Bessel functions of two complex mutually conjugated variables and their application in boundary-value problems of mathematical physics - Sukhorolskyi, M. A. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
We formulate boundary-value problems for the eigenvalues and eigenfunctions of the Helmholtz equation in simplyconnected domains by using two complex mutually conjugated variables. The systems of eigenfunctions of these problemsare orthogonal in the domain. They are formed by Bessel functions of complex variables and the powers of conformalmappings of the analyzed domains onto a circle. The boundary-value problems for the main equations of mathematicalphysics are formulated in an infinite cylinder with the use of complex and time variables. The solutions of the boundaryvalueproblems are obtained in the form of series in the systems of eigenfunctions. The Cauchy problem for the mainequations of mathematical physics with three independent variables is also considered.