On complex roots of an equation arising in the oblique derivative problem

The paper is concerned with the eigenvalue problem for the Laplace operator in a disc under the condition that the oblique derivative vanishes on the disc boundary. In a famous article by V.A. Il'in and E.I. Moiseev (Differential equations, 1994) it was found, in particular, that the root of any equation of the form with the Bessel function J_n ( μ ) determines the eigenvalue λ = μ ² of the problem. In our work we correct the information about the location of eigenvalues. It is specified explicit view of the corner, containing all the eigenvalues. It is shown that all the nonzero roots of the equation are simple and given a refined description of the set of their localization on the complex plane. To prove these facts we use the partial differential equations methods and also methods of entire functions theory. (paper)