Talk:Analytic capacity

Things to be added

2.3 Proof that H1(E)=0 implies gamma(E)=0 2.4 Proof that dim H>1 implies gamma(E)>0 Carolus m 23:19, 18 September 2007 (UTC)

Apparently the classical Painlevé problem has been recently solved by Xavier Tolsa. Someone who knows more about it than myself might like to add some words about this. 75.45.178.71 (talk) 06:59, 17 February 2009 (UTC)

mistakes

if we assume additionally that E is connected. In this case, $\mathbb {C} \setminus E$ is simply connected and open.

but isn't this simply false? consider E to be the closed unit disc or even a point. the complement is not simply connected. also, the switching between E and K in this article makes it look very bad and confusing. the whole article deserves a careful reread by an expert. - 99.249.178.235 01:00, 7 November 2007 (UTC)

Not quite. You consider the complement in the Riemann sphere (that was not quite clear, I have amended it accordingly). There is still an issue when you consider $S^{1}$ -- the complement is not simply connected, since it is connected. The trick is to apply the Riemann Mapping Theorem to the unbounded component of $\mathbb {C} \setminus E$ and map the rest to 0. I will amend the article accordingly and unify notation. Carolus m 16:25, 16 November 2007 (UTC)

This article switches freely from E to K and is thus very confusing. I concur that this article needs a careful review and cleanup by an expert. Rljacobson (talk) 03:56, 28 January 2008 (UTC)

I changed every E to K, which I'm pretty sure doesn't create any new problems. 75.45.178.71 (talk) 06:54, 17 February 2009 (UTC)

It is a bad idea to redefine the notion of simple connectivity. It is better to just make $U$ a subset of the Riemann sphere. Moreover, simple-connectivity is not really proven. The fact that the complement of $U$ is connected is proven in great detail, but to deduce that $U$ is simply connected you need to either do some work or quote some theorem. I will search for the right thing to quote there. Oded (talk) 15:16, 18 June 2008 (UTC)

The statements in the section Analytic capacity#Additional properties assuming finite connectivity seem to be plainly wrong. Oded (talk) 15:40, 18 June 2008 (UTC)

Alternate definition

The Encyclopedia of Math (bottom of article on Capacity of a set) gives an alternate definition:

$\gamma (K)=\sup \left|{\frac {1}{2\pi }}\int _{C}f(z)dz\right|$

where C is a contour enclosing K and the sup is taken over the same conditions as this article (f analytic outside K, f is bounded by 1, f(infty) vanishes). Seems plausible to me that this is a valid alternative definition, and (to me) it seems to be easier to grok. Hmmm. Whatever. I just added it to the article. 67.198.37.16 (talk) 06:30, 4 February 2024 (UTC)