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Mathieu’s eigenvalue problem −y′′(x) + 2e_0 cos(2x)y(x) = λy(x), 0 2k_0. Besides the diagonal band, A has two off-diagonal bands, and furthermore, a cross band appears when k_0 ≥ 2. The product formula, the recursion formulas of characteristic functions and a fictitious time integration method (FTIM) are developed to find the eigenvalues of Mathieu’s equation.

Analytic Solutions of the Eigenvalues of Mathieu’s Equation