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mathieu_rad_ce

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

Calling Sequence

y = mathieu_rad_ce( m, q, z, fun_or_der [, Arm] )

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)

Arm

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

y

value of the 'Ce' function or its first derivative

Description

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.

Examples

// 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)];

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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