Asymptotic expansion of the Mathieu and prolate spheroidal eigenvalues for large parameter c

The asymptotic expansions of the Mathieu eigenfunctions and the prolate spheroidal wave functions can be expanded in terms of the parabolic cylinder functions, from which their asymptotic eigenvalue can be expressed in an inverse power series of c, where the parameter c is proportional to the operating wave number. Analytical expressions of the eigenvalues, as well as those of the expansion coefficients of the eigenfunctions, are derived and verified with numerical results.