The Eigenvalues of Mathieu's Equation and their Branch Points | Zendy

Hunter C. | Zendy; Guerrieri B. | Zendy

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The Eigenvalues of Mathieu's Equation and their Branch Points

Author(s) -

Hunter C.,

Guerrieri B.

Publication year - 1981

Publication title -

studies in applied mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.164

H-Index - 46

eISSN - 1467-9590

pISSN - 0022-2526

DOI - 10.1002/sapm1981642113

Subject(s) - eigenvalues and eigenvectors , mathieu function , mathematics , matrix differential equation , mathematical analysis , type (biology) , sequence (biology) , series (stratigraphy) , variable (mathematics) , convergence (economics) , pure mathematics , differential equation , physics , ecology , paleontology , genetics , quantum mechanics , biology , economic growth , economics

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.

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