exist integral relations between functions of the first and second kind, for instance: valid for any complex and real . Asymptotic expansions The following asymptotic expansions hold for , , , and : Thus, the modified Mathieu functions decay exponentially for large real argument. Similar asymptotic expansions can be written down for and ; these also decay exponentially for large real argument. For the even and odd periodic Mathieu functions and the associated characteristic numbers one can also derive asymptotic expansions for large . For the characteristic numbers in particular, one has with approximately an odd integer, i.e. Observe the symmetry here in replacing and by and , which is a significant feature of the expansion. Terms of this expansion have been obtained explicitly up to and including the term of order . Here is only approximately an odd integer because in the limit of all minimum segments of the periodic potential become effectively independent harmonic oscillators (hence an odd integer). By decreasing , tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers (in quantum mechanics called eigenvalues) corresponding to even and odd periodic Mathieu functions. This splitting is obtained with boundary conditions (in quantum mechanics this provides the splitting of the eigenvalues into energy bands). The boundary conditions are: Imposing these boundary conditions on the asymptotic periodic Mathieu functions associated with the above expansion for one obtains The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e. Insertion of the appropriate expressions above yields the result For these are the eigenvalues associated with the even Mathieu eigenfunctions or (i.e. with upper, minus sign) and odd Mathieu eigenfunctions or (i.e. with lower, plus sign). The explicit and normalised expansions of the eigenfunctions can be found in or. Similar asymptotic expansions can be obtained for the solutions of other periodic differential equations, as for Lamé functions and prolate and oblate spheroidal wave functions. Applications Mathieu's differential equations appear in a wide range of contexts in engineering, physics, and applied mathematics. Many of these applications fall into one of two general categories: 1) the analysis of partial differential equations in elliptic geometries, and 2) dynamical problems which involve forces that are periodic in either space or time. Examples within both categories are discussed below. Partial differential equations Mathieu functions arise when separation of variables in elliptic coordinates is applied to 1) the Laplace