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mathieu_rad_mc

Compute even radial (modified) Mathieu function 'Mc' or its first derivative (kinds 1 and 2).

Calling Sequence

y = mathieu_rad_mc( m, q, z, fun_or_der[, kind, 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)

kind

kind of function: 1 (default) for besselj series, 2 for bessely series - optional

Arm

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

y

value of the 'Mc' function (kind=1 or kind=2) or its first derivative

Description

Mathieu_rad_mc computes even radial (modified) Mathieu function or its first derivative (kinds 1 and 2) using the following formulas [1].

Functions  and  are calculated by respectively 20.6.7 and 20.6.8 from [1] with multipliers, described by 20.4.13 [1] (so we always choose the numerically largest coefficients of the set  or  ).

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

During unit-testing of this function values of even radial (modified) function and its derivative were compared to the tables:
What tested Table(s)
Table I [2], 14.7(a) [4]
Table II [2], 14.7(b) [4]
Table I [3], 14.9(a) [4]
Table II [3], 14.9(b) [4]
The results are very close.

Examples

// Example: Mc0(1) and its derivative for comparison with Fig. 20.11-12 of Abramowitz-Stegun [1]
    f=scf(); mrgn = 50; font_sz = 4;
    sz = get(0, 'screensize_px') + [-1 mrgn-1 0 -2*mrgn ] ;
    f.figure_name = 'Mc0(1) and its derivative for comparison with Fig. 20.11-12 of Abramowitz-Stegun';

    z = linspace(0, 2, 100);
    fun = 1; der = 0;
    
    m = 0;
    subplot(1,2,1)
        plot(z, sqrt(%pi/2)*mathieu_rad_mc(m, 0.25, z, fun), z, sqrt(%pi/2)*mathieu_rad_mc(m, 0.75, z, fun), z, ...
        sqrt(%pi/2)*mathieu_rad_mc(m, 1.5, z, fun),z, sqrt(%pi/2)*mathieu_rad_mc(m, 3.75, z, fun),...
        z, sqrt(%pi/2)*mathieu_rad_mc(m, 5.25, z, fun))
        xgrid
        ty1 = '$\sqrt{{\pi}/2} Mc^{(1)}_0(z,q)$'; xlabel('$z$'); ylabel(ty1);
        legend('$q = 0.25$','$q = 0.75$','$q = 1.5$','$q = 3.75$','$q = 5.25$');
        
        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(1,2,2)
        plot(z, sqrt(%pi/2)*mathieu_rad_mc(m, 0.25, z, der), z, sqrt(%pi/2)*mathieu_rad_mc(m, 0.75, z, der), z,...
        sqrt(%pi/2)*mathieu_rad_mc(m, 1.5, z, der),z, sqrt(%pi/2)*mathieu_rad_mc(m, 3.75, z, der),...
        z, sqrt(%pi/2)*mathieu_rad_mc(m, 5.25, z, der))
        xgrid
        ty1 = '$\sqrt{{\pi}/2} Mc^{\ \prime\ (1)}_0(z,q)$'; xlabel('$z$'); ylabel(ty1);
        legend('$q = 0.25$','$q = 0.75$','$q = 1.5$','$q = 3.75$','$q = 5.25$', 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. G. Blanch and D. S. Clemm. Tables relating to the radial Mathieu functions. Volume 1. Functions of the First Kind. ARL, US Air Force. 1963. (online at HathiTrust).

3. G. Blanch and D. S. Clemm. Tables relating to the radial Mathieu functions. Volume 2. Functions of the Second Kind. ARL, US Air Force. 1963. (online at HathiTrust).

4. S. Zhang and J. Jin. Computation of Special Functions. New York, Wiley, 1996.


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