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Asymptotic behavior of the Heun equation and Heun functions. (Russian) Zbl 0742.34004

The paper deals with the “centenarian” Heun’s equation in the complex domain: (1)

w '' +p(z)w ' +q(z)w=0,p(z)= j=1 3 α j /(z-a j ),q(z)=(h-z)/(z-a 1 )(z-a 2 )(z-a 3 )·

Firstly, the author recalls the classification of the special cases of Heun’s equations and other basic notions, as a spectrum of Heun’s equation and Heun’s functions. Further, by the properties of the solution of (1), the spectrum Σ of Heun’s equation is characterized. For example, (h,)Σ iff the equation (1) has such a solution w(z), for which w ' (z)/w(z) is a rational function. With help of the Heun’s surface function, the properties of the spectrum are studied. Finally, the WKB-estimates of the solutions of (1) are derived.

Reviewer: A.Klíč (Praha)

MSC:
34M99Differential equations in the complex domain
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
34L20Asymptotic distribution of eigenvalues for OD operators