Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter

By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137-174], uniform asymptotic approximations are obtained in the first part of this paper for the Lamé and Mathieu functio...

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Datum:	2015
Hauptverfasser:	Ogilvie, K., Olde Daalhuis, A.B.
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spelling	irk-123456789-1471502019-02-14T01:24:04Z Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter Ogilvie, K. Olde Daalhuis, A.B. By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137-174], uniform asymptotic approximations are obtained in the first part of this paper for the Lamé and Mathieu functions with a large real parameter. These approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid in their respective real open intervals. In all cases explicit bounds are supplied for the error terms associated with the approximations. Approximations are also obtained for the large order behaviour for the respective eigenvalues. We restrict ourselves to a two term uniform approximation. Theoretically more terms in these approximations could be computed, but the coefficients would be very complicated. In the second part of this paper we use a simplified method to obtain uniform asymptotic expansions for these functions. The coefficients are just polynomials and satisfy simple recurrence relations. The price to pay is that these asymptotic expansions hold only in a shrinking interval as their respective parameters become large; this interval however encapsulates all the interesting oscillatory behaviour of the functions. This simplified method also gives many terms in asymptotic expansions for these eigenvalues, derived simultaneously with the coefficients in the function expansions. We provide rigorous realistic error bounds for the function expansions when truncated and order estimates for the error when the eigenvalue expansions are truncated. With this paper we confirm that many of the formal results in the literature are correct. 2015 Article Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter / K. Ogilvie, A.B. Olde Daalhuis // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E10; 34E05; 34E20 DOI:10.3842/SIGMA.2015.095 http://dspace.nbuv.gov.ua/handle/123456789/147150 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137-174], uniform asymptotic approximations are obtained in the first part of this paper for the Lamé and Mathieu functions with a large real parameter. These approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid in their respective real open intervals. In all cases explicit bounds are supplied for the error terms associated with the approximations. Approximations are also obtained for the large order behaviour for the respective eigenvalues. We restrict ourselves to a two term uniform approximation. Theoretically more terms in these approximations could be computed, but the coefficients would be very complicated. In the second part of this paper we use a simplified method to obtain uniform asymptotic expansions for these functions. The coefficients are just polynomials and satisfy simple recurrence relations. The price to pay is that these asymptotic expansions hold only in a shrinking interval as their respective parameters become large; this interval however encapsulates all the interesting oscillatory behaviour of the functions. This simplified method also gives many terms in asymptotic expansions for these eigenvalues, derived simultaneously with the coefficients in the function expansions. We provide rigorous realistic error bounds for the function expansions when truncated and order estimates for the error when the eigenvalue expansions are truncated. With this paper we confirm that many of the formal results in the literature are correct.
format	Article
author	Ogilvie, K. Olde Daalhuis, A.B.
spellingShingle	Ogilvie, K. Olde Daalhuis, A.B. Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Ogilvie, K. Olde Daalhuis, A.B.
author_sort	Ogilvie, K.
title	Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter
title_short	Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter
title_full	Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter
title_fullStr	Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter
title_full_unstemmed	Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter
title_sort	rigorous asymptotics for the lamé and mathieu functions and their respective eigenvalues with a large parameter
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/147150
citation_txt	Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter / K. Ogilvie, A.B. Olde Daalhuis // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 29 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT ogilviek rigorousasymptoticsforthelameandmathieufunctionsandtheirrespectiveeigenvalueswithalargeparameter AT oldedaalhuisab rigorousasymptoticsforthelameandmathieufunctionsandtheirrespectiveeigenvalueswithalargeparameter
first_indexed	2025-07-11T01:28:07Z
last_indexed	2025-07-11T01:28:07Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 095, 31 pages Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter? Karen OGILVIE and Adri B. OLDE DAALHUIS Maxwell Institute and School of Mathematics, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK E-mail: K.Ogilvie@sms.ed.ac.uk, A.OldeDaalhuis@ed.ac.uk Received August 03, 2015, in final form November 20, 2015; Published online November 24, 2015 http://dx.doi.org/10.3842/SIGMA.2015.095 Abstract. By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137–174], uniform asymptotic approximations are obtained in the first part of this paper for the Lamé and Mathieu functions with a large real parameter. These approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid in their respective real open intervals. In all cases explicit bounds are supplied for the error terms associated with the approximations. Approximations are also obtained for the large order behaviour for the respective eigenvalues. We restrict ourselves to a two term uniform approximation. Theoretically more terms in these approximations could be computed, but the coefficients would be very complicated. In the second part of this paper we use a simplified method to obtain uniform asymptotic expansions for these functions. The coefficients are just polynomials and satisfy simple recurrence relations. The price to pay is that these asymptotic expansions hold only in a shrinking interval as their respective parameters become large; this interval however encapsulates all the interesting oscillatory behaviour of the functions. This simplified method also gives many terms in asymptotic expansions for these eigenvalues, derived simultaneously with the coefficients in the function expansions. We provide rigorous realistic error bounds for the function expansions when truncated and order estimates for the error when the eigenvalue expansions are truncated. With this paper we confirm that many of the formal results in the literature are correct. Key words: Lamé functions; Mathieu functions; uniform asymptotic approximations; coa- lescing turning points 2010 Mathematics Subject Classification: 33E10; 34E05; 34E20 1 Introduction The Lamé equation is d2w dz2 + ( h− ν(ν + 1)k2 sn2(z, k) ) w = 0, (1.1) where h, k and ν are real parameters such that 0 < k < 1 and ν ≥ −1 2 , and sn(z, k) is a Jacobian elliptic function (see [24, § 22.2]). We consider the interval z ∈ (−K,K), where K = K(k) is Legendre’s complete elliptic integral of the first kind (see [24, § 19.2(ii)]). When h assumes the special values amν or bm+1 ν for m = 0, 1, . . . , Lamé’s equation admits even or odd periodic solutions denoted Ecmν (z, k2) or Esm+1 ν (z, k2) respectively. ?