Schwarz boundary value problem of complex partial differential equations for the inhomogeneous Cauchy-Riemann equation in the equilateral triangle

In this paper, we explicitly investigate the Schwarz boundary value problem of complex partial differential equations for an inhomogeneous Cauchy-Riemann equation on a polygon domain with distinct points of the equilateral triangle. By applying the technique of parquet reflection and selecting an arbitrary point of the equilateral triangle and its repeated reflections in all parts of the boundary, full page of complex spaces covered. In addition, the fundamental tool for solving the Schwarz boundary value problem from the complex partial differential equations for the Cauchy-Riemann equation is the Cauchy–Pompeiu integral representation formula. Thus, using the technique of parquet reflection and the Cauchy–Pompeiu integral representation formula, we accurately calculate the Schwarz-Poisson integral representation formula on the equilateral triangle and its different boundary portions. We also consider boundary behaviors of the Schwarz-type operator. Finally, we give an exact answer for the Schwarz boundary value problem of complex partial differential equations for an inhomogeneous Cauchy-Riemann equation on the equilateral triangle.