# Polar set

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The polar set of analytic function of the complex variable , , is the set of points in some domain of the complex space for which: a) is holomorphic everywhere in ; b) cannot be analytically continued to any point of ; and c) for every point there are a neighbourhood and a function , holomorphic in , for which the function , which is holomorphic in , can be holomorphically continued to . At every point one has . The polar set consists of the poles (cf. Pole (of a function)) of , for which , and the points of indeterminacy of , for which (it is assumed that and have no common factors that are holomorphic and vanish at ). Every polar set is an analytic set of complex dimension .

A polar set in potential theory is a set of points of the Euclidean space , , for which there exists a potential , , for some Borel measure , that takes the value at the points of and only at those points.

In the case of the logarithmic potential for and the Newton potential for , for a bounded set to be a polar set it is necessary and sufficient that is a set of type and has zero outer capacity. Here, in the definition of a polar set, one can replace "potential" by "superharmonic function" . The main properties of polar sets in this case are: a) the set which consists of a single point is a polar set; b) a countable union of polar sets is a polar set; c) any polar set has Lebesgue measure zero in ; and d) under a conformal mapping a polar set goes to a polar set.

For a local criterion for being polar see Thinness of a set.

#### References

 [1] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian) [2] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian) [3] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)