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Mathieu Functions Revisited: Matrix Evaluation and Generating Functions

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Mathieu Functions Revisited: Matrix Evaluation and Generating Functions ENSENANZA˜ REVISTA MEXICANA DE FISICA´ 48 (1) 67–75 FEBRERO 2002 Mathieu functions revisited: matrix evaluation and generating functions L. Chaos-Cador1 and E. Ley-Koo2 Instituto de F´ısica, Universidad Nacional Autonoma´ de Mexico´ Apartado postal 20-364, 01000 Mexico,´ D.F., Mexico e-mail: 1lorea@fisica.unam.mx, 2eleykoo@fisica.unam.mx Recibido el 24 de octubre de 2001; aceptado el 13 de noviembre de 2001 An updated didactical approach to Mathieu functions is presented as an alternative to the orthodox treatment. The matrix evaluation of the angular Mathieu functions is not only pertinent, but long over due, after seventy five years of matrix quantum mechanics and with the availability of computing tools. Plane waves are identified as generating functions of the Mathieu functions, showing their explicit expansions. Some mathematical and physical applications are illustrated. Keywords: Mathieu functions; matrix evaluation; generating functions Se presenta un enfoque actualizado y didactico´ para las funciones de Mathieu como una alternativa a la presentacion´ ortodoxa. La evaluacion´ matricial de las funciones de Mathieu angulares es no solamente pertinente sino retrasada despues´ de setenta y cinco anos˜ de mecanica´ cuantica´ matricial y con la disponibilidad de herramientas computacionales. Se identifican las ondas planas como funciones generadoras de las funciones de Mathieu, mostrando sus desarrollos expl´ıcitos. Se ilustran algunas aplicaciones matematicas´ y f´ısicas. Descriptores: Funciones de Mathieu; evaluacion´ matricial; funciones generadoras PACS: 02.30.Jr; 02.30.Hq; 02.30.Gp 1. Introduction Recent physical applications of the Mathieu functions have been discussed by Ruby [17]. His opening sentence Mathieu functions were first investigated by that author in is “with few exceptions, notably Morse and Feshbach, and his “Memoire sur le movement vibratoire d’une membrane Mathews and Walker, most authors of textbooks on mathe- de forme elliptique” in 1868 [1], evaluating the lowest order matics for science and engineering choose to omit any dis- characteristic numbers and corresponding angular functions cussion of the Mathieu equation”. His review of the solution in ascending powers of the intensity parameter. A decade of the Mathieu equation follows the orthodox treatment [5], later, Heine defined the periodic Mathieu angular functions involving the evaluation of the characteristic numbers from of integer order as Fourier cosine and sine series, without the continued fraction equation and then the evaluation of the evaluating the corresponding coefficients; obtained a tran- corresponding Fourier coefficients. scendental equation for the characteristic numbers expressed The authors of the present article have become involved in terms of an infinite continued fraction; and also demons- with Mathieu functions in connection with two recent works. trated that one set of periodic functions of integer order The first one was the review of the doctoral thesis “Formal could be expanded in a series of Bessel functions [2]. Flo- analysis of the propagation of invariant optical fields with quet published his mathematical work “Sur les equations elliptic symmetries” [18], in which the orthodox method of differentielles´ lineaires´ a´ coefficients periodiques” containing evaluating the Mathieu functions was also used. The second the theorem named after him [3], and Hill his memoir “On the one is about the bidimensional hydrogen molecular ion con- path of motion of the lunar perigee” introducing the determi- fined inside an ellipse, which can be solved in a matrix form nant named after him [4], both works having played relevant following the methods of quantum mechanics [19]. roles in subsequent investigations of the Mathieu functions The last two paragraphs contain the motivation to revisit and their extensions. the Mathieu functions. Section 2 is devoted to the separation For the purposes of the revisiting in the title of this pa- of the Helmholtz equation in elliptical coordinates leading per, reference is made to Chapter 20 on Mathieu functions to the canonical and modified forms of the Mathieu equa- by G. Blanch in Ref. 5. The reader is also referred to some tions. The orthodox method to construct the angular Mathieu of the classical books on mathematical methods [6–10] and functions is briefly described as a point of comparison. Its specific books on Mathieu functions [11–16] whose authors extension for the construction of the radial Mathieu func- made original contributions to establish mathematical pro- tions is also mentioned. In Sec. 3, their matrix evaluation is perties, to develop physical applications