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica- tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html mailto:K.Ogilvie@sms.ed.ac.uk mailto:A.OldeDaalhuis@ed.ac.uk http://dx.doi.org/10.3842/SIGMA.2015.095 http://www.emis.de/journals/SIGMA/OPSFA2015.html 2 K. Ogilvie and A.B. Olde Daalhuis Lamé’s equation first appeared in a paper by Gabriel Lamé in 1837 [14]. It appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates. Lamé functions have applications in antenna research, occur when studying bifurcations in chaotic Hamiltonian systems, and in the theory of Bose–Einstein condensates, to name a few (see [24, § 29.19]). Mathieu’s equation is d2w dz2 + ( λ− 2h2 cos 2z ) w = 0, (1.2) where λ and h are real parameters. We consider the interval z ∈ (0, π). When λ assumes the special values am or bm+1 for m = 0, 1, 2, . . . , Mathieu’s equation admits even or odd periodic solutions denoted cem(h, z) or sem+1(h, z) respectively. These functions first arose in physical applications in 1868 in Émile Mathieu’s study of vib- rations in an elliptic drum [15]. Since they have appeared in problems pertaining to vibrational systems, electrical and thermal diffusion, electromagnetic wave guides, elliptical cylinders in viscous fluids, and diffraction of sound and electromagnetic waves, to name a few. In general, they appear when studying solutions of differential equations that are separable in elliptic cylin- drical coordinates. For an insight to see how Mathieu functions appear in physical applications see [16]. We wish to obtain uniform asymptotic approximations for the Lamé and Mathieu functions, and asymptotic expansions for their respective eigenvalues, as the parameters ν in Lamé’s equa- tion and h in Mathieu’s equation become large. We denote for the moment the parameter h in Lamé’s equation to be hL, to avoid confusion with the parameter h in Mathieu’s equation. We have in the limit k → 0+ from [24, § 22.5(ii)] that lim k→0+ sn(z, k) = sin z, and lim k→0+ K(k) = π 2 , thus if ν → ∞ in such a way that as k → 0+, √ ν(ν + 1)k = 2h for some constant h, then we can rewrite the limit of Lamé’s equation in the form d2w dz2 + ( hL − 2h2 − 2h2 cos 2 (π 2 − z )) w = 0. Thus for ν = −1 2 + √ 1 4 + ( 2h k )2 we have lim k→0+ Ecmν ( z, k2 ) = cem ( h, π 2 − z ) , lim k→0+ Esm+1 ν ( z, k2 ) = sem+1 ( h, π 2 − z ) , (1.3) and am = lim k→0+ amν − 2h2, bm+1 = lim k→0+ bm+1 ν − 2h2. (1.4) With this in mind, we derive rigorous results for the Lamé functions and their eigenvalues, and deduce analogous results for Mathieu’s equation using this limiting relation. For a general overview of the Lamé and Mathieu equations, see [2, 5, 29]. For a more detailed study of Mathieu’s equation see also [17]. Whilst the results for asymptotic expansions of the Lamé functions and their respective eigenvalues for parameter ν large are not so abundant, analogous problems for the Mathieu functions and their eigenvalues for parameter h large have been studied extensively. The main results in the Lamé case can be found in [11, 12, 18, 19, 20]. These results are all formal, meaning they are not accompanied with any error analysis. None of the results about the functions have been published in [24, § 29.7], and only limited formal results Asymptotics for the Lamé and Mathieu Functions 3 about the corresponding eigenvalues can be found there. In [24, § 29.7(ii)] it is stated that one could derive from the results of [28] asymptotic approximations for the Lamé functions. However in that paper the results are given without much justification and the error bounds given for the approximations do not make sense in the intervals where the approximant is exponentially small. Here the results we give for the Lamé functions are new; we give a two term uniform asymptotic approximation for the Lamé functions in terms of parabolic cylinder functions for ν large complete with error bounds, and we show this holds uniformly in the interval z ∈ [0,K]. We also make a rigorous statement about the corresponding eigenvalues. We also give asymptotic expansions for the functions and eigenvalues in a shrinking neighbourhood of the origin, which correspond with the few formal results in the literature. The main results in the Mathieu case can be found in [3, 4, 6, 7, 8, 9, 10, 13, 17, 25, 26, 27, 28]. In [17, 28], error estimates are written down for a one term asymptotic approximation of the Mathieu functions but these are given without any justification, and the error does not make sense in the intervals where the approximant is exponentially small. In [13], similar results are given for the functions with similar issues for the error estimates; the results about the eigenvalues however seem reasonable, but methods of obtaining terms in the eigenvalue expansions seem cumbersome. The most satisfactory work thus far is contained in [4]. Here Dunster derives uniform asymptotic approximations for all complex values z when −2h2 ≤ λ ≤ (2 − d)h2, d > 0, with error bounds either included or available for all approximations. These approximations involve both elementary functions and Whittaker functions. He also includes rigorous statements related to the eigenvalues am and bm+1. The remaining papers stated include only formal results, without any satisfactory error analysis. Here we consider only the real interval z ∈ (0, π) as many physical applications are restricted to real variables. The results we give are uniform asymptotic approximations complete with error bounds in the interval z ∈ [ 0, π2 ] , given in the most natural form for the case we consider. Since we restrict ourselves to real variable analysis we can make stronger statements about the error bounds for the functions and their respective eigenvalues than given in [4]. We also give asymptotic expansions for the functions and eigenvalues in a shrinking neighbourhood of π 2 , which correspond with the formal results in the literature. 2 Overview This paper is written in two parts: In Part I we derive two term uniform asymptotic ap- proximations for the Lamé functions Ecmν ( z, k2 ) and Esm+1 ν ( z, k2 ) which hold for z ∈ [0,K], and rigorous approximations for the large order behaviour of their respective eigenvalues amν and bm+1 ν , m = 0, 1, . . . , as κ → ∞, where κ = √ ν(ν + 1)k. Treating the Mathieu functions and their respective eigenvalues as a special case of those in the Lamé case, we obtain simply uniform asymptotic approximations for the Mathieu functions cem(h, z) and sem+1(h, z) which hold for z ∈ [ 0, π2 ] , and rigorous approximations for the large order behaviour of their respective eigenvalues am and bm+1, as h→∞. Part II uses a simplified method to derive asymptotic expansions for both the Lamé and Mathieu functions and their respective eigenvalues. We can compute as many terms as we like in these expansions. The price to pay is that these function expansions only hold for z = O(κ−1/2) and z = π 2 +O(h−1/2), as κ→∞ and h→∞ respectively. These intervals at least encapsulate all of the interesting oscillatory behaviour of the functions. We give rigorous and realistic error bounds for the function expansions once truncated, along with order estimates for the error when the eigenvalue expansions are truncated. First, we will summarise the relevant properties of the Lamé and Mathieu functions and their respective eigenvalues in the upcoming section. 4 K. Ogilvie and A.B. Olde Daalhuis 3 Properties of Lamé and Mathieu functions Lamé functions We will summarise their important properties here, for a full treatment see [24, § 29]. These functions are either 2K-periodic or 2K-antiperiodic, depending on the parity ofm. The functions have exactly m zeros in the interval (−K,K) and their eigenvalues are ordered such that a0ν < a1ν < a2ν < · · · , b1ν < b2ν < b3ν < · · · , and interlace such that amν < bm+1 ν , bmν < am+1 ν . The eigenvalues coalesce such that amν = bmν , when ν = 0, 1, . . . ,m− 1. Since the Jacobian elliptic function dn(z, k) (see [24, § 22]) is even, we can rewrite the normali- sations given in [24, § 29.3] as∫ K −K dn(z, k) { Ecmν ( z, k2 )}2 dz = ∫ K −K dn(z, k) { Esm+1 ν ( z, k2 )}2 dz = π 2 , (3.1) To complete their definitions we have Ecmν ( K, k2 ) > 0, and dEsmν ( z, k2 ) dz ∣∣∣∣ z=K < 0. (3.2) They satisfy the orthogonality conditions for m 6= n, (n = 0, 1, . . . )∫ K −K Ecmν ( z, k2 ) Ecnν ( z, k2 ) dz = 0, and ∫ K −K Esm+1 ν ( z, k2 ) Esn+1 ν ( z, k2 ) dz = 0. We summarise their properties and give boundary conditions in Table 1. Table 1. Properties and boundary conditions for Lamé functions. eigenfunctions eigenvalues periodicity parity at z = K parity at z = 0 boundary conditions Ec2mν ( z, k2 ) a2mν period 2K even even w′(0) = w′(K) = 0 Ec2m+1 ν ( z, k2 ) a2m+1 ν antiperiod 2K even odd w(0) = w′(K) = 0 Es2m+1 ν ( z, k2 ) b2m+1 ν antiperiod 2K odd even w′(0) = w(K) = 0 Es2m+2 ν ( z, k2 ) b2m+2 ν period 2K odd odd w(0) = w(K) = 0 Mathieu functions We will summarise their important properties here, for a full treatment see [24, § 28]. These are either π-periodic or π-antiperiodic, depending on the parity of m. Both functions have m zeros in the interval (0, π) and their eigenvalues are ordered such that a0 < a1 < · · · → ∞, b1 < b2 < · · · → ∞, Asymptotics for the Lamé and Mathieu Functions 5 and interlace such that a0 < b1 < a1 < b2 < a2 < · · · . The normalisations in [24, § 28.2] can be rewritten as∫ π 0 {cem(h, z)}2 dz = ∫ π 0 {sem+1(h, z)}2 dz = π 2 , and to complete their definitions, the signs are determined by continuity from ce0(0, z) = 1√ 2 , cem(0, z) = cosmz, sem(0, z) = sinmz. We summarise their properties and give boundary conditions in Table 2. Table 2. Properties and boundary conditions for Mathieu functions. eigenfunctions eigenvalues periodicity parity of w(h, z) parity of w ( h, z + π 2 ) boundary conditions ce2m(h, z) a2m period π even even w′(0) = w′ (π/2) = 0 ce2m+1(h, z) a2m+1 antiperiod π even odd w′(0) = w (π/2) = 0 se2m+1(h, z) b2m+1 antiperiod π odd even w(0) = w′ (π/2) = 0 se2m+2(h, z) b2m+2 period π odd odd w(0) = w (π/2) = 0 Part I Uniform asymptotic approximations We wish to obtain uniform approximations for the Lamé functions Ecmν ( z, k2 ) and Esm+1 ν ( z, k2 ) , and rigorous approximations for their respective eigenvalues amν and bm+1 ν , for m = 0, 1, . . . . In [24, § 29.7], the first few terms are given for formal asymptotic expansions of the eigen- values amν and bm+1 ν as ν → ∞. These indicate that (1.1) will have two turning points, one either side of the origin, which coalesce symmetrically into the origin of the z-plane as ν →∞. In Section 4 we will use this fact and derive uniform approximations for solutions in terms of the parabolic cylinder functions U ( −1 2κσ 2, √ 2κζ ) and U ( −1 2κσ 2, √ 2κζ ) as ν → ∞, where κ = √ ν(ν + 1)k and ζ arises from a special complicated transformation of the variable z, and σ is related to the eigenvalue parameter h. Only when −1 2κσ 2 is exactly a negative half integer does this function decay exponentially on both the positive and negative real axis when the vari- able is large (see [24, § 12.9]). Respective of this, in Section 5 we get a rigorous approximation for the eigenvalues amν and bm+1 ν and use this in Section 6 to give an approximation in terms of the special parabolic cylinder functions Dm (√ 2κζ ) , where Dm(z) = U ( −m− 1 2 , z ) , and these decay exponentially for both large positive and large negative values of the variable, and have exactly m zeros in their oscillatory interval (again see [24, § 12.2]). Finally in Section 7 we use these results for the Lamé equation to obtain in a limiting form uniform approximations for the Mathieu functions cem(h, z) and sem+1(h, z), and rigorous results for their eigenvalues am and bm+1, as h→∞. 4 Uniform approximations for the Lamé functions The periodic coefficient in (1.1) is troublesome, thus we transform the independent variable to obtain an algebraic equation, and then transform the dependent variable to remove the 6 K. Ogilvie and A.B. Olde Daalhuis subsequent term in the first derivative; this is done by letting x = sn(z, k), w ( z, k2 ) = 1 ((1− x2) (1− k2x2))1/4 w̃ ( x, k2 ) , (4.1) and denoting κ = √ ν(ν + 1)k, we obtain Lamé’s equation in the form d2w̃ ( x, k2 ) dx2 = ( κ2 x2 − s2 (1− x2) (1− k2x2) + φ ( x, k2 )) w̃ ( x, k2 ) , (4.2) where s2 = h κ2 (4.3) and φ ( x, k2 ) = − 2k2 ( k2 + 1 ) x4 + ( k4 − 10k2 + 1 ) x2 + 2 ( 1 + k2 ) 4 (1− x2)2 (1− k2x2)2 . We correspondingly consider the interval x ∈ (−1, 1), where x = −1, 0, 1 corresponds to z = −K, 0,K. We now write κ→∞ to correspond to ν →∞. From formal asymptotic expansions given in [24, § 29.7] we deduce that s → 0 as κ → ∞, hence in this limit (4.2) has two coalescing turning points. The turning points of our equation are at x = ±s and x2 − s2 (1− x2) (1− k2x2) < 0, −s < x < s, thus we apply the theory of Case I in [21]. In this case uniform asymptotic approximations are in terms of the parabolic cylinder functions U ( −1 2κσ 2, √ 2κζ ) and U ( −1 2κσ 2, √ 2κζ ) . For the standard notation see [24, § 12.2]. Following Olver [21], new variables relating {x, w̃} to {ζ,W} are introduced by the appro- priate Liouville transformation given by W ( ζ, k2 ) = ẋ− 1 2 w̃ ( x, k2 ) , ẋ2 x2 − s2 (1− x2) (1− k2x2) = ζ2 − σ2 (4.4) the dot signifying differentiation with respect to ζ, where σ is defined by∫ s −s √ s2 − t2 (1− t2) (1− k2t2) dt = ∫ σ −σ √ σ2 − τ2dτ = π 2 σ2. (4.5) From this we denote that 0 < s < 1 corresponds to 0 < σ < σ∗, where σ∗ = 2 √ arcsin(k) πk . Since ζ = ±σ corresponds to x = ±s, integration of the second of (4.4) yields∫ −s x √ t2 − s2 (1− t2) (1− k2t2) dt = ∫ −σ ζ √ τ2 − σ2dτ, − 1 < x ≤ −s, ∫ x −s √ s2 − t2 (1− t2) (1− k2t2) dt = ∫ ζ −σ √ σ2 − τ2dτ, − s ≤ x ≤ s, Asymptotics for the Lamé and Mathieu Functions 7 ∫ x s √ t2 − s2 (1− t2) (1− k2t2) dt = ∫ ζ σ √ τ2 − σ2dτ, s ≤ x < 1. (4.6) These equations define ζ as a real analytic function of x. There is a one-to-one correspondence between the variables x and ζ, where ζ is an increasing function of x, and we denote ζ = −ζ∗, 0, ζ∗ to correspond to x = −1, 0, 1. It follows that x(ζ, σ) is analytic both in ζ and σ for ζ ∈ (−ζ∗, ζ∗) and σ ∈ (−σ∗, σ∗). Also ẋ is non-zero in these intervals. Performing the substitution t = sτ in (4.5) we expand the integral and obtain σ2 = s2 + 1 + k2 8 s4 + 3 + 2k2 + 3k4 64 s6 +O ( s8 ) , s→ 0, and by reversion s2 = σ2 − 1 + k2 8 σ4 − ( 1− k2 )2 64 σ6 +O ( σ8 ) , σ → 0. (4.7) In the critical case s = σ = 0 we have from the third of (4.6) arctanh(k) k = ∫ 1 0 t√ (1− t2)(1− k2t2) dt = ∫ ζ∗ 0 τdτ = 1 2ζ 2 ∗ , which gives ζ∗ = √ 2 arctanh(k) k . Thus we deduce that as s, σ → 0 ζ∗ → √ 2 arctanh(k) k . The transformed differential equation is now of the form d2W ( ζ, k2 ) dζ2 = ( κ2 ( ζ2 − σ2 ) + ψ ( ζ, k2 )) W ( ζ, k2 ) , (4.8) where ψ ( ζ, k2 ) = ẋ2φ ( x, k2 ) + ẋ1/2 d2 dζ2 ( ẋ−1/2 ) = 2σ2 + 3ζ2 4 (σ2 − ζ2)2 + σ2 − ζ2 4 ( −k2 + 1 + k2 ( 1− 3s2 ) s2 − x2 + 3 ( 1− 2 ( 1 + k2 ) s2 + 3k2s4 ) (s2 − x2)2 + 5s2 ( s2 − 1 ) ( 1− k2s2 ) (s2 − x2)3 ) . By construction, the apparent singularities in the above function at ζ = ±σ, corresponding to x = ±s, cancel each other out so that ψ ( ζ, k2 ) is well-behaved there. To check this one could expand ψ ( ζ, k2 ) around this point. One could also note that in (4.2), φ ( x, k2 ) has singularities at x = ±1, where as ψ ( ζ, k2 ) does not blow up there. However ζ and therefore ψ ( x, k2 ) will have branch point singularities there. For our advantage in the error analysis, we write our differential equation in the form d2W ( ζ, k2 ) dζ2 = ( κ2 ( ζ2 − σ̃2 ) + ψ̃ ( ζ, k2 ) ) W ( ζ, k2 ) , (4.9) 8 K. Ogilvie and A.B. Olde Daalhuis where ψ̃ ( ζ, k2 ) = ψ ( ζ, k2 ) − ψ ( 0, k2 ) = ψ ( ζ, k2 ) + σ4 − s4 2σ2s4 , and correspondingly σ̃2 = σ2 + σ4 − s4 2κ2σ2s4 , (4.10) where σ4 − s4 2σ2s4 = 1 + k2 8 +O ( s2 ) , s→ 0. (4.11) This gives ψ̃ ( 0, k2 ) = 0. (4.12) On inspection it follows that since s, σ and the variables x and ζ are all bounded as κ→∞, and the apparent singularities at x = ±s and ζ = ±σ cancel each other, we have for ζ ∈ [−ζ∗, ζ∗] ψ̃ ( ζ, k2 ) = O(1), uniformly in this limit. On applying Theorem I of [21, § 6] with u = κ, α = σ̃ and ζ2 = ζ∗, we obtain the following solutions of (4.9) W1 ( ζ, k2 ) = U ( −1 2κσ̃ 2, √ 2κζ ) + ε1 ( ζ, k2 ) , W2 ( ζ, k2 ) = U ( −1 2κσ̃ 2, √ 2κζ ) + ε2 ( ζ, k2 ) , (4.13) valid when ζ ∈ [0, ζ∗). Choosing as a consequence of (4.12) Ω(z) = z, and thus defining the variational operator as Va,b ( ψ̃ ) = ∫ b a ∣∣ψ̃ (t, k2) ∣∣ √ 2κt dt, the bounds obtained are ∣∣ε1 (ζ, k2)∣∣ ≤ M ( −1 2κσ̃ 2, √ 2κζ ) E ( −1 2κσ̃ 2, √ 2κζ ) [exp ( 1 2 √ π/κl1 ( −1 2κσ̃ 2 ) Vζ,ζ∗ ( ψ̃ )) − 1 ] , ∣∣ε2 (ζ, k2)∣∣ E ( −1 2κσ̃ 2, √ 2κζ ) ≤M ( −1 2κσ̃ 2, √ 2κζ ) [ exp ( 1 2 √ π/κl1 ( −1 2κσ̃ 2 ) V0,ζ ( ψ̃ )) − 1 ] , where l1 ( −1 2κσ̃ 2 ) = sup z∈(0,∞) { Ω(z)M2 ( −1 2κσ̃ 2, z ) Γ ( 1 2 (1 + κσ̃2) ) } . The functions E, M and later N are defined in [21, § 5.8]. It follows from (4.12) and the evenness of ψ̃ ( ζ, k2 ) that V0,ζ ( ψ̃ ) = O ( κ−1/2 ) and Vζ,ζ∗ ( ψ̃ ) = O ( κ−1/2 ) as κ→∞. Asymptotics for the Lamé and Mathieu Functions 9 From [21, § 5.8] we have M2 ( −1 2κσ̃ 2, z ) = O ( z−1 ) as z →∞, (4.14) thus clearly l1 ( −1 2κσ̃ 2 ) is bounded as κ→∞. Hence ε1 ( ζ, k2 ) = M ( −1 2κσ̃ 2, √ 2κζ ) E ( −1 2κσ̃ 2, √ 2κζ ) O (κ−1) , ε2 ( ζ, k2 ) = E ( −1 2κσ̃ 2, √ 2κζ ) M ( −1 2κσ̃ 2, √ 2κζ ) O ( κ−1 ) . (4.15) Applying similar analysis to the above, again from Theorem 1 in [21, § 6], we obtain ∂ε1 ( ζ, k2 ) ∂ζ = N ( −1 2κσ̃ 2, √ 2κζ ) E ( −1 2κσ̃ 2, √ 2κζ )O(κ−1/2), ∂ε2 ( ζ, k2 ) ∂ζ = E ( −1 2κσ̃ 2, √ 2κζ ) N ( −1 2κσ̃ 2, √ 2κζ ) O ( κ−1/2 ) . (4.16) We can extend this analysis to include the point ζ∗ since ζ and ψ ( ζ, k2 ) are bounded there. 