A note on Mathieu functions
Glasgow mathematical journal, Tome 2 (1957) no. 3, pp. 132-134
Voir la notice de l'article provenant de la source Cambridge University Press
The Mathieu functions of integral order [1] are the solutions with period π or 2π of the equationThe eigenvalues associated with the functions ceN and seN, where N is a positive integer, denoted by aN and bN respectively, reduce toaN = bN = N2when q is zero. The quantities aN and bN can be expanded in powers of q, but the explicit construction of high order coefficients is very tedious. In some applications the quantity of most interest is aN – bN, which may be called the “width of the unstable zone“. It is the object of this note to derive a general formula for the leading term in the expansion of this quantity, namelySuppose first that N is an odd integer. Then there is an expansionwhereThese functions π satisfyandOn Substituting (3) in (1), one obtains the algebraic equationwhereExplicitly,{11} = q{lm} = 0 otherwise.
Bell, M. A note on Mathieu functions. Glasgow mathematical journal, Tome 2 (1957) no. 3, pp. 132-134. doi: 10.1017/S204061850003358X
@article{10_1017_S204061850003358X,
author = {Bell, M.},
title = {A note on {Mathieu} functions},
journal = {Glasgow mathematical journal},
pages = {132--134},
year = {1957},
volume = {2},
number = {3},
doi = {10.1017/S204061850003358X},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S204061850003358X/}
}
[1] 1.McLachlan, N. W., The theory and application of Mathieu functions (Oxford, 1947). Google Scholar
[2] 2.Brillouin, L., J. de Phys., 4 (1933), 1. Google Scholar
[3] 3.Wigner, E. P., Math. u. naturw. Anz. Ungar. Akad. Wiss., 53 (1935), 475. Google Scholar
Cité par Sources :