A note on Mathieu functions

Abstract

The Mathieu functions of integral order [1] are the solutions with period n or 2ir of the equation^ 0(1)The eigenvalues associated with the functions ce lV and se iV , where N is a positive integer, denoted by a N and b N respectively, reduce to when q is zero. The quantities a N and b N 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 a N -b$, which may be called the " width of the unstable zone " . It is the object of this note to derive a general formula for the leading term in the expansion of this quantity, namely qSuppose first that N is an odd integer. Then there is an expansion The equation (7) is solved by the method, well-known in mathematical physics, of Brillouin [2] and Wigner [3]. Imposing the normalisation a$ = l (to all orders in q), one may rewrite (7) as at https://www.cambridge.org/core/terms. https://doi

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