5 An interlude: eigenvalues Thus we obtain from (4.13), (4.4) and (4.1) that for z ∈ [0,K] Ecmν ( z, k2 ) = Cmν ( ζ2 − (σmν )2 x2 − (smν )2 )1/4 ( Wm ν,1 ( ζ, k2 ) + ηmν,cW m ν,2 ( ζ, k2 )) , Esm+1 ν ( z, k2 ) = Sm+1 ν ( ζ2 − (σmν )2 x2 − (smν )2 )1/4 ( Wm ν,1 ( ζ, k2 ) + ηm+1 ν,s Wm ν,2 ( ζ, k2 )) , where Cmν , Sm+1 ν , ηmν,c and ηm+1 ν,s are constants, and smν , σmν , Wm ν,1 and Wm ν,2 are defined with respect to either h = amν or h = bm+1 ν in the previous section. Let’s first consider m odd. In correspondence with the boundary conditions we require Ecmν ( 0, k2 ) = dEcmν ( z, k2 ) dz ∣∣∣∣ z=K = 0, (5.1) Esm+1 ν ( 0, k2 ) = Esm+1 ν ( K, k2 ) = 0. (5.2) The requirement at z = K in (5.1) gives 1 2ζ∗W m ν,1 ( ζ∗, k 2 ) + ( ζ2∗ − (σmν )2 )dWm ν,1 ( ζ, k2 ) dζ ∣∣∣∣ ζ=ζ∗ + ηmν,c ( 1 2ζ∗W m ν,2 ( ζ∗, k 2 ) + ( ζ2∗ − (σmν )2 )dWm ν,2 ( ζ, k2 ) dζ ∣∣∣∣ ζ=ζ∗ ) = 0, thus from (4.16), (4.15) and (4.13), and since as κ→∞ (see [24, § 12.9]) U ( −1 2κσ̃ 2, √ 2κζ∗ ) ∼ e− κζ2∗ 2 (√ 2κζ∗ )1 2 ( κσ̃2 − 1 ) , 10 K. Ogilvie and A.B. Olde Daalhuis U ′ ( −1 2κσ̃ 2, √ 2κζ∗ ) ∼ − √ κ 2 ζ∗U ( −1 2κσ̃ 2, √ 2κζ∗ ) , U ( −1 2κσ̃ 2, √ 2κζ∗ ) ∼ √ 2 π Γ ( 1 2 ( 1 + κσ̃2 )) e κζ2∗ 2 (√ 2κζ∗ )−1 2 ( κσ̃2 + 1 ) , U ′ ( −1 2κσ̃ 2, √ 2κζ∗ ) ∼ √ κ 2 ζ∗U ( −1 2κσ̃ 2, √ 2κζ∗ ) , necessarily ηmν,c is exponentially small in this limit. Note that here we have assumed that κσ̃2 = O(1) as κ→∞, and we show later, in (5.3), that this assumption is justified. Similarly the requirement at z = K in (5.2) gives Wm ν,1 ( ζ∗, k 2 ) + ηm+1 ν,s Wm ν,2 ( ζ∗, k 2 ) = 0, then necessarily ηm+1 ν,s is exponentially small as κ→∞. Hence for both these cases by considering the requirements at z = 0 we have the condition U ( −1 2κ (σ̃mν )2 , 0 ) +O ( κ−1 ) = 2(κ(σ̃mν )2−1)/4√π Γ ( 1 4 ( 3− κ (σ̃mν )2 )) +O ( κ−1 ) = 0. and to satisfy this we require 1 2κ (σ̃mν )2 = j + 1 2 +O ( κ−1 ) , where j is an odd integer. Let’s now consider m even. In correspondence with the boundary conditions we require dEcmν ( z, k2 ) dz ∣∣∣∣ z=0 = dEcmν ( z, k2 ) dz ∣∣∣∣ z=K = 0, dEsm+1 ν ( z, k2 ) dz ∣∣∣∣ z=0 = Esm+1 ν ( K, k2 ) = 0. By similar reason to the m odd case, when considering the boundary condition at z = K both ηmν,c and ηm+1 ν,s are necessarily exponentially small as κ→∞. Hence by using the boundary condition at z = 0 we have the condition U ′ ( −1 2κ (σ̃mν )2 , 0 ) +O ( κ−1 ) = − 2(κ(σ̃mν )2+1)/4√π Γ ( 1 4 ( 1− κ (σ̃mν )2 )) +O ( κ−1 ) = 0. To satisfy this we require that 1 2κ (σ̃mν )2 = j + 1 2 +O ( κ−1 ) , where j is an even integer. As κ→∞, the zeros of U ( −1 2κ (σ̃mν )2 , √ 2κζ ) tend to the zeros of Dj (√ 2κζ ) , the parabolic cylinder functions with exactly j zeros in it’s oscillatory interval (see [24, § 12.11]). Thus in correspondence with the number of zeros of the Lamé functions and those of Dj (√ 2κζ ) , we deduce that j = m and as such from (4.7) and (4.10) we have that (σmν )2 = 2m+ 1 κ +O ( κ−2 ) . (5.3) We deduce from (4.7) and (4.3) that amν = (2m+ 1)κ+O(1), bm+1 ν = (2m+ 1)κ+O(1), κ→∞. (5.4) Asymptotics for the Lamé and Mathieu Functions 11 6 Approximations in terms of parabolic cylinder D functions The Lamé functions decay exponentially on either side of the oscillatory interval in (−K,K). Our approximant in Section 4, U ( −1 2κσ̃ 2, √ 2κζ ) , displays the appropriate exponentially decreasing behaviour when ζ is large and positive, but when the variable is large and negative it becomes exponentially large. If our argument −1 2κσ̃ 2 had been exactly a negative half-integer, which it is not, then the approximant would have exhibited this wanted exponentially decaying behaviour when the variable is both large and positive and large and negative. With respect to (5.3) we define (ωmν )2 = (σmν )2 − 2m+ 1 κ = O ( κ−2 ) (κ→∞), (6.1) thus it makes sense to split up (4.8) so that d2Wm ν ( ζ, k2 ) dζ2 = ( κ2 ( ζ2 − 2m+ 1 κ ) + ψ̂mν ( ζ, k2 )) Wm ν ( ζ, k2 ) , (6.2) where ψ̂mν ( ζ, k2 ) = ψmν ( ζ, k2 ) − κ2 (ωmν )2 . From (6.1) and since smν , σmν and the variables x and ζ are all bounded as κ→∞, we have for ζ ∈ [−ζ∗, ζ∗] ψ̂mν ( ζ, k2 ) = O(1), in this limit uniformly. Now our approximant will have the desired property of decaying expo- nentially on either side of the oscillatory interval for large positive and large negative ζ. We obtain the solutions for (6.2) Wm ν,1 ( ζ, k2 ) = Dm (√ 2κζ ) + εmν,1 ( ζ, k2 ) , Wm ν,2 ( ζ, k2 ) = Dm (√ 2κζ ) + εmν,2 ( ζ, k2 ) , (6.3) valid when ζ ∈ [0, ζ∗], where we denoted for convenience Dm(t) = U ( −m− 1 2 , t ) . We have that ψ̂mν ( 0, k2 ) = −κ2 (ωmν )2 + ψmν ( 0, k2 ) = −κ2 (σmν )2 + (2m+ 1)κ+ (smν )4 − (σmν )4 2 (σmν )2 (smν )4 In Part II, Section 9, we will show that (σmν )2 = 2m+ 1 κ − 1 + k2 8κ2 +O ( κ−3 ) . We combine this with (4.11) and obtain ψ̂mν ( 0, k2 ) = O ( κ−1 ) (κ→∞). (6.4) This time in the error analysis we choose it so that Ω(z) = 1 + z, 12 K. Ogilvie and A.B. Olde Daalhuis and, hence, take as the variational operator Va,b ( ψ̃mν ) = ∫ b a ∣∣ψ̃mν (t, k2) ∣∣ 1 + √ 2κt dt. The bounds for the errors are∣∣εmν,1 (ζ, k2)∣∣ ≤ Mm (√ 2κζ ) Em (√ 2κζ ) [exp ( 1 2 √ π/κlm,1Vζ,ζ∗ ( ψ̃mν )) − 1 ] ,∣∣εmν,2 (ζ, k2)∣∣ ≤ Em (√ 2κζ ) Mm (√ 2κζ ) [ exp ( 1 2 √ π/κlm,1V0,ζ ( ψ̃mν )) − 1 ] , where lm,1 = sup z∈(0,∞) { Ω(z)M2 m(z) Γ(m+ 1) } . We define for convenience E ( −m− 1 2 , z ) = Em(z), M ( −m− 1 2 , z ) = Mm(z), N ( −m− 1 2 , z ) = Nm(z). We have the bounds Va,b ( ψ̃mν ) = ∫ b a ∣∣ψ̃mν (t, k2)∣∣ 1 + √ 2κt dt ≤ ∫ b a ∣∣ψ̃mν (0, k2)∣∣ 1 + √ 2κt dt+ ∫ b a ∣∣ψ̃mν (t, k2)− ψ̃mν (0, k2)∣∣√ 2κt dt, thus from (6.4) we have V0,ζ ( ψ̂mν ) = O ( κ−1/2 ) and Vζ,ζ∗ ( ψ̂mν ) = O ( κ−1/2 ) as κ→∞. From (4.14) clearly lm,1 is a bounded constant as κ→∞. Hence we have εmν,1 ( ζ, k2 ) = Mm (√ 2κζ ) Em (√ 2κζ ) O (κ−1) , εmν,2 ( ζ, k2 ) Em (√ 2κζ ) = Mm (√ 2κζ ) O ( κ−1 ) . (6.5) Applying similar analysis to the above, again from Theorem 1 in [21, § 6], we obtain ∂ε1 ( ζ, k2 ) ∂ζ = Nm (√ 2κζ ) Em (√ 2κζ )O(κ−1/2), ∂ε2 ( ζ, k2 ) /∂ζ Em (√ 2κζ ) = Nm (√ 2κζ ) O ( κ−1/2 ) . Thus from (6.3), (4.4) and (4.1) we obtain Ecmν ( z, k2 ) = Cmν ( ζ2 − (σmν )2 x2 − (smν )2 )1/4 × ( Dm (√ 2κζ ) + εmν,1 ( ζ, k2 ) + ηmν,c ( Dm (√ 2κζ ) + εmν,2 ( ζ, k2 ))) , Esm+1 ν ( z, k2 ) = Sm+1 ν ( ζ2 − (σmν )2 x2 − (smν )2 )1/4 × ( Dm (√ 2κζ ) + εmν,1 ( ζ, k2 ) + ηm+1 ν,s ( Dm (√ 2κζ ) + εmν,2 ( ζ, k2 ))) , valid as κ→∞ for z ∈ [0,K], where εmν,1 ( ζ, k2 ) and εmν,2 ( ζ, k2 ) are given in (6.5), and ηmν,c and ηm+1 ν,s are constants to be determined. Since we have the boundary conditions dEcmν ( z, k2 ) dz ∣∣∣∣ z=K = 0, Esm+1 ν ( K, k2 ) = 0, we obtain that necessarily both ηmν,c and ηm+1 ν,s are O ( e−κζ 2 ∗κm+1/2 ) as κ → ∞. Only when ζ nears the endpoint the contribution from the Dm (√ 2κζ ) + εmν,2 ( ζ, k2 ) term is comparable to Dm (√ 2κζ ) + εmν,1 ( ζ, k2 ) . These functions will be made unique by their normalisation. Asymptotics for the Lamé and Mathieu Functions 13 A second term in the approximation in terms of parabolic cylinder D functions In [21, § 11] uniform asymptotic expansions are included. The two term version is of the form Wm ν,1 ( ζ, k2 ) ∼ A0Dm (√ 2κζ ) + B0(ζ, κ) κ2 d dζ Dm (√ 2κζ ) , in which we have taken A0 = 1 and for B0(ζ, κ) we have B0(ζ, κ) = ∫ ζ σ̃ ψ̂mν ( t, k2 ) 2 √ (ζ2 − σ̃2) (t2 − σ̃2) dt, where σ̃ = √ (2m+ 1)/κ. This coefficient depends on κ and its dominant part is B0(ζ) = lim κ→∞ B0(ζ, κ) = 1 2ζ ∫ ζ 0 lim κ→∞ ψ̂mν ( t, k2 ) t dt. (6.6) To simplify the integrand we take the final equation in (4.6) and let κ→∞. The result is 1 2ζ 2 = ∫ x 0 tdt√ (1− t2) (1− k2t2) = 1 k ln ( 1 + k√ 1− k2x2 + k √ 1− x2 ) . This relation can be inverted x2 = k−1 sinh ( kζ2 ) − ( 1 + k−2 ) sinh2 ( kζ2/2 ) . We use this in lim κ→∞ ψ̂mν ( ζ, k2 ) = 3 4ζ2 + ζ2 4 ( k2 ( x2 + x4 ) + x2 − 3 x4 ) + 1 + k2 8 , and are able to evaluate the integral in (6.6) and obtain 32ζB0(ζ) = ( k2 + 1 ) ln ( 1 4ζ 2C(ζ, k) ) − 3 ( k2 − 1 )2 2C(ζ, k) + 3k coth ( kζ2/2 ) + 2k2ζ2 − 6 ζ2 , where C(ζ, k) = 2k coth ( kζ2/2 ) − k2 − 1. Thus our two term approximation is Ecmν ( z, k2 ) ∼ Cmν ( ζ2 − (σmν )2 x2 − (smν )2 )1/4( Dm (√ 2κζ ) + B0(ζ) κ2 d dζ Dm (√ 2κζ )) , as κ→∞ and similarly for Esm+1 ν ( z, k2 ) . Normalisation constant We now want an approximation for Cmν and Sm+1 ν . Due to the complicated nature of the mapping between x and ζ, it is not simple to express one in terms of the other. With respect to (3.1) we consider the integral (Cmν )2 ∫ K −K dn(z, k) √ ζ2 − (σmν )2 x2 − (smν )2 D2 m (√ 2κζ ) dz. Using only this first term in