Several boundary value problems arising from the extensions of analytic functions

In this paper, with the properties of $\beta$-Cauchy type integrals and the Plemelj formula for the corresponding singular integrals, the Hilbert boundary value problems for $\beta$-analytic functions were first discussed on the unit disc in $\mathbb{C}$, and the concrete representations of the solutions were obtained. Thereafter, on the basis of the results for the Riemann boundary value problems for analytic functions, several Riemann problems for higher-order complex partial differential systems and $(\lambda ,1)$ bi-analytic functions were investigated for the bicylinder in ${\mathbb{C}}^2$. The solutions to the problems and the corresponding solvable conditions were obtained.