Abstract
The parqueting-reflection method is applied to a nonregular domain and the harmonic Green function for the half hexagon is constructed. The related Dirichlet problem for the Poisson equation is solved explicitly.
Highlights
The basic boundary value problems for the second-order complex partial differential equations are the harmonic Dirichlet and Neumann problems for the Laplace and Poisson equations
Reflection of the point z ∈ P defines the poles of the meromorphic function in the hexagons P1, . . . , P6
Each hexagon includes 3 poles and 3 zeros. Continuation of these operations reveals that all the points have the same coefficients of rotation: 1, −(1/2)(1 + i√3), −(1/2)(1 − i√3), and displacement 3m + i√3n, m + n ∈ 2Z
Summary
The basic boundary value problems for the second-order complex partial differential equations are the harmonic Dirichlet and Neumann problems for the Laplace and Poisson equations. Difficulties arise since the elliptic integrals appearing in the formula imply complicated computations and need to be solved numerically. As analogue to this formula, another method can be applied which gives the covering of the entire complex plane C by reflection of the given domain D at its boundary. Our aim is to find the solution of the Dirichlet boundary value problem for the Poisson equation through the Poisson integral formula. It is known that the Poisson kernel function is an analogue of the Cauchy kernel for the analytic functions and the Poisson integral formula solves the Dirichlet problem for the inhomogeneous Laplace equation. The later one provides the solution to the harmonic Dirichlet problem which is shown in the last part
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