an expansion for the solution, we can only obtain coefficients for up to 1/h terms. Any further terms in an expansion for the solution would contribute to further 14 K. Ogilvie and A.B. Olde Daalhuis terms in the normalisation constant expansion, which we will not consider here. Since the first terms in their respective function uniform approximations are the same for large κ, it will be the same for both Cmν and Sm+1 ν . First we perform the transformation x = sn(z, k) to obtain (Cmν )2 ∫ 1 −1 √ ζ2 − (σmν )2( x2 − (smν )2 ) (1− x2) D2 m (√ 2κζ ) dx, then another transformation to the ζ variable to obtain (Cmν )2 ∫ ζ∗ −ζ∗ ζ2 − (σmν )2 x2 − (smν )2 √ 1− k2x2D2 m (√ 2κζ ) dζ. As the oscillatory behaviour of Dm (√ 2κζ ) happens in a shrinking region of the origin as κ→∞, we seek to approximate the integral around this point to get an approximation for Cmν . Thus we consider an expansion of the form x = ∞∑ k=1 ckζ k, (6.7) as ζ → 0, and substituting this into the relation dx dζ = √( ζ2 − (σmν )2 ) (1− x2) (1− k2x2) x2 − (smν )2 . (6.8) From this we can determine, by matching (6.7) and (6.8), the ck terms, and obtain x = σmν smν ζ + ( (σmν )4 − (smν )4 6(smν )5σmν − ( 1 + k2 ) (σmν )3 6(smν )3 ) ζ3 +O ( ζ5 ) , as ζ → 0. From this we obtain ζ2 − (σmν )2 x2 − (smν )2 √ 1− k2x2 = (σmν )2 (smν )2 + 2 (σmν )4 − 2 (smν )4 − k2 (smν )2 (σmν )4 2 (smν )6 ζ2 +O ( ζ4 ) , as ζ → 0. With respect to this, we consider as a first approximation for Cmν the integral (Cmν )2 ∫ ζ∗ −ζ∗ ( (σmν )2 (smν )2 + (2σmν )4 − 2 (smν )4 − k2 (smν )2 (σmν )4 2 (smν )6 ζ2 ) D2 m (√ 2κζ ) dζ. and letting t = √ 2κζ, we have the approximation (Cmν )2√ 2κ ∫ ∞ −∞ ( 1 + ( 1 + k2 ) (2m+ 1) + t2 − k2t2 8κ ) D2 m(t)dt ∼ (Cmν )2m!√ κ/π ( 1 + 2m+ 1 4κ ) . In correspondence with (3.2) and that for large z, Dm(z) > 0, D′m(z) < 0 (see [24, § 12.9(i)], we deduce from (3.1) that as κ→∞ Cm Sm+1 } ∼ (πκ)1/4√ 2m! ( 1− 2m+ 1 8κ ) . Asymptotics for the Lamé and Mathieu Functions 15 7 A special case: Mathieu functions We obtain rigorous uniform results for Mathieu’s equation, using the limiting arguments given in (1.3) and (1.4). Thus (reader can check the details, we will just summarise) we have the uniform approximations as h→∞ cem(h, z) = Cm ( ζ2 − σ2m x2 − s2m )1/4 × ( Dm ( 2 √ hζ ) + εm,1(ζ) + ηcm ( Dm (√ 2hζ ) + εm,2(ζ) )) , sem+1(h, z) = Sm+1 ( ζ2 − (σmν )2 x2 − (smν )2 )1/4 × ( Dm ( 2 √ hζ ) + εm,1(ζ) + ηsm+1 ( Dm (√ 2hζ ) + εm,2(ζ) )) , valid for ζ ∈ [0, ζ∗], i.e., z ∈ [0, π2 ], as h→∞, where εm,1(ζ) = Mm (√ 2hζ ) Em (√ 2hζ ) O (h−1) , εm,2(ζ) = Em (√ 2hζ ) Mm (√ 2hζ ) O ( h−1 ) both ηcm and ηsm+1 are O ( e−2hζ 2 ∗hm+1/2 ) as h→∞, the relationship between z and x is defined by x = cos z and by evaluating the integrals (4.6), the relationship between x and ζ is defined by ∫ −sm x √ t2 − s2m (1− t2) (1− k2t2) dt = ∫ −σm ζ √ τ2 − σ2mdτ, − 1 < x ≤ −sm, ∫ x −sm √ s2m − t2 (1− t2) (1− k2t2) dt = ∫ ζ −σm √ σ2m − τ2dτ, − sm ≤ x ≤ sm, ∫ x sm √ t2 − s2m (1− t2) (1− k2t2) dt = ∫ ζ σm √ τ2 − σ2mdτ, sm ≤ x < 1, where s2m = λm + 2h2 4h2 , σ2m = s2m 2F1 ( 1 2 , 1 2 2 ; s2m ) , and λm corresponds to either am or bm+1 depending on the solution we are considering, and Cm Sm+1 } ∼ ( πh 2m!2 )1/4( 1− 2m+ 1 16h ) . We have that s2m = m+ 1 2 h +O ( h−2 ) as h→∞, thus from (4.3) and (1.4) we obtain am = −2h2 + (4m+ 2)h+O(1), bm+1 = −2h2 + (4m+ 2)h+O(1), as h → ∞. One should note that this result about the eigenvalues was given formally in [17, § 2.33, Satz 5 & Satz 6]. 16 K. Ogilvie and A.B. Olde Daalhuis Two term approximations are given by cem(h, z) ∼ Cm ( Dm ( 2 √ hζ ) + B0(ζ) h2 d dζ Dm ( 2 √ hζ )) , sem+1(h, z) ∼ Sm+1 ( Dm ( 2 √ hζ ) + B0(ζ) h2 d dζ Dm ( 2 √ hζ )) , where for B0(ζ) we have the relation 256B0(ζ) = 3ζ 4− ζ2 − 2 ζ ln ( 1− ζ2 4 ) . Part II Uniform asymptotic expansions In Part I, uniform asymptotic approximations were given for the Lamé and Mathieu functions when a parameter became large. A third term in an asymptotic expansion could not be computed due to the complicated nature of the transformation of the independent variable. In Section 8 of this part we employ a simpler transformation than in Part I such that we can construct formal asymptotic expansions for the Lamé functions and their corresponding eigenvalues in the forms wmν (t) ∼ Dm(t) ∞∑ s=0 As(t) κs +D′m(t) ∞∑ s=0 Bs(t) κs , h ∼ (2m+ 1)κ+ ∞∑ s=0 µs κs , where t = √ 2κ sn (z, k), the coefficients As(t) and Bs(t) are just polynomials, and the µs are constants in terms of m and k. The function expansions will clearly only make sense when t = O(1) as κ→∞. In Section 9, we analyse the asymptotic expansions for the eigenvalues and give an order estimate for the error corresponding to the truncated eigenvalue expansion. Then in Section 10 we use these results to give rigorous and realistic error bounds for the function expansions upon truncation. In Sections 11 and 12 we identify our expansions with the Lamé functions and then summarise our results. Finally in Section 13 we give analogous results for the Mathieu functions and their corresponding eigenvalues as a special limiting case of those given for the Lamé functions and their corresponding eigenvalues. 8 Uniform asymptotic expansions of the Lamé functions In this section we construct formal asymptotic expansions for solutions of Lamé’s equation and corresponding eigenvalues. We do this by performing a simpler transformation than employed in Part I. The oscillatory behaviour of the Lamé functions happens in a shrinking neighbourhood of the origin as κ → ∞, and it can be shown that around the origin ζ behaves approximately like sn(z, k). Thus the variable in the parabolic cylinder function around this point behaves approximately like √ 2κ sn(z, k). This motivates the next simpler transformation. t-plane Letting t = √ 2κ sn(z, k) in (1.1) we have d2w dt2 + 2h− t2κ 4κ w − 1 2κ ((( 1 + k2 ) t2 − k2t4 2κ ) d2 dt2 + t ( 1 + k2 − k2t2 κ ) d dt ) w = 0, Asymptotics for the Lamé and Mathieu Functions 17 where z ∈ (−K,K) corresponds to t ∈ ( − √ 2κ, √ 2κ ) . Supposing in accordance with (5.4) that h 2κ = m+ 1 2 + n∑ s=1 µs κs + µ̃n+1 κn+1 , (8.1) where n is a positive integer and µ̃n can be re-expanded in a sensible manner. We can then write Lamé’s equation in the form d2w dt2 + ( m+ 1 2 − t2 4 ) w + 1 κ {( −1 2 ( 1 + k2 ) t2 + k2t4 4κ ) d2 dt2 + t 2 ( −1− k2 + k2t2 κ ) d dt + n∑ s=0 µs+1 κs + µ̃n+1 κn+1 } w = 0. (8.2) This equation is split in such a way that constructing a formal asymptotic expansion in terms of parabolic cylinder functions Dm(t) in the form wmν ( t, k2 ) = Dm(t) ∞∑ s=0 As(t) κs +D′m(t) ∞∑ s=0 Bs(t) κs (8.3) seems sensible. However one should observe that this splitting only makes sense when t = O(1) as h→∞, hence this formal expansion is only sensible for this range of t. We denote this range as (−t∗, t∗). One should note that whilst this appears to be a new ansatz, there are similar expansions given in the literature for the Mathieu functions. These expansions are given in terms of Dm−4j(t) for j ∈ Z instead of in terms of Dm(t) and D′m(t), but using the recurrence relations for the parabolic cylinder functions it is obvious these expansions are equivalent up to normalisation. An expansion in the form we have given allows us to differentiate easily and hence seems the most natural in this case, also allowing us to perform rigorous error analysis. We seek solutions which are either even or odd respective to the parity of m. Since Dm(t) is either even or odd respective to when m is either even or odd, we deduce that As(t) and Bs(t) must be even and odd respectively. Substituting (8.3) into (8.2) and equating powers of κ, we have the recurrence relations for As(t) and Bs(t) 2A′s(t) +B′′s (t)− t+ tk2 2 ( 2tA′s−1(t) +As−1(t) + tB′′s−1(t) +B′s−1(t) + t̃(m)tBs−1(t) ) + k2t3 4 ( 2tA′s−2(t) + 2As−2(t) + tB′′s−2(t) + 2B′s−2(t) + t̃(m)tBs−2(t) ) + s∑ j=1 µjBs−j(t) = 0, (8.4) A′′s(t) + 2t̃(m)B′s(t) + t 2 Bs(t) + s∑ j=1 µjAs−j(t)− t+ tk2 2 ( tA′′s−1(t) +A′s−1(t) + t2 2 Bs−1(t) + t̃(m) ( tAs−1(t) + 2tB′s−1(t) +Bs−1(t) )) + k2t3 4 ( tA′′s−2(t) + 2A′s−2(t) + t2 2 Bs−2(t) + t̃(m) ( tAs−2(t) + 2tB′s−2(t) + 2Bs−2(t) )) = 0, (8.5) where t̃(m) = 1 4 t 2−m− 1 2 . Neither of these relations separately determine solutions for As(t) or Bs(t) from previous coefficients, thus we differentiate (8.4) to obtain an expression for A′′s(t) and substitute it into (8.5); this gives the third order inhomogeneous differential equation for Bs(t) B′′′s (t)− ( t2 − 4m− 2 ) B′s(t)− tBs(t) = bs(t), (8.6) 