Compute odd radial (modified) Mathieu function 'Ms' or its first derivative (kinds 1 and 2).
y = mathieu_rad_ms( m, q, z, fun_or_der[, kind, 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)
kind of function: 1 (default) for besselj series, 2 for bessely series - optional
expansion coefficients for odd angular and radial Mathieu function for the same m and q (optional, for speed).
value of the 'Mc' function (kind=1 or kind=2) or its first derivative
Mathieu_rad_ms computes odd radial (modified) Mathieu function or its first derivative (kinds 1 and 2) using the following formulas [1].
Functions
and
are calculated by respectively 20.6.10 and 20.6.9 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 odd radial (modified) function and its derivative were compared to the tables:
What tested | Table(s) |
---|---|
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Table III [2], 14.8(a) [4] |
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Table IV [2], 14.8(b) [4] |
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Table III [3], 14.10(a) [4] |
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Table IV [3], 14.10(b) [4] |
// Example: Ms1(2) for comparison with Fig. 20.13 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 = 'Ms1(2) for comparison with Fig. 20.13 of Abramowitz-Stegun'; z = linspace(0, 2, 100); fun = 1; m = 1; kind = 2; plot(z, sqrt(%pi/2)*mathieu_rad_ms(m, 0.25, z, fun, kind), z, sqrt(%pi/2)*mathieu_rad_ms(m, 0.75, z, fun, kind), z, ... sqrt(%pi/2)*mathieu_rad_ms(m, 1.5, z, fun, kind),z, sqrt(%pi/2)*mathieu_rad_ms(m, 3.75, z, fun, kind),... z, sqrt(%pi/2)*mathieu_rad_ms(m, 5.25, z, fun, kind)) xgrid ty1 = '$\sqrt{{\pi}/2} Ms^{(2)}_1(z,q)$'; xlabel('$z$'); ylabel(ty1); legend('$q = 0.25$','$q = 0.75$','$q = 1.5$','$q = 3.75$','$q = 5.25$',pos=4); h = gca(); h.margins=[0.15 0.05 0.05 0.12]; 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. 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.