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First, I know that the Green function of 2D Laplace operator is given by $$G(z,w)\propto \ln\frac{|z-w|}{|z-\bar{w}|}.$$ Second, I also understand how can I obtain the Green function on unit disk, $$G_D(z,w)\propto \ln\frac{|z-w|}{|1-\bar{w}z|}.$$ Third, I know that there is the function that is closely related to the 2D Green functions, Poisson kernel, $$P(z,w)=\frac{1-|z|^2}{|w-z|^2}.$$ Recently I have found the statement [see p. 4, eq. (1.10) of Wolfgang Woess notes 'Euclidean unit disk, hyperbolic plane and homogeneous tree: a dictionary'] that the Poisson kernel can be represented as the following ratio of two Green functions on disk, $$P(z,w)=\lim\limits_{\xi\rightarrow w}\frac{G_D(z,\xi)}{G_D(0,\xi)},\quad (*)$$ and the author claims that this representation is called Martin kernel.

However, I have the vague feeling that there is a typo in $(*)$ exression: naively, the left hand side does not contain logs, whereas logs are in the right hand side.

So, my questions are:

  1. Is it possible to represent the Poisson kernel $P(z,w)$ in terms of Green functions?
  2. How does the Green functions on disk relate to CFT or QFT?
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  • $\begingroup$ Minor comment to the post (v3): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$
    – Qmechanic
    Commented Feb 27, 2021 at 10:18

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The short answer is this:

Poisson kernel is the normal derivative of the Green function for the laplacian on the unit disk.

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