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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)

${w}^{\text{'}\text{'}}+p\left(z\right){w}^{\text{'}}+q\left(z\right)w=0,\phantom{\rule{1.em}{0ex}}p\left(z\right)=\sum _{j=1}^{3}{\alpha }_{j}/\left(z-{a}_{j}\right),\phantom{\rule{1.em}{0ex}}q\left(z\right)=\left(h-\ell z\right)/\left(z-{a}_{1}\right)\left(z-{a}_{2}\right)\left(z-{a}_{3}\right)·$

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, $\left(h,\ell \right)\in {\Sigma }$ iff the equation (1) has such a solution $w\left(z\right)$, for which ${w}^{\text{'}}\left(z\right)/w\left(z\right)$ 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 Differential equations in the complex domain 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE) 34L20 Asymptotic distribution of eigenvalues for OD operators