A note on Mathieu functions

Abstract

The Mathieu functions of integral order [1] are the solutions with period π or 2π of the equation The eigenvalues associated with the functions ce N and se N , where N is a positive integer, denoted by a N and b N respectively, reduce to a N = b N = N 2 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 N , 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 Suppose first that N is an odd integer. Then there is an expansion where These functions π satisfy and On Substituting (3) in (1), one obtains the algebraic equation where Explicitly, {11} = q { lm } = 0 otherwise.