b, p a b a b, (.56) and satisfy the reflection forulae U(b, z) = sin( b)u(b, z)+ U(b, z) = b + b U(b, z), cos( b) U(b, z)+sin( a)u(b, z). b (.57) One can then see that when b = for N 0 we have U(, z) =( ) U(,z), U(, z) =( )+ U(,z). (.58) This paraeter case is of special iportance and we use the given notation U(,z)=D (z), (.59) and define U(,z)=D (z). (.60) The requireent N 0 is not necessary for this notation. When N 0 the parabolic

30 .5. Preliinaries: Parabolic cylinder functions 0 cylinder function D (z) can be written in ters of Herite polynoials by the relation D (z) =e 4 z He (z) = n/ e 4 z H (z/ p ). (.6) These functions satisfy the orthogonality relation Z D (z)d n (z)dz = n! p, (.6) and the following recurrence relations are satisfied by both U(b, z) and U(b, z) zw(b, z) W (b,z)+(b + )W (a +,z)=0, (.63) W 0 (b, z) zw(b, z)+w (b,z)=0. I D 0 (z) D 0 (z) D (z) D (z) D (z) D (z) D 3 (z) D 3 (z) Figure.: Plots for assorted parabolic cylinder D and D functions. These functions exhibit the following behaviour as z! U (b, z) z b e 4 z, U 0 (b, z) z b e 4 z, U (b, z) q b z b e 4 z, U 0 (b, z) p b z b+ e 4 z. (.64)

31 .5. Preliinaries: Parabolic cylinder functions When b U(b, z) has no real zeros, 3 <b< U(b, z) has no positive real zeros, 3 n <b< n +, n =,,.. U(b, z) has n positive real zeros, b = n, n =,,.. U(b, z) has n real zeros in the interval [ p b, p b]. (.65) Auxiliary functions The following is a suary of definitions given in [7], which are relevant in this part of the thesis. Denote (b) to be the largest real root of the equation U(b, z) =U(b, z). (.66) For b apple 0, E(b, z) = (0apple x apple (b)), E(b, z) = s U(b, z) U(b, z) (x (b)). (.67) When b is fixed, E(b, x) is non-decreasing on the interval [0, ). For b apple 0 and x 0, we define U(b, z) =E (b, z)m(b, z)sin (b, z), U 0 (b, z) =E (b, z)n(b, z)sin!(b, z), U(b, z) =E(b, z)m(b, z) cos (b, z), U 0 (b, z) =E(b, z)n(b, z) cos!(b, z), (.68) where E (b, z) =/E(b, z). Thus for 0 apple x apple (b), q M(b, z) = U (b, z)+u (b, z), q N(b, z) = U 0 (b, z)+u 0 (b, z), (b, z) = arctan{u(b, z)/u(b, z)},!(b, z) = arctan{u 0 (b, z)/u 0 (b, z)}, (.69)

32 .5. Preliinaries: Parabolic cylinder functions whereas for x (b) q M(b, z) = U(b, z)u(b, z), (b, z) = 4, s U 0 (b, z)u (b, z)+u 0 (b, z)u N(b, z) = (b, z), U(b, z)u(b, z) ( ) U 0 (b, z)u(b, z)!(b, z) = arctan U 0. (b, z)u(b, z) (.70) For large x, M(b, z) s 8 4 q ( b) ( b)x, N(b, z) x ( ) /4. (.7) These hold for fixed b and also when b ranges over the copact interval (, 0].