and to evaluate nu- formulated, recognizing that the output from the matrix dia- merical values of such functions. Humbert’s monograph con- gonalization provides simultaneously the values of both cha- tains the bibliography up to 1924 [11], and McLachlan’s book racteristic numbers and Fourier coefficients. In Sec. 4 another provides a historical introduction and updated bibliography gap in the study of the Mathieu functions is also filled in. It through 1947 [15]. is well known that the majority of the special functions can 68L.CHAOS-CADORANDE.LEY-KOO be obtained from generating functions [5], but the Mathieu The substitution of Eq. (6) or (7) in the differential Eq. (5) functions have been so far an exception. We have identified leads to three-term recurrence relations for the Fourier coeffi- 2r+p 2r+p that plane waves are also their generating functions showing cients A2s+p(q) or B2s+p (q). Their explicit forms appear as the corresponding series expansions in products of radial and Eqs. 20:2:5–20:2:11 in Ref. 5. The ratios of consecutive coef- angular Mathieu functions. In the final section a discussion is ficients of the same parity then satisfy two types of continued made of illustrative mathematical and physical applications fractions, appearing as Eqs. 20:2:12–20:2:20 in Ref. 5. The of some of our results. convergence of the Fourier series of Eqs. (6) and (7) requires that the coefficients Am and Bm vanish as m ! 1, which 2. Helmholtz equation in elliptical coordinates implies the vanishing of the respective infinite continued frac- tions. The corresponding roots are the characteristic numbers Elliptical coordinates (0 · u < 1; 0 · v · 2¼), are de- a2r; a2r+1; b2r; b2r+1 from Eqs. 20:2:21–20:2:24 in Ref. 5. fined through the transformation equations to Cartesian coor- Ince developed the numerical methods to evaluate the char- dinates: acteristic numbers accurately, and from them the ratios of the Fourier coefficients, and finally the coefficients themselves x = c cosh u cos v; taking into account the normalization of Eqs. (6) and (7) [20]. y = c sinh u sin v; (1) Ince’s method became the orthodox method to evaluate Math- ieu functions. where c is the common semifocal distance of confocal el- The orthonormalization for the Mathieu functions in lipses with eccentricities 1= cosh u, and confocal hyperbolas Eqs. (6) and (7) is the same as that of the cosine and sine with eccentricities 1= cos v. functions, namely, Helmholtz equation in these coordinates takes the form Z 2¼ ½ · ¸ ¾ 2 @2 @2 cem(v; q)cen(v; q) dv = ¼±mn; (8) 2 0 2 2 + 2 +k Ã(u; v)=0; (2) c (cosh 2u¡cos 2v) @u @v Z 2¼ and is separable. In fact, it admits factorizable solutions sem(v; q)sen(v; q) dv = ¼±mn: (9) 0 Ã(u; v) = U(u)V (v); (3) This normalization convention leads to the conditions on the respective Fourier coefficients: where the respective factors satisfy the ordinary differential equations 2r 2 2r 2 2r 2 2(A0 ) + (A2 ) + ::: + (A2s) + ::: = 1; (10) · ¸ d2 2r+1 2 2r+1 2 2r+1 2 + 2q cosh 2u U(u) = aU(u); (4) (A1 ) + (A3 ) + ::: + (A2s+1) + ::: = 1; (11) du2 2r 2 2r 2 2r 2 · ¸ (B2 ) + (B4 ) + ::: + (B2s ) + ::: = 1; (12) d2 ¡ 2q cos 2v V (v) = ¡aV (v): (5) 2r+1 2 2r+1 2 2r+1 2 dv2 (B1 ) + (B3 ) + ::: + (B2s+1 ) + ::: = 1: (13) Here q = k2c2=4 is the intensity parameter and a is the se- Notice the factor of 2 in the first term of Eq. (10), associ- paration constant. Equation (5) is the canonical form of Ma- ated with the normalization to 2¼ of cos mv for m = 0. thieu’s equation, and Eq. (4) is the modified form. They are In the limit in which q ! 0, Eqs. (6) and (7) guarantee connected through the substitutions v ! iu and V ! U, so that the Mathieu functions tend to the respective cosine and that the elliptical or radial Mathieu functions U can be ob- sine functions of the same order. Mathieu made the analy- tained by analytical continuation of the hyperbolic or angular sis of the vibrations of an elliptic membrane for small values Mathieu functions V . of q obtaining power series in this parameter for the charac- The periodic angular Mathieu functions can be expressed teristic values and the periodic functions, corresponding to as Fourier series of four different types: Eqs. 20:2:25–20:2:26 and 20:2:27–20:2:28 in Ref. 5, respec- tively. In the context of quantum mechanics these expansions X1 £ ¤ ce (v; q) = A2r+p(q) cos (2s + p)v ; (6) can be understood as the result of applying perturbation the- 2r+p 2s+p ory in different orders of approximation, taking the circular s=0 membrane as the nonperturbed starting problem. X1 £ ¤ se (v; q) = B2r+p(q) sin (2s + p)v ; (7) The radial Mathieu functions are obtained by analytical 2r+p 2s+p continuation from those of Eqs. (6) and (7): s=0 when p = 0 their period is ¼, and when p = 1 their period Ce2r+p(z; q) = ce2r+p(iz; q) is 2¼. The characteristic numbers for the even solutions of X1 £ ¤ Eq.
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