Compute even radial (modified) Mathieu function of the first kind 'Ce' or its first derivative.
y = mathieu_rad_ce( m, q, z, fun_or_der [, Arm] )
the order of Mathieu function
the value of q parameter (can be positive, negative or complex number)
an argument
calculate function (1) or first derivative (0)
expansion coefficients for even angular and radial Mathieu function for the same m and q (optional, for speed).
value of the 'Ce' function or its first derivative
Mathieu_rad_ce computes even radial (modified) Mathieu function of the first kind
or its first derivative
using the following formulas [1, 2] from known
function (or its derivative).
Functions
and
are calculated by 20.6.15 [1] with multipliers
and
, described in (3)-(4) [2, p. 368-369]:
The formulations for derivatives were found symbolically from the equations above.
During unit-testing of this function values of even radial (modified) function of the first kind
were compared to Tables 1-6 from [3]. The results are very close.
// Example: Ce0 and Ce1 (for comparison with Fig. 4a, p. 236 of the article J. C. Gutiérrez-Vega, // et al. Mathieu functions, a visual approach [4]) with first derivative f=scf(); mrgn = 50; font_sz = 4; sz = get(0, 'screensize_px') + [-1 mrgn-1 0 -2*mrgn ] ; f.figure_name = 'Ce0 and Ce1 (for comparison with Fig. 4a, p. 236 of the article'+... ' J. C. Gutiérrez-Vega, et al. Mathieu functions, a visual approach) with first derivative'; z = linspace(0, 2, 100); fun = 1; der = 0; m = 0; subplot(2,2,1) plot(z, mathieu_rad_ce(m, 1, z, fun), 'r', z, mathieu_rad_ce(m, 2, z, fun), 'b',... z, mathieu_rad_ce(m, 3, z, fun), 'g') xgrid ty1 = '$Ce_0(z,q)$'; xlabel('$z$'); ylabel(ty1); legend('$q = 1$', '$q = 2$', '$q = 3$'); h = gca(); h.margins=[0.15 0.05 0.05 0.2]; 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(2,2,3) plot(z, mathieu_rad_ce(m, 1, z, der), 'r-.', z, mathieu_rad_ce(m, 2, z, der), 'b-.',... z, mathieu_rad_ce(m, 3, z, der), 'g-.') xgrid ty2 = '$Ce^{\ \prime}_0(z,q)$'; xlabel('$z$'); ylabel(ty2); legend('$q = 1$', '$q = 2$', '$q = 3$',pos=2); h = gca(); h.margins=[0.15 0.05 0.05 0.2]; 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; m = 1; subplot(2,2,2) plot(z, mathieu_rad_ce(m, 1, z, fun), 'r', z, mathieu_rad_ce(m, 2, z, fun), 'b',... z, mathieu_rad_ce(m, 3, z, fun), 'g') xgrid ty1 = '$Ce_1(z,q)$'; xlabel('$z$'); ylabel(ty1); legend('$q = 1$', '$q = 2$', '$q = 3$'); h = gca(); h.margins=[0.15 0.05 0.05 0.2]; 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(2,2,4) plot(z, mathieu_rad_ce(m, 1, z, der), 'r-.', z, mathieu_rad_ce(m, 2, z, der), 'b-.',... z, mathieu_rad_ce(m, 3, z, der), 'g-.') xgrid ty2 = '$Ce^{\ \prime}_1(z,q)$'; xlabel('$z$'); ylabel(ty2); legend('$q = 1$', '$q = 2$', '$q = 3$',pos=2); h = gca(); h.margins=[0.15 0.05 0.05 0.2]; 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_position=[sz(1) sz(2)]; f.figure_size=[sz(3) sz(4)]; | ![]() | ![]() |
R.Coisson and G. Vernizzi, Parma University
N. O. Strelkov, NRU MPEI
1. M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions, Dover, New York, 1965.
2. N. W. McLachlan. Theory and Application of Mathieu Functions, Oxford Univ. Press, 1947.
3. E. T. Kirkpatrick. Tables of Values of the Modified Mathieu Functions. Mathematics of Computation, Vol. 14, No. 70 (Apr., 1960), pp. 118-129. (online at AMS).
4. J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, "Mathieu functions, a visual approach", American Journal of Physics, 71 (233), 233-242. An introduction to applications (online at http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/modern/matheiu0.pdf).