Integral representations for a second-order system with a supersingular point (Q2388001)

Let \(D\) be a connected domain containing \(z=0\) as interior point. The author considers in \(D\setminus \{0\}\) the system \[ \frac{\partial^{2}u}{\partial\bar{z}^{2}}+\frac{a}{r^{n}}\frac{\partial u}{\partial\bar{z}}+\frac{b}{r^{2n}}u=\frac{f}{r^{n}}, \] where \(a, b\) and \(f\) are given complex-valued functions, \[ n\geq 2, \quad \partial_{\bar{z}}=(\partial_{x}+\partial_{y})/2, \quad r=\sqrt{x^{2}+y^{2}}. \] He finds various integral representations of solutions of the system depending on two arbitrary analytic functions of the complex variable \(z\). These representations depend on the roots of the characteristic equation and are valid under certain relationships between the coefficients of the system.