The Eigenvalues of Mathieu's Equation and their Branch Points

@article{Hunter1981TheEO,
  title={The Eigenvalues of Mathieu's Equation and their Branch Points},
  author={Christopher Hunter and Bruno Guerrieri},
  journal={Studies in Applied Mathematics},
  year={1981},
  volume={64},
  pages={113-141}
}
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… 
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The solutions of the Mathieu equation with a complex variable and at least one parameter large
also commonly known as the equation of the elliptic cylinder functions, is too well known to require any introduction. Its solutions govern problems of the greatest diversity in astronomy and
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Mathieu's differential equation, y" + (a - 2q cos 2x)y 0, admits of solutions of period xr or 27r for four countable sets of characteristic values, a(q), which can be ordered as aT(q), r = O, 1, *.