The Eigenvalues of Mathieu's Equation and their Branch Points

  title={The Eigenvalues of Mathieu's Equation and their Branch Points},
  author={Christopher Hunter and Bruno Guerrieri},
  journal={Studies in Applied Mathematics},
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… 
On the double points of a Mathieu equation
Quadratic Growth of Convergence Radii for Eigenvalues of Two-Parameter Sturm–Liouville Equations
Abstract The n th eigenvalue μ of the equation y ″+( μ + λr ( x ))  y =0, a ⩽ x ⩽ b , subject to self-adjoint boundary conditions admits a power series expansion into powers of λ for sufficiently
Mathieu functions for purely imaginary parameters
Exceptional points of the eigenvalues of parameter-dependent Hamiltonian operators
We calculate the exceptional points of the eigenvalues of several parameter-dependent Hamiltonian operators of mathematical and physical interest. We show that the calculation is greatly facilitated
Singular points from Taylor series
A simple and accurate method is developed for calculating singular points from Taylor series. It consists of finding the least‐squares deviation of the Taylor coefficients from a proposed asymptotic
Mathieu functions and numerical solutions of the Mathieu equation
  • R. Coisson, G. Vernizzi, Xiaoke Yang
  • Mathematics, Computer Science
    2009 IEEE International Workshop on Open-source Software for Scientific Computation (OSSC)
  • 2009
A numerical algorithm which allows a flexible approach to the computation of all the Mathieu functions is described, and an elegant and compact matrix notation is used which can be readily implemented on any computing platform.
On the Growth of Convergence Radii for the Eigenvalues of the Mathieu Equation
It is proved that the convergence radii ρn of the eigenvalues of the Mathieu equation satisfy lim inf ρn/n2 > kk′K2 = 2.0418., where the modulus k of the complete elliptic integrals is determined by
Uniform asymptotic approximation of Mathieu functions
Uniform asymptotic approximations are derived for solutions of Mathieu's equation —Y = {2qcos(2z) ajw, for a and q real, and z complex. These are uniformly valid for q large and a lying in the


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, *.