18 K. Ogilvie and A.B. Olde Daalhuis where bs(t) = s∑ j=1 µj ( 2As−j(t)−B′s−j(t) ) + 1 + k2 2 [ 3tA′s−1(t) +As−1(t) + t2B′′′s−1(t) + 3tB′′s−1(t) +B′s−1(t) − 1 2 t 3Bs−1(t)− t̃(m)t2 ( 2As−1(t) + 3B′s−1(t) )] − k2t2 2 [ 3tA′s−2(t) + 3As−2(t) + 1 2 t 2B′′′s−2(t) + 3tB′′s−2(t) + 3B′s−2(t) − 1 4 t 3Bs−2(t)− 1 2 t 2t̃(m) ( 2As−2(t) + 3B′s−2(t) )] . (8.7) Once Bs(t) is determined, we can use (8.4) to determine As(t). There will be freedom in choosing the integration constants in the As(t) terms, with identification of our solutions made unique by their normalisation. General coefficients Bs(t) and As(t) Using variation of parameters we obtain the general solution for Bs(t) Bs(t) = b1s(t)D 2 m(t) + b2s(t)D 2 m(t) + b3s(t)Dm(t)Dm(t), where Dm(t) = U ( −m− 1 2 , t ) (for U see formula (12.2.21) in [24]), and b1s(t) = ∫ bs(t)W{D 2 , DD} W{D2, D 2 , DD} dt+ c1, b2s(t) = − ∫ bs(t)W{D2, DD} W{D2, D 2 , DD} dt+ c2, b3s(t) = ∫ bs(t)W{D2, D 2} W{D2, D 2 , DD} dt+ c3. By expanding the Wronskians we derive the relations W{D2, DD} =W{D,D}D2 m(t), W{D2 , DD} = −W{D,D}D2 m(t), W{D2, D 2} = 2W{D,D}Dm(t)Dm(t), W{D2, D 2 , DD} = −2W{D,D}3. Then without loss of generality we can rewrite the indefinite integrals in { bis(t) }3 i=1 as definite integrals from 0 to t, and since Bs(t) is supposed to be an odd function we have to take c1 = c2 = 0. Thus Bs(t) = π 4m!2 ∫ t 0 bs(τ) ( Dm(τ)Dm(t)−Dm(τ)Dm(t) )2 dτ + c3Dm(t)Dm(t). (8.8) Although we consider t to be O(1) as κ → ∞, we still want an expansion which exhibits the correct behaviour at the endpoints of the interval. Expanding the squared term in (8.8) and splitting it into three separate integrals, large variable asymptotics for the parabolic cylinder functions (see [24, § 12.9]) tells us that the terms involving D2 m(t) or Dm(t)Dm(t) grow no faster than polynomials when t becomes large. Since Dm(t)2 ∼ 2m!2 π et 2/2t−2m−2, t→∞, to ensure the boundedness of our formal expansion as t becomes large, µs is determined uniquely by the condition that∫ ∞ 0 bs(τ)D2 m(τ) dτ = 0. (8.9) Asymptotics for the Lamé and Mathieu Functions 19 This condition will also ensure boundedness as t → −∞. Note that in the next section we will derive an alternative method to compute the µs terms which we use to compute the coefficients as it is simpler. Consider first s = 0. From (8.7) we see that b0(t) = 0, thus ensuring that B0(t) is odd we obtain the general solution B0(t) = c3Dm(t)Dm(t), and then A0(t) = −1 2c3 ( D′m(t)Dm(t) +Dm(t)D ′ m(t) ) + c. Rearranging, on the κ0 level of the asymptotic expansion (8.3) we have( 1 2c3W(Dm, Dm) + c ) Dm(t). Thus having this term in B0(t) equates to B0(t) = 0 and an extra constant term in A0(t), and since we have freedom in the arbitrary constant terms in As(t), we can take c3 = 0 without loss of generality. For simplicity we take A0(t) = 1 and adopt the convention As(t) = 0 for s ≥ 1. With these choices we have b1(t) = 2µ1 + 1 + k2 4 ( 2− t2 ( t2 − 4m− 2 )) , and expressing the Dm(t) in (8.9) via [24, formula (12.7.2)] in terms of Hermite polynomials we can evaluate the integral in (8.9) and obtain µ1 = −1 8 ( 1 + k2 ) ( 1 + 2m+ 2m2 ) . In the case s = 1, it can be shown using integration by parts that for µ1 which satisfies (8.9) π 4m!2 ∫ t 0 b1(τ) ( Dm(τ)Dm(t)−Dm(τ)Dm(t) )2 dτ = 1 + k2 16 ( t3 − (2m+ 1)t+ (−1)m 2m+ 1 m! √ π 2 Dm(t)Dm(t) ) . Thus clearly in this case, if the correct c3 is chosen in (8.8) then B1(t) is exactly an odd polynomial. Using similar observations as in the s = 0 case, we also note that if this multiple of DmDm is included in the B1(t) term, the expansion can be rearranged so that this term is instead represented in the constant term of A1(t). Taking B1(t) to be exactly an odd polynomial, we get that A1(t) is an even polynomial. If one would go through the details to compute the representation for B1(t) as a polynomial plus this multiple of DmDm, one would see that it would appear for all s ≥ 1 that if the previous As(t) and Bs(t) terms are all polynomials, then the As(t) and Bs(t) terms can be represented as polynomials. Polynomial coefficients Bs(t) and As(t) To obtain explicit expressions for As(t) and Bs(t) we try substituting in polynomial expansions with undetermined coefficients. Take A0(t) = 1 and B0(t) = 0, then for s ≥ 1 we consider As(t) and Bs(t) in the form As(t) = ∞∑ i=1 as,it 2i and Bs(t) = ∞∑ i=0 bs,it 2i+1. 20 K. Ogilvie and A.B. Olde Daalhuis Substituting these into (8.6) and (8.4) we obtain the recurrence relations for coefficients (2i+ 3)(2i+ 2)(2i+ 1)bs,i+1 + (4m+ 2)(2i+ 1)bs,i − 2ibs,i−1 + 2 s−1∑ j=0 µs−j (( i+ 1 2 ) bj,i − aj,i ) − ( 1 + k2 ) [( 3i+ 1 2 ) as−1,i + 1 2 (2i+ 1)3 bs−1,i + ( m+ 1 2 ) ( as−1,i−1 + ( 3i− 3 2 ) bs−1,i−1 ) − 1 4as−1,i−2 − ( 3 4 i− 7 8 ) bs−1,i−2 ] + k2 [( 3i− 3 2 ) as−2,i−1+ 2i ( i2− 1 4 ) bs−2,i−1+ (2m+ 1) ( 1 4as−2,i−2+ ( 3 4 i− 9 8 ) bs−2,i−2 ) − 1 8as−2,i−3 − ( 3 8 i− 13 16 ) bs−2,i−3 ] = 0, (8.10) 2(2i+ 2)as,i+1 + (2i+ 3)(2i+ 2)bs,i+1 + s−1∑ j=0 µs−jbj,i − 1 2 ( 1 + k2 ) ( (4i+ 1)as−1,i + (2i+ 1)2bs−1,i − ( m+ 1 2 ) bs−1,i−1 + 1 4bs−1,i−2 ) (8.11) + k2 ( i ( i− 1 2 ) bs−2,i−1 + ( i− 1 2 ) as−2,i−1 − 1 8(2m+ 1)bs−2,i−2 + 1 16bs−2,i−3 ) = 0. From these recurrence relations it is observed that only a finite number of the as,i and bs,i are non-zero. The orders are displayed in Table 3. Table 3. The orders of the polynomials. s As(t) Bs(t) even 4s 4s− 3 odd 4s− 2 4s− 1 In this manner the coefficients are determined recursively. We deduce by considering the difference of (8.10) and (8.11) in the case i = 0 that µs = (2m+ 1)bs,0 − 2as,1. (8.12) In the case that s is even then we start with i = 2s − 1 in (8.10) to determine bs,2s−2. Then we take i = 2s − 2 and determine bs,2s−3, and so on. Once every power of t is eliminated, we are left with a constant equation which we must make zero by specifying µs; in this manner the eigenvalue terms are determined uniquely. These µs terms are the same as those specified by the condition (8.9), since it is required for solutions Bs(t) which grow no faster than poly- nomials. Once the coefficients in the polynomial expression for Bs(t) are determined, and the corresponding µs, then the coefficients in the polynomial expression for As(t) can be deter- mined from (8.11). In the case that s is odd then we have to start with i = 2s in (8.10) to determine bs,2s−1. Returning to the z-plane: odd solutions In [24, § 28.8] expansions for the odd Mathieu functions are given which exhibit the correct odd behaviour, which an expansion in the form (8.3) would not when transformed back into the z-plane. We want something similar for the Lamé functions. Once we transform our formal expansion back into the z-plane, over the whole real line they behave like the even Lamé functions Ecmν ( z, k2 ) . We want to construct a similar expansions which when transformed back into the z-plane behave like the odd Lamé functions Esm+1 ν ( z, k2 ) . We need our expansion to have the property wmν ( −K, k2 ) = wmν ( K, k2 ) = 0. Asymptotics for the Lamé and Mathieu Functions 21 The Jacobi elliptic function cn(z, k) is even around the origin and odd around z = −K and z = K. Hence again letting t = √ 2κ sn(z, k) we consider a formal expansion for a solution of (8.2) of the form wmν ( z, k2 ) = cn(z, k) ( Dm(t) ∞∑ s=0 Ps(t) κs +D′m(t) ∞∑ s=1 Qs(t) κs ) . This has the correct behaviour at z = −K and z = K. Again we will require that Ps(t) is even and Qs(t) is odd. By writing cn ( arcsn ( t√ 2κ , k ) , k ) = √ 1− t2 2κ , we can express the formal solution in the form wmν ( z, k2 ) = √ 1− t2 2κ ( Dm(t) ∞∑ s=0 Ps(t) κs +D′m(t) ∞∑ s=1 Qs(t) κs ) . Expanding this square root, we can rewrite this formal expansion as wmν ( z, k2 ) = Dm(t) ∞∑ s=0 As(t) κs +D′m(t) ∞∑ s=1 Bs(t) κs , where the connection between Ps(t) and As(t), and Qs(t) and Bs(t) is given by As(t) = s∑ j=0 (1 2 j )( − t 2 2 )j Ps−j(t), Bs(t) = s∑ j=0 (1 2 j )( − t 2 2 )j Qs−j(t), where ( a b ) is the generalised binomial coefficient. Thus we determine the Ps(t) and Qs(t) terms by connection with the As(t) and Bs(t) derived previously. By connection with the previous case we get the same eigenvalue expansion to all orders for both amν and bm+1 ν . 