Definition:Barrier
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Definition
A complex function $\varphi \in \map \C {\overline \Omega}$ is a barrier for $\Omega$ at $z \in \partial \Omega$ if and only if:
- $\varphi$ is subharmonic
- $\map \varphi z = 0$
- $\varphi < 0$ on $\partial \Omega \setminus \set z$
This article, or a section of it, needs explaining. In particular: Define $\map \C {\overline \Omega}$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
This article, or a section of it, needs explaining. In particular: Define $\Omega$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
This article, or a section of it, needs explaining. In particular: Define $\partial \Omega$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
This article, or a section of it, needs explaining. In particular: What is the domain and range of $\varphi$? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
This article, or a section of it, needs explaining. In particular: If $\varphi$ is a complex function, its range is also complex, and so cannot be ordered, so the meaning of $\varphi < 0$ needs to be explained carefully You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Also see
- Definition:Regular Boundary Point: a boundary point $z \in \partial \Omega$ which has a barrier for $\Omega$ at $z \in \partial \Omega$.