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Mathieu Functions for Purely Imaginary Parameters View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 236 (2012) 4513–4524 Contents lists available at SciVerse ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Mathieu functions for purely imaginary parameters C.H. Ziener a,d,∗, M. Rückl b,c, T. Kampf c, W.R. Bauer d, H.P. Schlemmer a a German Cancer Research Center - DKFZ, Department of Radiology, Im Neuenheimer Feld 280, 69120 Heidelberg, Germany b Humboldt-University Berlin, Department of Physics, Newtonstraße 15, 12489 Berlin, Germany c Julius-Maximilians-Universität Würzburg, Experimentelle Physik 5, Am Hubland, 97074 Würzburg, Germany d Julius-Maximilians-Universität Würzburg, Medizinische Klinik I, Oberdürrbacher Straße 6, 97080 Würzburg, Germany article info a b s t r a c t Article history: For the Mathieu differential equation y00.x/CTa−2q cos.x/Uy.x/ D 0 with purely imaginary Received 23 December 2011 parameter q D is, the characteristic value a exhibits branching points. We analyze the Received in revised form 24 April 2012 properties of the Mathieu functions and their Fourier coefficients in the vicinity of the branching points. Symmetry relations for the Mathieu functions as well as the Fourier Keywords: coefficients behind a branching point are given. A numerical method to compute Mathieu Mathieu functions functions for all values of the parameter s is presented. Numerical computation ' 2012 Elsevier B.V. All rights reserved. 1. Introduction Many problems in physics can be expressed as partial differential equations and the main task that remains is to find mathematical solutions of these equations that fit the physical initial and boundary conditions. One of the most popular strategies to solve these partial differential equations is the separation of variables. In 1868 Mathieu analyzed the vibrations of an elliptically shaped membrane and reached a periodic differential equation [1] which is named after him [2]. In general the Mathieu differential equation is written in the form y00.x/ CTa − 2q cos.x/Uy.x/ D 0. Later more and more applications of the Mathieu equation to physical problems have been found [3] which led to a thorough analysis of this equation for the real parameter q. Comprehensive textbooks on Mathieu functions (the solution of the Mathieu differential equation) were written in [4–8]. In the well known pocketbook of Abramowitz and Stegun [9] comprehensive material on the Mathieu functions written by Blanch can be found. Newer results are presented by Wolf in the NIST Handbook of Mathematical functions [10] or in the recently published book of Mechel [11]. Mathieu functions for real parameters are visualized in the work of Vega et al. [12]. However, until now the Mathieu functions for a purely imaginary parameter q D is where i is the imaginary unit and s is real are rarely discussed. Mulholland and Goldstein [13] first analyzed this case and found that the first adjacent characteristic values coincide at s ≈ 1:47. A further study on purely imaginary parameters was given in [14]. In 1969 Blanch and Clemm published detailed tables of characteristic values for complex parameters [15]. However, to calculate the course of the Mathieu functions the knowledge of the characteristic values only is not sufficient. In general the Mathieu functions can be expressed in terms of a Fourier series. The characteristic values and the corresponding Fourier coefficients obey a recurrence relation which can be rewritten in terms of a Matrix eigenvalue ∗ Corresponding author at: German Cancer Research Center - DKFZ, Department of Radiology, Im Neuenheimer Feld 280, 69120 Heidelberg, Germany. Tel.: +49 6221 42 2524. E-mail address: [email protected] (C.H. Ziener). 0377-0427/$ – see front matter ' 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2012.04.023 4514 C.H. Ziener et al. / Journal of Computational and Applied Mathematics 236 (2012) 4513–4524 problem [16]. A comprehensive review on this issue is given in [17]. While the eigenvalues of the matrix are the characteristic values, the eigenvectors contain the Fourier coefficients of the Mathieu functions. This provides a valuable tool for numerical implementations of the Mathieu functions as fast and accurate solvers for eigenvalues and eigenvectors of matrix problems are available. Thus, this way of implementation is used in many numerical routines for computation of the Mathieu functions. However, in the case of a purely imaginary parameter the characteristic values become complex. Thus the natural ordering of real numbers cannot be used to order the characteristic values. Nevertheless, also the Fourier coefficients of the Mathieu functions become complex. This problem is not well discussed in the literature