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mathieu_ang_se

Compute odd angular Mathieu function 'se' or its first derivative.

Calling Sequence

y = mathieu_ang_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_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.

Examples

// 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)];

See Also

Authors

R.Coisson and G. Vernizzi, Parma University

N. O. Strelkov, NRU MPEI

Bibliography

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.


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