Bessel Functions of Two Complex Mutually Conjugate Variables and Their Application in Boundary-Value Problems of Mathematical Physics

We formulate boundary-value problems for the eigenvalues and eigenfunctions of the Helmholtz equation in simply connected domains by using two complex mutually conjugate variables. The systems of eigenfunctions of these problems are orthogonal in the domain. They are formed by Bessel functions of complex variables and the powers of conformal mappings of the analyzed domains onto a circle. The boundary-value problems for the main equations of mathematical physics are formulated in an infinite cylinder with the use of complex and time variables. The solutions of the boundary-value problems are obtained in the form of series in the systems of eigenfunctions. The Cauchy problem for the main equations of mathematical physics with three independent variables is also considered.