An iterative real sequence based on and providing a plane parqueting and harmonic Green functions

Abstract

The real monotone decreasing sequence $\begin{array}{rl}& m_1=\sqrt{3},m_2=\frac{2}{\sqrt{3}},(\sqrt{m_1^2-1}=\sqrt{2}),\\ & m_{k+2}=\frac{{\alpha }_k{m}_{k+1}+{\text{β}}_k}{{\text{β}}_k{m}_{k+1}+{\alpha }_k},{\alpha }_k=m_k{m}_{k+1}-1,\\ & {\text{β}}_k=m_k-m_{k+1},k\in \mathbb{N},\end{array}$ converging to 1 is used to construct a parqueting of the complex plane leading to the harmonic Green function and the Poisson kernel for the countably many domains of the parqueting clustering at the point 1.

Keywords:

• Parqueting-reflection principle
• plane circular rectangle
• harmonic Green function
• Poisson kernel
• Poisson equation
• boundary value problems

AMS SUBJECT CLASSIFICATIONS:

• 30E25
• 30G20
• 31A10
• 31A25
• 35C15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author “Bibinur Shupeyeva” was supported by the Re-Invitation Programme for Former Scholarship Holders, 2019 (57440919) of the German Academic Exchange Service (DAAD) in the fall term 2019 at FU Berlin.