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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References

SHOWING 1-8 OF 8 REFERENCES
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…
Deducing the Properties of Singularities of Functions from their Taylor Series Coefficients
The asymptotic behavior of the coefficients in the Taylor series expansion of an analytic function about some point is governed by the nature of the singularity of the function that lies closest to…
The double points of Mathieu’s differential equation
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, *.…
Instability intervals of Hill's equation
Mathieusche Funktionen und Sphäroidfunktionen
XC.The characteristic numbers of the Mathieu equation with purely imaginary parameter
XXII.—Researches into the Characteristic Numbers of the Mathieu Equation—(Third Paper).
Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique.