Painlevé problem

The problem of characterizing removable sets (cf. Removable set) for a class of bounded single-valued analytic functions of the complex variable . Let be a compact set in the complex plane such that is a domain. One has to determine the minimal conditions on under which any bounded single-valued analytic function in can be continued analytically to and thus is a constant. P. Painlevé [1] stated a sufficient condition: The linear Hausdorff measure of should vanish (such sets are sometimes called Painlevé sets); however, his arguments contain some errors (see [2], [3]). A necessary and sufficient condition on is that the analytic capacity of vanishes (Ahlfors' theorem). An example has been constructed of a set with zero analytic capacity but having positive linear measure [5].

References

[1]	P. Painlevé, "Leçons sur la théorie analytique des équations différentielles, professées à Stockholm (1895)" , Paris (1897)
[2]	L. Zoretti, "Sur les fonctions analytiques uniformes qui possèdent un ensemble parfait discontinu de points singuliers" J. Math. Pure Appl. , 1 (1905) pp. 1–51
[3]	L. Zoretti, "Leçons sur la prolongement analytique" , Gauthier-Villars (1911)
[4]	L. Ahlfors, "Bounded analytic functions" Duke Math. J. , 14 (1947) pp. 1–11
[5]	A.G. Vitushkin, "Example of a set of positive length but of zero analytic capacity" Dokl. Akad. Nauk SSSR , 127 : 2 (1959) pp. 246–249 (In Russian)

References

[a1]	J.B. Garnett, "Analytic capacity and measure" , Lect. notes in math. , 297 , Springer (1972)
[a2]	L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1957)
[a3]	W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976) pp. 229ff