Compute odd angular Mathieu function 'se' or its first derivative.
y = mathieu_ang_se( m, q, z, fun_or_der [, Brm] )
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 odd angular and radial Mathieu function for the same m and q (optional, for speed).
value of the 'se' function or its first derivative
Mathieu_ang_se computes odd angular Mathieu function
or its first derivative
using the following formulas [1]:
functions -
derivatives -
During unit-testing of this function values of odd angular function
and its derivative
were compared to well-known tables: Tables 20.1 from [2], Tables 14.6(a), 14.6(b) from [3]. The results are very close.
// Example: Odd periodic Mathieu functions and their first derivatives f=scf(); mrgn = 50; font_sz = 4; sz = get(0, 'screensize_px') + [-1 mrgn-1 0 -2*mrgn ] ; f.figure_name = 'Odd periodic Mathieu functions and their first derivatives'; z = linspace(0, 2*%pi, 100); fun = 1; der = 0; m = 1; subplot(2,2,1) plot(z, mathieu_ang_se(m,-3,z,fun), z, mathieu_ang_se(m,-1,z,fun),... z, mathieu_ang_se(m,0,z,fun), z, mathieu_ang_se(m,1,z,fun), z, mathieu_ang_se(m,3,z,fun)) xgrid legend('$q = -3$', '$q = -1$', '$q = 0$', '$q = 1$', '$q = 3$'); ty = '$se_1(z,q)$'; xlabel('$z$'); ylabel(ty); 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_ang_se(m,-3,z,der), z, mathieu_ang_se(m,-1,z,der),... z, mathieu_ang_se(m,0,z,der), z, mathieu_ang_se(m,1,z,fun), z, mathieu_ang_se(m,3,z,der)) xgrid legend('$q = -3$', '$q = -1$', '$q = 0$', '$q = 1$', '$q = 3$'); ty = '$se^{\ \prime}_1(z,q)$'; xlabel('$z$'); ylabel(ty); 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 = 2; subplot(2,2,2) plot(z, mathieu_ang_se(m,-3,z,fun), z, mathieu_ang_se(m,-1,z,fun),... z, mathieu_ang_se(m,0,z,fun), z, mathieu_ang_se(m,1,z,fun), z, mathieu_ang_se(m,3,z,fun)) xgrid legend('$q = -3$', '$q = -1$', '$q = 0$', '$q = 1$', '$q = 3$'); ty = '$se_2(z,q)$'; xlabel('$z$'); ylabel(ty); 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_ang_se(m,-3,z,der), z, mathieu_ang_se(m,-1,z,der),... z, mathieu_ang_se(m,0,z,der), z, mathieu_ang_se(m,1,z,fun), z, mathieu_ang_se(m,3,z,der)) xgrid legend('$q = -3$', '$q = -1$', '$q = 0$', '$q = 1$', '$q = 3$'); ty = '$se^{\ \prime}_2(z,q)$'; xlabel('$z$'); ylabel(ty); 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. 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.