9 An interlude: error analysis for the eigenvalue expansions In this section we match our eigenvalue expansions with the eigenvalues amν and bm+1 ν , and give order estimates for the expansions on truncation. The theory of this subsection is included in [17], where it is employed for Mathieu’s equation. For this we consider Sturm–Liouville theory; for a fuller treatment on Sturm–Liouville theory see [1]. We have the differential equation d dx [ p(x) dy dx ] + (λw(x)− q(x)) y = 0 and we consider p(x), w(x) > 0, and p(x), p′(x), q(x) and w(x) to be continuous functions over a finite real interval [a, b]. When (a, b) is bounded and p(x) does not vanish on [a, b], this is a regular SL problem, otherwise it is singular. Consider the regular problem. We want to find special values of λ called eigenvalues for which there exists a non-trivial solution satisfying the separated boundary conditions α1y(a) + α2y ′(a) = 0, α2 1 + α2 2 > 0, β1y(b) + β2y ′(b) = 0, β21 + β22 > 0. (9.1) 22 K. Ogilvie and A.B. Olde Daalhuis The regular SL theory states that these eigenvalues are real and can be ordered such that λ0 < λ1 < · · · < λν < · · · → ∞ and these eigenvalues λν correspond to unique eigenfunctions yν(x) with exactly ν zeros in (a, b). Normalised eigenfunctions form an orthonormal basis such that (yn, ym) = ∫ b a yn(x)ym(x)w(x)dx = δmn in the Hilbert space L2 [[a, b], w(x)dx]. The following theorem is Theorem 1 in [17, § 1.52] Theorem 9.1. Consider SL to be a self-adjoint differential operator defined on a subspace A of L2 [[a, b], w(x)dx] containing all the twice differentiable functions which satisfy the boundary conditions (9.1) and corresponding λν such that (SL + λν)yν(x) = 0. Let ỹ(x) ∈ A be such that (ỹ, ỹ) = 1. Now define a remainder function R(x) such that (SL + λ)ỹ(x) = R(x) for some constant λ. If R(x) ∈ L2 [[a, b], w(x)dx] then min k \|λ− λk\| < √ (R,R). Proof. We have that (R, yk) = ((SL+ λ)ỹ, yk) = (ỹ, SLyk) + λ(ỹ, yk) = (λ− λk)(ỹ, yk), and using this in 1 = (ỹ, ỹ) = ∑ k \|(ỹ, yk)\|2 = ∑ k \|(R, yk)\|2 \|λ− λk\|2 < (R,R) mink \|λ− λk\|2 , we have that min k \|λ− λk\| < √ (R,R). � Eigenvalues of Lamé’s equation We now apply this theory to Lamé’s equation to obtain order estimates upon truncation of asymptotic expansions of the eigenvalues amν and bm+1 ν . Special eigenvalues amν correspond to non-trivial solutions satisfying the separated boundary conditions d dz Ecmν ( z, k2 ) ∣∣∣∣ z=−K = d dz Ecmν ( z, k2 ) ∣∣∣∣ z=K = 0 and these eigenvalues are real and can be ordered such that a0ν < a1ν < · · · < amν < · · · → ∞. Asymptotics for the Lamé and Mathieu Functions 23 The functions Ecmν ( z, k2 ) have exactly m zeros in (−K,K). The normalised eigenfunctions wmν ( z, k2 ) form an orthonormal basis such that ( wiν , w j ν ) = ∫ K −K wiν ( z, k2 ) wjν ( z, k2 ) dz = δij in the Hilbert space L2 [[−K,K], dz]. Let Lν be the operator Lν := d2 dz2 − κ2 sn2(z, k) so that (Lν + amν )wmν ( z, k2 ) = 0. Letting t = √ 2κ sn(z, k), we define the truncated expansions corresponding to wmν ( z, k2 ) as wmν,n ( z, k2 ) = cmν,n ( Dm(t) n∑ s=0 As(t) κs +D′m(t) n∑ s=0 Bs(t) κs ) , (9.2) where the As(t) and Bs(t) terms were derived previously, and cmν,n is defined to be a function of κ so that∫ K −K ( wmν,n ( z, k2 ))2 dz = 1. (9.3) Then we can write the derivative of wmν,n ( z, k2 ) with respect to z as dwmν,n ( z, k2 ) dz = cmν,n √ 2κ cn(z, k) dn(z, k) × Dm(t) n∑ s=0 A′s(t) + ( t2 4 −m− 1 2 ) Bs(t) κs +D′m(t) n∑ s=0 As(t) +B′s(t) κs  , where the dash represents differentiation with respect to t. Thus clearly wmν,n ( z, k2 ) ∈ A since cn(−K, k) = cn(K, k) = 0. We also define the truncated eigenvalue expansions amν,n = (2m+ 1)κ+ 2 n−1∑ s=0 µs+1 κs , where the µs were derived previously and define Rmν,n ( z, k2 ) such that( Lν + amν,n ) wmν,n ( z, k2 ) = Rmν,n ( z, k2 ) . We consider the operator Lν acting on Dm(t) and derive Lν (Dm(t)) = ( (2m+ 1) ( 1 + k2 2 t2 − κ ) − ( k2(2m+ 1) + κ ( 1 + k2 )) t4 4κ + k2t6 8κ ) Dm(t) + ( − ( 1 + k2 ) t+ k2t3 κ ) D′m(t). (9.4) 24 K. Ogilvie and A.B. Olde Daalhuis Using the recurrence relations tDm(t) = Dm+1(t) +mDm−1(t), D′m(t) = mDm−1(t)− 1 2 tDm(t), (9.5) we can rewrite wmν,n ( z, k2 ) given in (9.2) such that for s ∈ {0, . . . , n}, As(t)Dm(t) and Bs(t)D ′ m(t) are sums of varying orders of parabolic cylinder functions with constant coefficients dependent only on m and k; it then follows from (9.4) that with the solution rewritten in this form, we have Rmν,n ( z, k2 ) = cmν,nκ −n [(· · · ) + κ−1(· · · ) + · · · ] , where the terms inside the brackets are sums of varying orders of parabolic cylinder functions with constant coefficients dependent only on k and m. Note that this remainder term will be finite sum and clearly Rmν,n ( z, k2 ) ∈ L2 [[−K,K], dz]. As wmν,n ( z, k2 ) ∈ A we have that min k ∣∣amν,n − akν∣∣2 <√(Rmν,n, Rmν,n). In Part I, Section 5, we proved that amν = (2m+ 1)κ+O(1) as κ→∞, hence for κ large enough, necessarily min k ∣∣amν,n − akν∣∣2 = ∣∣amν,n − amν ∣∣2 and thus∣∣amν,n − amν ∣∣ <√(Rmν,n, Rmν,n). We need an order estimate for ( Rmν,n, R m ν,n ) . The integrals we must consider then are of the form I = ∫ K −K Di (√ 2κ sn(z, k) ) Dj (√ 2κ sn(z, k) ) dz, where i, j ∈ N0. Performing the substitution t = √ 2κ sn(z, k) we obtain I = 1√ 2κ ∫ √2κ − √ 2κ 1√ 1− t2 2κ √ 1− k2t2 2κ Di(t)Dj(t)dt. Since when t is large Dm is exponentially small it follows that I = O ( κ− 1 2 ) , as κ→∞. Considering the first term of the expansion for wmν,n ( z, k2 ) in (9.2) we have ( cmν,n )2 ∫ K −K D2 m (√ 2κ sn(z, k) ) dz = ( cmν,n )2 √ 2κ ∫ √2κ − √ 2κ 1√ 1− t2 2κ √ 1− k2t2 2κ D2 m(t)dt ∼ ( cmν,n )2 √ 2κ ∫ ∞ −∞ D2 m(t) dt = ( cmν,n )2 m! √ π κ , Asymptotics for the Lamé and Mathieu Functions 25 as κ → ∞. Hence it follows from (9.3) that cm,n = O(κ1/4) as κ → ∞. Similar observations would give us that∫ K −K (Rm,n(z))2 dz = O ( κ−2n ) , and so amν − amν,n = O ( κ−n ) , as κ→∞. The error analysis for bmν,n would in the same manner give the order estimate bm+1 ν − bm+1 ν,n = O ( κ−n ) as κ→∞. Hence we have from (8.1) that amν = (2m+ 1)κ+ 2 n−1∑ s=0 µs+1 κs +O ( κ−n ) . In the same manner we would deduce that bm+1 ν = (2m+ 1)κ+ 2 n−1∑ s=0 µs+1 κs +O ( κ−n ) . This result also follows from the difference between amν and bm+1 ν being exponentially small as κ→∞ (see [24, § 28.8]). 10 Error analysis for the uniform asymptotic expansions of the Lamé functions We now use the results of Section 9 to obtain strict and realistic error bounds for the functions expansions derived in Section 8. Define the differential operator Lν := d2 dt2 +m+ 1 2 − t2 4 + 1 2κ (( −t2 ( 1 + k2 ) + k2t4 2κ ) d2 dt2 − t ( 1 + k2 − k2t2 κ ) d dt + h− κ(2m+ 1) ) , and consider t ∈ (−t∗, t∗). We have the truncated expansion corresponding to an even solution wmν,n ( t, k2 ) = Dm(t) n∑ s=0 As(t) κs +D′m(t) n∑ s=1 Bs(t) κs , such that we have the exact solution wmν ( t, k2 ) = wmν,n ( t, k2 ) + εmν,n ( t, k2 ) . (10.1) We define the remainder term Rmν,n ( t, k2 ) such that Lν ( wmν,n ( t, k2 )) = Rmν,n ( t, k2 ) and split the eigenvalue such that h 2κ = m+ 1 2 + n∑ s=1 µs κs + µ̃n+1 κn+1 , 26 K. Ogilvie and A.B. Olde Daalhuis and note we just proved that µ̃n+1 = O(1) as κ → ∞. Since the coefficients As(t) and Bs(t) satisfy (8.4) and (8.5) it follows that Rmν,n ( t, k2 ) = O ( κ−n−1 ) , as κ→∞. Applying Lν to (10.1) we obtain( εmν,n )′′ + ( m+ 1 2 − t2 4 ) εmν,n = 1 1− t2 2κ ( 1 + k2 − k2t2 2κ )[ t 2κ ( 1 + k2 − k2t2 κ )( εmν,n )′ + ( t2 ( −1− k2 + k2t2 2κ )( 2m+ 1− t2 2 ) − h+ κ (2m+ 1) ) εmν,n 2κ −Rmν,n ] , and denoting the right hand side of this equation Ωm ν,n ( t, k2 ) , by use of variation of parameters we have εmν,n ( t, k2 ) = √ π/2 m! ∫ t∗ t [ Dm(t)Dm(τ)−Dm(τ)Dm(t) ] Ωm ν,n ( τ, k2 ) dτ. In accordance with [22, § 6.10.2] we define J(τ) = 1, H(τ) = 1 + k2 − k2τ2 2κ and K (t, τ) = √ π/2 m! ( Dm(t)Dm(τ)−Dm(τ)Dm(t) ) , φ(τ) = −Rmν,n ( τ, k2 ) 1− τ2 2κH(τ) , ψ0(τ) = κ (2m+ 1)− h− τ2H(τ) ( 2m+ 1− τ2 2 ) 2κ− τ2H(τ) , ψ1(τ) = τ ( 1 + k2 − k2τ2 κ ) 2κ− τ2H(τ) , Φ(t) = ∫ t∗ t \|φ(τ) dτ \| , Ψ0(t) = ∫ t∗ t \|ψ0(τ) dτ \| , Ψ1(t) = ∫ t∗ t \|ψ1(τ) dτ \| . Since we consider t ∈ (−t∗, t∗) where t∗ = O(1) as κ → ∞, the error analysis is much simpler than the analysis Olver uses in [21] as we have \|K (t, τ) \| ≤ k0, and \|∂K (t, τ) /∂t\| ≤ k1, where k0 and k1 are O(1) as κ→∞. Thus again in accordance with [22, § 6.10.2] we define P0(t) = k0, Q(τ) = 1, P1(t) = k1 (10.2) (we do not define P2(t) as we do not need to bound \|∂2K(t, τ)/∂t2\| to carry out our analysis), and finally the constants κ̃ = 1, κ̃0 = k0, κ̃1 = k1. (10.3) Hence it follows from Theorem 10.1 in [22, § 6.10.2] that∣∣εmν,n(t, k2)∣∣ ≤ P0(t)κ̃Φ(t) exp [ κ̃0Ψ0(t) + κ̃1Ψ1(t) ] . (10.4) Since h− κ(2m+ 1) = O(1) for both h = amν and h = bm+1 ν we obtain Φ(t) = O ( κ−n−1 ) , Ψ0(t) = O ( κ−1 ) , Ψ1(t) = O ( κ−1 ) , (10.5) as