and leads to problems in current numerical implementations of the Mathieu functions. Especially the exact knowledge of the Fourier coefficients and their behavior in the vicinity of the branching points is mandatory to compute the Mathieu functions for purely imaginary parameters accurately. In this work we give a detailed analysis of Mathieu functions for purely imaginary parameters with emphasis on the characteristic values and the Fourier coefficients. In Section2, we recover the general theory of Mathieu functions including the matrix representation of the corresponding recurrence relation. In general, the matrix is tridiagonal and infinite. To obtain analytical approximations for the first two characteristic values and Fourier coefficients we consider a truncated 2×2 matrix system as the simplest nontrivial case. Furthermore, we rewrite the well known approximations for the Mathieu functions and their characteristic values for small purely imaginary parameters, which are valid before the branching points. This enables us to determine the correct sign of the eigenvector as well as the correct order of the characteristic values and the Fourier coefficients behind the branching point. In Section3 we use this results to develop a numerical algorithm which correctly implements the Mathieu functions and the corresponding characteristic values as well as the Fourier coefficients. Since the main idea of the implementation is the same for all four kinds of the Mathieu functions, we focus on the even π-periodic Mathieu functions. Summary and conclusions are given in Section4. 2. Theory In this section, we summarize the formulas for the calculation of the even π-periodic Mathieu functions ce2m(φ; q/, which satisfy the Mathieu differential equation @φφce2m(φ; q/ C [a2m.q/ − 2q cos.2φ/] ce2m(φ; q/ D 0: (1) Since we focus on purely imaginary parameters, it is reasonable to introduce the quantity q D is; (2) where s > 0 is a purely real value. We recall the general theory of Mathieu functions in terms of the parameter s. Approximations for the characteristic values and Fourier coefficients are given. We present symmetry relations for the Fourier coefficients and the Mathieu functions. 2.1. Mathieu functions and its Fourier expansion The even π-periodic Mathieu functions for a purely imaginary parameter q D is satisfy the differential equation @φφce2m(φ; is/ C [a2m.is/ − 2is cos.2φ/] ce2m(φ; is/ D 0; (3) where a2m are the corresponding characteristic values. The Mathieu functions obey the orthogonality relation Z 2π dφ ce2m(φ; is/ce2m0 (φ; is/ D πδmm0 : (4) 0 The Mathieu functions can be expressed in the Fourier series 1 D X .2m/ ce2m(φ; is/ A2r cos.2rφ/; (5) rD0 .2m/ where the Fourier coefficients A2r also depend on the parameter s. Introducing the Fourier expansion given in Eq. (5) in the considered Mathieu differential equation (3), one yields the following recurrence relation: .2m/ D .2m/ a2mA0 isA2 ; (6) h i − .2m/ D .2m/ C .2m/ [a2m 4] A2 is 2A0 A4 and (7) h i − 2 .2m/ D .2m/ C .2m/ ≥ a2m 4r A2r is A2r−2 A2rC2 for r 2: (8) C.H. Ziener et al. / Journal of Computational and Applied Mathematics 236 (2012) 4513–4524 4515 Fig. 1. Real part and imaginary part of the characteristic values in dependence on the parameter s. The results were obtained by solving Eq. (9). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) For numerical evaluation it is reasonable to rewrite the recurrence relation as an eigenvalue problem of an infinite tridiagonal matrix in the form [17]: p p p 0 .2m/1 0 .2m/1 0 0 2is 1 2A0 2A0 p .2m/ .2m/ A A B 2is 4 is C B 2 C B 2 C B C B .2m/ C B .2m/ C is 16 is B A4 C B A4 C B C B .2m/ C B .2m/ C B is 36 is C B A C B A C B C 6 D a 6 ; (9) B : : : C B : C 2m B : C B :: :: :: C B : C B : C B C B : C B : C 2 B .2m/ C B .2m/ C B is 4r is C B A C B A C @ A @ 2r A @ 2r A :: : : : : : .2m/ where the eigenvalues are the characteristic values a2m. The eigenvectors contain the Fourier coefficients A2r . For the numerical computation of the Mathieu functions it is advantageous to solve the matrix equation (9). Numerical methods to solve the eigenvalue problem of an infinite complex symmetric tridiagonal matrix approximately are available [16]. Due to the orthogonality relation, the Fourier coefficients have to obey the normalization 1 h i2 h i2 .2m/ C X .2m/ D 2 A0 A2r 1: (10) rD1 A further orthonormality relationship between the Fourier coefficients is given in [18] 1 X .2m/ .2m/ δ0pδ0r A A D δpr − for p; r D 0; 1; 2;:::: (11) 2p 2r 2 mD0 Especially for p D 0 D r we obtain 1 2 X h .2m/i 1 A D (12) 0 2 mD0 and for p D r ≥ 1: 1 h i2 X .2m/ D A2r 1: (13) mD0 These relations can be used to estimate the number of coefficients which are necessary for a required numerical accuracy.
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