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I am not a (complex) analyst but it seems that some of the questions I am working on are related to the following concepts:

  1. logarithmic capacity
  2. transfinite diameter
  3. Green's function of a compact set
  4. System of Fekete points
  5. upper regularization

I have looked at various books but some of them are very old (e.g., Hille's Analytic function theory which was published in 1962), and most of them just explain the subject briefly. I am looking for references which are more on the advanced side rather than elementary, to be able to find in them the results I need.

Any suggestions?

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  • $\begingroup$ The question is not the resources of the Book you want to read, Do you have patience to read the book? $\endgroup$
    – Jame Ake
    Commented Oct 8, 2013 at 11:38

6 Answers 6

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I would say these concepts rather belong to the field of potential theory. You will find most of the definitions and a fairly advanced treatment of the subject in Logarithmic Potentials with External Fields by Saff and Totik.

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I really recommend the book Potential Theory in the complex plane" by Thomas Ransford.

It's a very nice book with exercises and it covers each of the 5 points you mentioned.

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  • $\begingroup$ +1. I had intended, while in Quebec, to work my way through the book properly... ;) $\endgroup$
    – Yemon Choi
    Commented Dec 2, 2011 at 2:14
  • $\begingroup$ Although to be fair, Tom never intended the book as a reference text. It might still be useful to the OP as an indication of where to look next and what one might expect $\endgroup$
    – Yemon Choi
    Commented Dec 2, 2011 at 7:55
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For the first two (maybe 3): L. Ahlfors: Conformal Invariants: Topics in Geometric Function Theory

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Conformal radius of a domain and Transfinite diameter seem to have most of these terms; see also http://en.wikipedia.org/wiki/Conformal_radius .

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Goluzin, Geometric theory of functions of a complex variable, contains a very comprehensive discussion of transfinite diameter, Fekete points etc. Ransford's book mentioned above is also very nice.

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J. B. Garnett, D. E. Marshall: Harmonic Measure And also Carleson's (I don't remember the name of the book) book contains at least first two.

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