33 Rigorous asyptotics for the Laé equation with a large paraeter This chapter and the next concerns linear ordinary di erential equations with periodic coe cients. These equations ainly arise fro the study of partial di erential equations where new coordinates are introduced and then the equations are separated into several ordinary di erential equations in the new coordinate syste. Solutions of interest in applications have to satisfy certain boundary conditions at special surfaces. For a satisfactory discussion of such equations see [8]. In this chapter we discuss the Laé equation and in particular we present rigorous results for the Laé functions, and their corresponding eigenvalues as a paraeter in its di erential equation becoes large. In preparation for the analysis we will discuss the properties of these functions and give a concise literature review related to the relevant probles.. Properties We will suarise the iportant properties here, for a fuller treatent see ([] 9). Laé s equation is d w dz + h ( + )k sn (z,k ) w =0, (.) and we consider h, k and to be real paraeters such that 0 <k< and, and sn(z,k) is the Jacobian elliptic sine function (see [].). We consider always

34 .. Properties 4 the interval z [ K, K] unless stated otherwise, K = Z 0 dz p ( z )( k z ) (.) is Legendre s coplete elliptic integral of the first kind (see [] 9.(ii)). assues the special values a or b + When h for N 0, Laé s equation adits even or odd periodic solutions denoted Ec z,k or Es + z,k respectively. These functions are either K-periodic or K-antiperiodic, depending on the parity of. For a suary of these properties, including the boundary conditions which the functions satisfy, see Table.. Ec 0 z,0.7 Es z,0.7 Ec z,0.7 Es z,0.7 Ec z,0.7 Es 3 z,0.7 Ec 3 z,0.7 Es 4 z,0.7 Figure.: Plots for assorted Laé functions in [ K, K]. These functions have exactly zeros in the interval ( are ordered such that K, K) and their eigenvalues a 0 <a <a <..., b <b <b 3 <..., (.3) interlace such that a <b +, b <a +, (.4)

35 .. Properties 5 and coalesce such that a = b, when =0,,..,. (.5) Since the Jacobian elliptic function dn(z,k) (see [] ) is even, we can rewrite the noralisations given in ([] 9.3) as Z K K dn(z,k) Ec z,k dz = Z K K dn(z,k) Es + z,k dz =. (.6) To coplete their definitions we have Ec K, k > 0, and des z,k dz z=k < 0. (.7) They satisfy the orthogonality conditions for 6= n, (n N 0 ) and Z K K Z K K Ec z,k Ec n z,k dz =0, (.8) Es + z,k Es n+ z,k dz =0. (.9) We suarise their properties and give boundary conditions in Table.. Table.: properties and boundary conditions for Laé functions Eigenfunctions Eigenvalues Periodicity Parity Boundary conditions at z =0,K Ec z,k a Period K even, even w 0 (0) = w 0 (K) =0 Ec + z,k a + Es + z,k b + Es + z,k b + Antiperiod K even, odd w(0) = w 0 (K) =0 Antiperiod K odd, even w 0 (0) = w(k) =0 Period K odd, odd w(0) = w(k) = 0 When becoes large the oscillatory region of the Laé functions in [ K, K] shrinks into a neighbourhood of the origin, and otherwise the functions are exponentially sall approaching the endpoints at z = ±K. Foral results given in the literature (see [] 9.7) indicate that Es + z,k and Ec z,k behave asyptotically the sae as!.

36 .. Properties 6 Ec 0 5 z,0.7 Es 5 z,0.7 Ec 5 z,0.7 Es 5 z,0.7 Ec 5 z,0.7 Es 3 5 z,0.7 Ec 3 5 z,0.7 Es 4 5 z,0.7 Figure.: Plots for assorted Laé functions in [ K, K]. Figure. shows that even for as large as 5, Es + z,k and Ec z,k already look very asyptotically siilar. Foral results given in the literature (see [] 9.7) also indicate that the di erence between the corresponding eigenvalues b + and a is exponentially sall as!. a 4 b 3 a 3 a b a b a 0 Figure.3: The first few eigenvalues as a function of for k =0.7. Figure.3 shows that as becoes large, b + and a tend to each other asyptotically. It also shows that as becoes larger, ust becoe larger to see that they are asyptotically the sae in this liit.

