The method of axisymmetric generalized analytical functions in the analysis of dynamic processes (Q1190881)

Let \(D\subset\mathbb{C}\) be a bounded domain with boundary \(\partial D\) and \(f(x,y)=0\) for \(z:=x+iy\in\partial D\), where \(f\) is a harmonic function in \(D\). The author introduces an axisymmetric generalized analytic function \(w=\varphi+i\psi/f\) (i.e. complex potential for axisymmetric field) by the system \[ {\partial\varphi\over\partial x}={1\over f}{\partial\psi\over\partial y},\quad{\partial\varphi\over\partial y}=- {1\over f}{\partial\psi\over\partial x}\quad\text{for } z\in D. \] He applies Polojii's theory of \(p\)-analytic functions (\textit{N. Polojii} [Theory and applications of \(p\)-analytic and \((p,q)\)-analytic functions (Russian) (Naukova Dumka) (1973; Zbl 0257.30040)]) and obtains a generalized Cauchy theorem, Morera's theorem, etc. Finally, some applications to three-dimensional boundary problems in a piecewise- homogeneous medium are given.