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mathieu_rad_se

Compute odd radial (modified) Mathieu function of the first kind 'Se' or its first derivative.

Calling Sequence

y = mathieu_rad_se( m, q, z, fun_or_der [, Brm] )

Parameters

m

the order of Mathieu function

q

the value of q parameter (can be positive, negative or complex number)

z

an argument

fun_or_der

calculate function (1) or first derivative (0)

Brm

expansion coefficients for odd angular and radial Mathieu function for the same m and q (optional, for speed).

y

value of the 'Se' function or its first derivative

Description

Mathieu_rad_se computes odd 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 (5)-(6) [2, p. 369]:

The formulations for derivatives were found symbolically from the equations above.

During unit-testing of this function values of odd radial (modified) function of the first kind  were compared to Tables 7-12 from [3]. The results are very close.

Examples

// Example: Se1 (for comparison with Fig. 4b, p. 236 of the article J. C. Gutiérrez-Vega,
//			et al. Mathieu functions, a visual approach [4]) and Se2 with first derivative
    f=scf(); mrgn = 50; font_sz = 4;
    sz = get(0, 'screensize_px') + [-1 mrgn-1 0 -2*mrgn ] ;
    f.figure_name = 'Se1 (for comparison with Fig. 4b, p. 236 of the article'+...
                    ' J. C. Gutiérrez-Vega, et al. Mathieu functions, a visual approach) and Se2 with first derivative';

    z = linspace(0, 2, 100);
    fun = 1; der = 0;

    m = 1;
    subplot(2,2,1);
        plot(z, mathieu_rad_se(m, 1, z, fun), 'r', z, mathieu_rad_se(m, 2, z, fun), 'b',...
        z, mathieu_rad_se(m, 3, z, fun), 'g')
        xgrid
        ty1 = '$Se_1(z,q)$'; 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(2,2,3)
        plot(z, mathieu_rad_se(m, 1, z, der), 'r-.', z, mathieu_rad_se(m, 2, z, der), 'b-.',...
        z, mathieu_rad_se(m, 3, z, der), 'g-.')
        xgrid
        ty1 = '$Se^{\ \prime}_1(z,q)$'; xlabel('$z$'); ylabel(ty1)
        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 = 2;
    subplot(2,2,2);
        plot(z, mathieu_rad_se(m, 1, z, fun), 'r', z, mathieu_rad_se(m, 2, z, fun), 'b',...
        z, mathieu_rad_se(m, 3, z, fun), 'g')
        xgrid
        ty1 = '$Se_2(z,q)$'; 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_se(m, 1, z, der), 'r-.', z, mathieu_rad_se(m, 2, z, der), 'b-.',...
        z, mathieu_rad_se(m, 3, z, der), 'g-.')
        xgrid
        ty1 = '$Se^{\ \prime}_2(z,q)$'; xlabel('$z$'); ylabel(ty1)
        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)];

See Also

Authors

R.Coisson and G. Vernizzi, Parma University

N. O. Strelkov, NRU MPEI

Bibliography

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).


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