Compute even radial (modified) Mathieu function of the second kind 'Fey' or its first derivative.
y = mathieu_rad_fey( 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 'Fey' function or its first derivative
Mathieu_rad_fey computes even radial (modified) Mathieu function of the second kind
or its first derivative
using the following formulas [1, 2] from known
function (or its derivative).
Functions
and
are calculated by generalization of 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 second kind
were compared with calculated from Tables 14.3, 14.5, 14.9 from [3]. The results are very close.
// Example: Fey0 and Fey1 (for comparison with Fig. 4c, 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 = 'Fey0 and Fey1 (for comparison with Fig. 4c, 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_fey(m, 1, z, fun), 'r', z, mathieu_rad_fey(m, 2, z, fun), 'b',... z, mathieu_rad_fey(m, 3, z, fun), 'g') xgrid ty1 = '$Fey_0(z,q)$'; 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_fey(m, 1, z, der), 'r-.', z, mathieu_rad_fey(m, 2, z, der), 'b-.',... z, mathieu_rad_fey(m, 3, z, der), 'g-.') xgrid ty2 = '$Fey^{\ \prime}_0(z,q)$'; xlabel('$z$'); ylabel(ty2); legend('$q = 1$', '$q = 2$', '$q = 3$',pos=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; m = 1; subplot(2,2,2); plot(z, mathieu_rad_fey(m, 1, z, fun), 'r', z, mathieu_rad_fey(m, 2, z, fun), 'b',... z, mathieu_rad_fey(m, 3, z, fun), 'g') xgrid ty1 = '$Fey_1(z,q)$'; xlabel('$z$'); ylabel(ty1); legend('$q = 1$', '$q = 2$', '$q = 3$',pos=4); 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_fey(m, 1, z, der), 'r-.', z, mathieu_rad_fey(m, 2, z, der), 'b-.',... z, mathieu_rad_fey(m, 3, z, der), 'g-.') xgrid ty2 = '$Fey^{\ \prime}_1(z,q)$'; xlabel('$z$'); ylabel(ty2); legend('$q = 1$', '$q = 2$', '$q = 3$',pos=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; 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. S. Zhang and J. Jin. Computation of Special Functions. New York, Wiley, 1996.
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).