37 .. Previous Results 7. Previous Results For a general overview of the theory concerning Laé functions and their corresponding eigenvalues see [8], [9] and [0]. In this part of the thesis we will be investigating two types of results. First the asyptotic expansions of the Laé functions and their respective eigenvalues for large, and secondly the exponentially sall di erence between the eigenvalues b + and a in the sae liit. The results for asyptotic expansions of the Laé functions and their respective eigenvalues for paraeter large is not so abundant. The ain results can be found in [], [], [3], [4], and [5], and are all foral, eaning they are not accopanied with any error analysis. None of the results about the functions have been published in ([] 9.7), and none are in a particularly satisfactory for. In ([] 9.7 (ii)) it is stated that one could derive fro the results of [6] asyptotic approxiations for the Laé functions. However in that paper the results are given without uch justification and the error bounds given for the approxiations do not ake sense in the intervals where the approxiant is exponentially sall. In ([] 9.7) only liited foral results about the corresponding eigenvalues can be found, and are given as follows; As!, a papple 0 apple apple... (.0) where apple = p ( + )k, p = +, 0 = 3 ( + k )( + p ), = p 6 (( + k ) (p + 3) 4k (p + 5)). (.) The sae Poincaré expansion holds for b +,sincethedi erence between b + a is exponentially sall, which has been given in [7] as and

38 .3. Chapter outline 8 a = ( k ) / k +/! p (8k ) +3/ ( + O( / )) (!). +k b + (.) This result is reasonable but due to issues in the paper it was derived fro we will be able to iprove it, and give another ter in the expansion. We wish to obtain rigorous results for these types of probles concerning the Laé equation..3 Chapter outline We now present the analysis concerning the aforeentioned probles. We obtain unifor approxiations for the Laé functions Ec z,k and Es + z,k as! on the interval [0,K], in ters of parabolic cylinder U functions, using the theory developed by Olver in [7]. Using this ethod we are also able to give rigorous approxiations for the eigenvalues a and b + in this liit. By a judicious renaing of paraeters we then give unifor asyptotic approxiations for the Laé functions in ters of the ore natural parabolic cylinder D functions. Then we give forally a second ter in the asyptotic approxiation for the Laé functions. We then derive unifor asyptotic expansions for the Laé functions, which hold only in a shrinking neighbourhood for the origin as becoes large, but encapsulate all the interesting oscillatory behaviour of the functions. The coe are polynoials and we can copute as any as we like. cients in the expansions Siultaneously we give asyptotic expansions for the eigenvalues, where again, we can copute as any ters as we like, and provide order estiates for the error when the expansions are truncated. b + Finally we give an expression for the exponentially sall di erence between a as!, and are able to say ore, and give stronger results, than has been previously given. Results given in the next two sections are published in the authors paper [8]. and

39 .4. Unifor asyptotic approxiations for the Laé functions 9.4 Unifor asyptotic approxiations for the Laé functions We first give the ain result of this section, and later give the proof after building up soe achinery. Theore.. Let apple = p ( + )k, N 0 and 0 <k<. Then for z [0,K] as apple! Ec z,k = C Es + z,k = S + ( ) x (s ) ( ) x (s )! /4 D p apple +,(,k ) +,c! /4 D p apple +,(,k ) p D apple +,(,k ), p +,s D + apple +,(,k ), (.3) where,( ) = M E p apple p apple O apple, p p,( ) =E apple M apple O apple, = N E p apple p O apple apple /,,k /@ p p = N apple O apple /, E (.5) with the order ters in the errors di ering depending on whether we consider Ec or Es + and +,s, the functions U, U, E, M and N are all defined in section.5, both,c q are O e apple apple +/ arctanh(k),with k, asapple!, the relationship between z and x is defined by x =sn(z,k), (.6)