κ→∞. Then substituting expressions from (10.5), (10.3) and the first of (10.2) into (10.4) we have εmν,n ( t, k2 ) = O ( κ−n−1 ) as κ→∞, for t ∈ (−t∗, t∗) . Asymptotics for the Lamé and Mathieu Functions 27 11 Identification of solutions Now we identify the solutions derived in Section 10 with the Lamé functions. We give the identification for t ∈ (−t∗, t∗) Ecmν ( z, k2 ) = Cmν ( Dm(t) n∑ s=0 As(t) κs +D′m(t) n∑ s=0 Bs(t) κs + 1 2 ( εmν,n ( t, k2 ) + (−1)mεmν,n ( −t, k2 ))) , Esm+1 ν ( z, k2 ) = Sm+1 ν √ 1− t2 2κ ( Dm(t) n∑ s=0 Ps(t) κs +D′m(t) n∑ s=0 Qs(t) κs + 1 2 ( εmν,n ( t, k2 ) + (−1)mεmν,n ( −t, k2 ))) , where the errors are defined according to h = amν or h = bm+1 ν respectively. We consider now just Cmν since we will obtain the same asymptotic expansion from Sm+1 ν by construction. To obtain an asymptotic expansion for these constants we consider with respect to (3.1) the integral (Cmν )2 ∫ K −K dn(z, k) ( Dm(t) ∞∑ s=0 As(t) κs +D′m(t) ∞∑ s=0 Bs(t) κs )2 dz. In the integral we let t = √ 2κ sn(z, k) and obtain (Cmν )2√ 2κ ∫ √2κ − √ 2κ 1√ 1− t2 2κ ( Dm(t) ∞∑ s=0 As(t) κs +D′m(t) ∞∑ s=0 Bs(t) κs )2 dt. Since the parabolic cylinder functions are exponentially small when the variable is large, we consider the integral from −∞ to ∞ and express the integral in the form (Cmν )2√ 2κ ∫ ∞ −∞ ∞∑ s=0 κ−s s∑ j=0 ( −1 2 j )( t2/2 )j s−j∑ i=0 ( Ai(t)As−j−i(t)D 2 m(t) + 2Ai(t)Bs−j−i(t)Dm(t)D′m(t) +Bi(t)Bs−j−i(t) ( D′m(t) )2) dt. Then from (3.1) we have the formal asymptotic expansions for the normalisation constants Cmν Sm+1 ν } ∼ (πκ)1/4√ 2m! ( 1 + ∞∑ s=1 ηs κs )−1/2 . To obtain analytic expressions for the ηs terms we need expressions for integrals of the form p(m,n) = ∫ ∞ −∞ tnD2 m(t)dt, q(m,n) = ∫ ∞ −∞ tn ( D′m(t) )2 dt, r(m,n) = ∫ ∞ −∞ tn+1Dm(t)D′m(t)dt. We have the identities p(m, 2n+ 1) = q(m, 2n+ 1) = r(m, 2n+ 1) = 0 for all m, n, and p(m, 0) = ∫ ∞ −∞ D2 m(t)dt = m! √ 2π and ∫ ∞ −∞ Dm(t)Dn(t)dt = 0 for m 6= n, 28 K. Ogilvie and A.B. Olde Daalhuis and using (9.5) we deduce the expression p(m, 2) = (2m+ 1)m! √ 2π. Using integration by parts we obtain the recurrence relation for p(m,n) p(m,n) = 2(n− 1)(2m+ 1) n p(m,n− 2) + (n− 3)(n− 2)(n− 1) n p(m,n− 4). We know p(m, 0) and p(m, 2) thus consequently can determine p(m,n) for any n recursively. Similarly using integration by parts we obtain expressions for the other integrals in terms of p(m,n): q(m,n) = n+ 3 4(n+ 1) p(m,n+ 2)− ( m+ 1 2 ) p(m,n), r(m,n) = 2m+ 1 n+ 2 p(m,n+ 2)− n+ 4 2(n+ 2)(n+ 3) p(m,n+ 4). 12 Summary of results for the Lamé functions and their respective eigenvalues Let κ = k √ ν(ν + 1) and t = √ 2κ sn (z, k). Then for m a non-negative integer and z = O ( κ−1/2 ) as κ→∞ Ecmν ( z, k2 ) = Cmν ( Dm(t) n∑ s=0 As(t) κs +D′m(t) n∑ s=0 Bs(t) κs +O ( κ−n−1 )) , Esm+1 ν ( z, k2 ) cn(z, k) = Sm+1 ν ( Dm(t) n∑ s=0 Ps(t) κs +D′m(t) n∑ s=0 Qs(t) κs +O ( κ−n−1 )) , where Cmν Sm+1 ν } ∼ (πκ)1/4√ 2m! ( 1 + ∞∑ s=1 ηs κs )−1/2 . Both As(t) and Ps(t) are even polynomials, and both Bs(t) and Qs(t) are odd polynomials. These polynomials are found recursively and we give here the first two terms: A0 = 1, A1 = k2 + 1 32 t2, B0 = 0, B1 = k2 + 1 16 ( t3 − (2m+ 1)t ) , P0 = 1, P1 = k2 + 9 32 t2, Q0 = 0, Q1 = B1, η1 = 3− k2 16 (2m+ 1). Correspondingly we have the eigenvalue expansion as κ→∞ amν bm+1 ν } = (2m+ 1)κ+ 2 n−1∑ s=0 µs+1 κs +O ( κ−n ) , as κ→∞. where the order term will be different in both cases. The µs terms are constant coefficients which depend on k and m, found recursively with the eigenfunction expansions. We give here the first two terms: µ1 = −k 2 + 1 8 ( 1 + 2m+ 2m2 ) , µ2 = −2m+ 1 32 (( k2 − 1 )2 ( 1 +m+m2 ) − 2k2 ) . These eigenvalue coefficients match the formal results given in [24, § 29.7]. Asymptotics for the Lamé and Mathieu Functions 29 13 A special case: Mathieu functions Mathieu’s equation Similarly to the Lamé case, if one considers the two term uniform approximation for the Mathieu functions you would observe that the oscillatory behaviour of the Mathieu functions happen in a shrinking neighbourhood of the z = π 2 as h → ∞. It can be shown that in a shrinking neighbourhood of this point, ζ behaves approximately like cos(z). Thus the variable in the parabolic cylinder function around this point behaves approximately like √ 2h cos(z). This would motivate a much simpler transformation. Letting t = 2 √ h cos z in (1.2) we present Mathieu’s equation in the algebraic form d2w dt2 + ( λ+ 2h2 4h − t2 4 ) w − 1 4h ( t2 d2 dt2 + t d dt ) w = 0, and since we proved that λ+ 2h2 4 − h ( m+ 1 2 ) = O(1), as h→∞ for the special eigenvalues λ which emit the Mathieu functions we have Mathieu’s equation in the form d2w dt2 + ( m+ 1 2 − t2 4 ) w + 1 h ( − t 2 4 d2 dt2 − t 4 d dt + λ+ 2h2 4 − h ( m+ 1 2 )) w = 0. In a similar manner to the previous sections coefficients in the expansions for the functions and eigenvalues can be computed using the same ansatz, although they would only make sense asymptotically in a shrinking neighbourhood of z = π 2 . However these are also realised by considering the Mathieu functions as a special case of the Lamé functions. Summary of results as a special case of Lamé’s equation We obtain rigorous uniform results for Mathieu’s equation, using the limiting arguments given in (1.3) and (1.4). Thus (reader can check the details, we will just summarise) letting t = 2 √ h cos z, for m ≥ 0 and z = π 2 +O(h−1/2) we have as h→∞ cem(h, am, z) = Cm ( Dm(t) n∑ s=0 As(t) hs +D′m(t) n∑ s=0 Bs(t) hs +O ( h−n−1 )) , sem+1(h, bm+1, z) sin z = Sm+1 ( Dm(t) n∑ s=0 Ps(t) hs +D′m(t) n∑ s=0 Qs(t) hs +O ( h−n−1 )) , (13.1) where Cm Sm+1 } ∼ ( πh 2(m!)2 )1/4 ( 1 + ∞∑ s=1 ηs hs )−1/2 . Both As(t) and Pm+1,s(t) are even polynomials, and both Bs(t) and Qm+1,s(t) are odd polyno- mials. These polynomials are found recursively and we give here the first few terms: A0 = 1, A1 = t2 26 , B0 = Q0 = 0, B1 = Q1 = t3 25 − (1 + 2m) t 25 , P0 = 1, P1 = 9t2 26 , 30 K. Ogilvie and A.B. Olde Daalhuis A2 = t8 213 − (1 + 2m) t6 211 + ( 9 + 10m+ 10m2 ) t4 212 + ( 5 + 6m− 12m2 − 8m3 ) t2 212 , B2 = t5 28 − (1 + 2m) 5t3 211 − ( 11 + 20m+ 20m2 ) t 211 , P2 = t8 213 − (1 + 2m) t6 211 + ( 113 + 10m+ 10m2 ) t4 212 + ( 5 + 6m− 12m2 − 8m3 ) t2 212 , Q2 = t5 27 − (1 + 2m) 13t3 211 − ( 11− 20m− 20m2 ) t 211 , η1 = 6m+ 3 32 . Correspondingly we have the eigenvalue expansion as h→∞ am bm+1 } = −2h2 + 4h n∑ s=0 µs hs +O ( h−n ) , (13.2) where the order term will be different in both cases. The µs terms are constant coefficients which depend on m, found recursively with the eigenfunction expansions. We give here the first few terms: µ0 = m+ 1 2 , µ1 = − 1 16 ( 1 + 2m+ 2m2 ) , µ2 = − 1 128 ( 1 + 3m+ 3m2 + 2m3 ) . These function and eigenvalue coefficients match the formal results given in [24, § 28.8] and [17, § 2], and various other papers discussed at the start of this paper. 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[29] Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. http://dx.doi.org/10.1017/S0370164600025864 http://dx.doi.org/10.1017/S0370164600025864 http://dx.doi.org/10.1017/S0370164600020058 http://dx.doi.org/10.1017/S0370164600020071 http://dx.doi.org/10.1007/978-3-662-00941-3 http://dx.doi.org/10.1002/mana.19660310108 http://dx.doi.org/10.1002/mana.19660310108 http://dx.doi.org/10.1002/mana.19660320106 http://dx.doi.org/10.1002/mana.19660320106 http://dx.doi.org/10.1002/mana.19660320305 http://dx.doi.org/10.1098/rsta.1975.0023 http://dx.doi.org/10.1098/rsta.1975.0023 http://dlmf.nist.gov/ http://dx.doi.org/10.1090/S0002-9947-1949-0033395-6 http://dx.doi.org/10.1090/S0002-9947-1949-0033395-6 http://dx.doi.org/10.1090/S0002-9947-1959-0103995 http://dx.doi.org/10.1090/S0002-9947-1959-0103995 http://dx.doi.org/10.1137/0145011 1 Introduction 2 Overview 3 Properties of Lamé and Mathieu functions I Uniform asymptotic approximations 4 Uniform approximations for the Lamé functions 5 An interlude: eigenvalues 6 Approximations in terms of parabolic cylinder D functions 7 A special case: Mathieu functions II Uniform asymptotic expansions 8 Uniform asymptotic expansions of the Lamé functions 9 An interlude: error analysis for the eigenvalue expansions 10 Error analysis for the uniform asymptotic expansions of the Lamé functions 11 Identification of solutions 12 Summary of results for the Lamé functions and their respective eigenvalues 13 A special case: Mathieu functions References

Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter

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