Compute expansion coefficients 'Arm' and eigenvalue 'am' for even angular and radial Mathieu functions.
[ Arm, am ] = mathieu_Arm( m, q )
the order of Mathieu function
the value of q parameter (can be positive, negative or complex number)
expansion coefficients for even Mathieu functions
characteristic value for even Mathieu functions
Mathieu_Arm computes expansion coefficients and eigenvalue for even angular and radial Mathieu functions.
This function formes tri-diagonal matrix as an equaivalent to known recurrence relations [1-2]:
The size of matrix is calculated from values of q and m by the formulas from Zhang & Jin [3].
The matrixes have the following structure [4-6]:
and
This function is based on fcoef1 program, developed by Da Ma [7]. Using Scilab spec function mathieu_Arm founds at once expansion coefficients
and eigenvalue
.
During unit-testing of this function expansion coefficients and characteristic values were compared to well-known tables: Tables 20.1, 20.2 from [2], Tables 14.1, 14.3 from [3]. The results are very close.
// Example: Expansion coefficients Arm as continous functions of q f=scf(); mrgn = 50; font_sz = 4; f.figure_name='Expansion coefficients Arm as continous functions of q'; sz = get(0, 'screensize_px') + [-1 mrgn-1 0 -2*mrgn ] ; q = linspace(0, 40, 40); Arm2n = zeros(6, length(q)); Arm2n1 = zeros(6, length(q)); for i=1:length(q) A2ns = mathieu_Arm(0, q(i)); A2n1s = mathieu_Arm(1, q(i)); Arm2n(1:6, i) = A2ns(1:6); Arm2n1(1:6, i) = A2n1s(1:6); end subplot(1,2,1) plot(q', Arm2n'); legend('$A_0^{(0)}$', '$A_2^{(0)}$', '$A_4^{(0)}$', '$A_6^{(0)}$', '$A_8^{(0)}$', '$A_{10}^{(0)}$'); xlabel('$q$'); ylabel('$A_m^{(0)}$'); xgrid h = gca(); h.margins=[0.15 0.05 0.05 0.12]; h.font_size = font_sz - 1; h.x_label.font_size=font_sz; h.y_label.font_size=font_sz; h.children(1).font_size = font_sz; subplot(1,2,2); plot(q', Arm2n1'); legend('$A_1^{(1)}$', '$A_3^{(1)}$', '$A_5^{(1)}$', '$A_7^{(1)}$', '$A_9^{(1)}$', '$A_{11}^{(1)}$'); xlabel('$q$'); ylabel('$A_m^{(1)}$'); xgrid h = gca(); h.margins=[0.15 0.05 0.05 0.12]; h.font_size = font_sz - 1; h.x_label.font_size=font_sz; h.y_label.font_size=font_sz; h.children(1).font_size = font_sz; f.figure_size=[sz(3) sz(4)]; f.figure_position=[sz(1) sz(2)]; | ![]() | ![]() |
R.Coisson and G. Vernizzi, Parma University
N. O. Strelkov, NRU MPEI
1. N. W. McLachlan, Theory and Application of Mathieu Functions, Oxford Univ. Press, 1947.
2. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
3. S. Zhang and J. Jin. Computation of Special Functions. New York, Wiley, 1996.
4. J. J. Stamnes and B. Spjelkavik. New method for computing eigenfunctions (Mathieu functions) for scattering by elliptical cylinders. Pure Appl. Opt. 4 251–62, 1995.
5. L. Chaos-Cador, E. Ley-Koo. Mathieu functions revisited: matrix evaluation and generating functions. Revista Mexicana de Fisica, Vol. 48, p.67-75, 2002.
6. Julio C. Gutiérrez-Vega, “Formal analysis of the propagation of invariant optical fields in elliptic coordinates,”Ph. D. Thesis, INAOE, México, 2000. (online at http://homepages.mty.itesm.mx/jgutierr/).
7. Da Ma. General Mathieu functions with arbitrary parameters V1.0 - File Exchange - MATLAB Central.