40 .4. Unifor asyptotic approxiations for the Laé functions 30 and x and by Z s x Z x s Z x s s t (s ) Z ( t )( k t ) dt = s (s ) t ( t )( k t ) dt = s t (s ) ( t )( k t ) dt = Z Z q ( ) d ( <xapple s ), q ( ) d ( s apple x apple s ), q ( ) d (s apple x<), (.7) where (s ) = h apple, ( ) = Z s s s (s ) t ( t )( k t dt, (.8) ) where h corresponds to either a or b + the even or odd solution solution, and depending on whether we are considering C S + ( apple)/4 p! +. (.9) 8apple Again as apple! a =( + )apple + O(), b + =( + )apple + O(). (.0).4. Approxiations in ters of parabolic cylinder U functions In (.), we denote the solution w such that d w (z,k ) dz + h ( + )k sn (z,k ) w (z,k )=0, (.) where h corresponds to either a or b +. The periodic coe cient in (.) is troublesoe, thus to obtain unifor asyptotic approxiations for w we transfor the independent variable to obtain an algebraic equation, and then transfor the dependent variable to reove the subsequent ter in the first derivative. This is done by letting x =sn(z,k), w (z,k )= ( x )( k x ) /4 ew (x, k ), (.)

41 .4. Unifor asyptotic approxiations for the Laé functions 3 and denoting apple = p ( + )k, we obtain the di erential equation corresponding to the Laé s functions in the for d ew (x, k ) dx = apple x (s ) ( x )( k x ) + x, k! ew (x, k ), (.3) where s is defined by the first of (.8) and x, k = k (k + )x 4 +(k 4 0k + )x + ( + k ) 4( x ) ( k x ). (.4) We note that the foral asyptotic expansions given in ([] 9.7) indicate that h = O(apple) as apple!. (.5) Corresponding to the transforation we consider the interval x [, ], where x =, 0, corresponds to z = K, 0,K. We deduce fro (.5) and the first of (.8) that s! 0 as apple!,henceinthis liit (.3) has two turning points which coalesce into the origin. The turning points of our equation lie at x = ±s and x (s ) ( x )( k x ) < 0 ( s <x<s ), (.6) thus we apply the theory of Case I in [7], which is detailed in section.4. In this case unifor asyptotic approxiations are in ters of the parabolic cylinder functions U( apple ( ), p apple ) and U( apple ( ), p apple ), where is defined in ters of s. Following Olver, new variables relating {x, ew } to {,W } are introduced by the appropriate Liouville transforation given by W (,k )=ẋ ew (x, k ), ẋ x (s ) ( x )( k x ) = ( ), (.7) the dot signifying di erentiation with respect to. It then follows that is defined as in the second of (.8). Fro this we denote that 0 <s < corresponds to 0 < <,, where, = r arcsin(k). (.8) k

42 .4. Unifor asyptotic approxiations for the Laé functions 3 Since = ± corresponds to x = ±s, integration of the second of (.7) yields (.7). These equations define as a real analytic function of x. There is a one-to-one correspondence between the variables x and, where is an increasing function of x, and we denote =, 0, to correspond to x =, 0,. It follows that x(, )is analytic both in and for [, ] and (,,, ). Also ẋ is non-zero in these intervals. Perforing the substitution t = s in the second of (.8), we expand the integral and obtain ( ) =(s ) + +k 8 and by reversion (s ) k +3k 4 64 (s ) =( ) +k ( ) 4 k 8 64 In the critical case s = = 0 we have fro the third of (.7) (s ) 6 + O((s ) 8 ) (s! 0), (.9) ( ) 6 + O ( ) 8 (! 0). (.30) arctanh(k) k = Z 0 t p ( t )( k t ) dt = Z 0 d =, (.3) which gives = Thus we deduce that as s,! 0 r arctanh(k). (.3) k The transfored di erential equation is now of the for where r arctanh(k)!. (.33) k d W (,k ) d = apple ( ) +,k W (,k ), (.34),k =ẋ x, k +ẋ d d ẋ