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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) \[ w''+p(z)w'+q(z)w=0,\quad p(z)=\sum_{j=1}^ 3 \alpha_ j/(z-a_ j),\quad q(z)=(h-\ell z)/(z-a_ 1)(z-a_ 2)(z-a_ 3). \] 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 \(\Sigma\) of Heun’s equation is characterized. For example, \((h,\ell)\in\Sigma\) 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:

34M99 Ordinary differential equations in the complex domain
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
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