1981/04/01 by C. Hunter, B. Guerrieri · 43 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Convergence (economics) #Differential equation #Eigenvalues and eigenvectors #Mathematical analysis #Mathematics #Mathieu function #Matrix differential equation #Nonlinear Waves and Solitons #Physics #Pure mathematics #Sequence (biology) #Series (stratigraphy) #Type (biology) #Variable (mathematics)
paper · doi:10.1002/sapm1981642113
published in Studies in Applied Mathematics 64(2), 113-141 (Wiley)
openalex publication_date 1981/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A comprehensive account is given of the behavior of the eigenvalues of Mathieu's equation as functions of the complex variable q . The convergence of their small‐ q expansions is limited by an infinite sequence of rings of branch points of square‐root type at which adjacent eigenvalues of the same type become equal. New asymptotic formulae are derived that account for how and where the eigenvalues become equal. Known asymptotic series for the eigenvalues apply beyond the rings of branch points; we show how they can now be identified with specific eigenvalues.