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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:

logarithmic capacity
transfinite diameter
Green's function of a compact set
System of Fekete points
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?

flag	asked Feb 7 2011 at 15:29 Hadi 387●1●6

Thanks everyone for the helpful references. – Hadi Feb 17 2011 at 23:52

Andrey Rekalo · Answer 1 · 2011-02-07 15:50:04Z UTC

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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.

answered Feb 7 2011 at 15:50

Andrey Rekalo
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Malik Younsi · Answer 2 · 2011-12-02 01:56:50Z UTC

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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.

answered Dec 2 2011 at 1:56

Malik Younsi
584●1●7●22

	+1. I had intended, while in Quebec, to work my way through the book properly... ;) – Yemon Choi Dec 2 2011 at 2:14
	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 – Yemon Choi Dec 2 2011 at 7:55

Gerald Edgar · Answer 3 · 2011-02-07 15:47:38Z UTC

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For the first two (maybe 3): L. Ahlfors: Conformal Invariants: Topics in Geometric Function Theory

answered Feb 7 2011 at 15:47

Gerald Edgar
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George Lowther · Answer 4 · 2011-12-02 01:34:15Z UTC

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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 .

edited Dec 2 2011 at 1:34

George Lowther

answered Feb 7 2011 at 15:43

Charles Matthews
9,800●13●40

Alexandre Eremenko · Answer 5 · 2012-08-04 10:47:48Z UTC

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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.

answered Aug 4 at 10:47

Alexandre Eremenko
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makman · Answer 6 · 2011-02-20 02:20:51Z UTC

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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.

answered Feb 20 2011 at 2:20

makman
1

Reference for complex analysis jargon

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