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The polar set of analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p0734401.png" /> of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p0734402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p0734403.png" />, is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p0734404.png" /> of points in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p0734405.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p0734406.png" /> for which: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p0734407.png" /> is holomorphic everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p0734408.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p0734409.png" /> cannot be analytically continued to any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344010.png" />; and c) for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344011.png" /> there are a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344012.png" /> and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344013.png" />, holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344014.png" />, for which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344015.png" />, which is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344016.png" />, can be holomorphically continued to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344017.png" />. At every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344018.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344019.png" />. The polar set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344020.png" /> consists of the poles (cf. [[Pole (of a function)|Pole (of a function)]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344022.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344023.png" />, and the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344024.png" /> of indeterminacy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344025.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344026.png" /> (it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344028.png" /> have no common factors that are holomorphic and vanish at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344029.png" />). Every polar set is an [[Analytic set|analytic set]] of complex dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344030.png" />.
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The polar set of analytic function $f(z)$ of the complex variable $z=(z_1,\dots,z_n)$, $n\geq1$, is the set $P$ of points in some domain $D$ of the complex space $\mathbf C^n$ for which: a) $f(z)$ is holomorphic everywhere in $D\setminus P$; b) $f(z)$ cannot be analytically continued to any point of $P$; and c) for every point $a\in P$ there are a neighbourhood $U_a$ and a function $q_a(z)\not\equiv0$, holomorphic in $U_a$, for which the function $p_a(z)=q_a(z)f(z)$, which is holomorphic in $D\cap\{U_a\setminus P\}$, can be holomorphically continued to $U_a$. At every point $a\in P$ one has $q_a(a)=0$. The polar set $P$ consists of the poles (cf. [[Pole (of a function)|Pole (of a function)]]) $a\in P$ of $f(z)$, for which $p_a(a)\neq0$, and the points $a\in P$ of indeterminacy of $f(z)$, for which $p_a(a)=0$ (it is assumed that $p_a(z)$ and $q_a(z)$ have no common factors that are holomorphic and vanish at $a$). Every polar set is a complex analytic variety (by which we mean the set of common zeros of a finite set of holomorphic functions) of complex dimension $n-1$.
  
A polar set in potential theory is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344031.png" /> of points of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344033.png" />, for which there exists a potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344035.png" />, for some Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344036.png" />, that takes the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344037.png" /> at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344038.png" /> and only at those points.
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A polar set in potential theory is a set $E$ of points of the Euclidean space $\mathbf R^n$, $n\geq2$, for which there exists a potential $U_\mu(x)$, $x\in\mathbf R^n$, for some Borel measure $\mu$, that takes the value $+\infty$ at the points of $E$ and only at those points.
  
In the case of the [[Logarithmic potential|logarithmic potential]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344039.png" /> and the [[Newton potential|Newton potential]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344040.png" />, for a bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344041.png" /> to be a polar set it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344042.png" /> is a set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344043.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344044.png" /> which consists of a single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344045.png" /> is a polar set; b) a countable union of polar sets is a polar set; c) any polar set has Lebesgue measure zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344046.png" />; and d) under a conformal mapping a polar set goes to a polar set.
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In the case of the [[Logarithmic potential|logarithmic potential]] for $n=2$ and the [[Newton potential|Newton potential]] for $n\geq3$, for a bounded set $E$ to be a polar set it is necessary and sufficient that $E$ is a set of type $G_\delta$ 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 $\{a\}$ which consists of a single point $a\in\mathbf R^n$ is a polar set; b) a countable union of polar sets is a polar set; c) any polar set has Lebesgue measure zero in $\mathbf R^n$; 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|Thinness of a set]].
 
For a local criterion for being polar see [[Thinness of a set|Thinness of a set]].
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====Comments====
 
====Comments====
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344047.png" /> as described under 2) is usually called a complete polar set. A (not necessarily complete) polar set is defined as a subset of a complete polar set. A bounded set is polar if and only if it has zero outer capacity.
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A set $E$ as described under 2) is usually called a complete polar set. A (not necessarily complete) polar set is defined as a subset of a complete polar set. A bounded set is polar if and only if it has zero outer capacity.
  
 
The sets described under 1) are also called pole sets, or sets of poles, which avoids confusion, cf. [[#References|[a4]]] and [[Meromorphic function|Meromorphic function]].
 
The sets described under 1) are also called pole sets, or sets of poles, which avoids confusion, cf. [[#References|[a4]]] and [[Meromorphic function|Meromorphic function]].
  
In parabolic potential theory, a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344048.png" /> is polar if and only if there exists an open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344050.png" /> and, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344051.png" />, a positive supercaloric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344052.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344053.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344054.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344055.png" />, see [[#References|[a3]]]. Again, points are polar and a countable union of polar sets is polar. Any polar set is totally thin but, in contrast with classical potential theory, not every totally thin set is polar. A similar theory of polarity holds in harmonic spaces, see [[#References|[a2]]], or in the more general case of balayage spaces, cf. [[#References|[a1]]]. In probabilistic potential theory, a Borel set is polar if its first hitting time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344056.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073440/p07344057.png" /> a.s.
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In parabolic potential theory, a set $A$ is polar if and only if there exists an open covering $\mathcal W$ of $A$ and, for any $V\in\mathcal W$, a positive supercaloric function $u_V$ on $V$ such that $u_V=\infty$ on $A\cap V$, see [[#References|[a3]]]. Again, points are polar and a countable union of polar sets is polar. Any polar set is totally thin but, in contrast with classical potential theory, not every totally thin set is polar. A similar theory of polarity holds in harmonic spaces, see [[#References|[a2]]], or in the more general case of balayage spaces, cf. [[#References|[a1]]]. In probabilistic potential theory, a Borel set is polar if its first hitting time $T_A$ satisfies $T_A=\infty$ a.s.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bliedtner,  W. Hansen,  "Potential theory. An analytic and probabilistic approach to balayage" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.L. Doob,  "Classical potential theory and its probabilistic counterpart" , Springer  (1983)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Grauert,  K. Fritzsche,  "Several complex variables" , Springer  (1976)  (Translated from German)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Whitney,  "Complex analytic varieties" , Addison-Wesley  (1972)  pp. Chapt. 8</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bliedtner,  W. Hansen,  "Potential theory. An analytic and probabilistic approach to balayage" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.L. Doob,  "Classical potential theory and its probabilistic counterpart" , Springer  (1983)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Grauert,  K. Fritzsche,  "Several complex variables" , Springer  (1976)  (Translated from German)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Whitney,  "Complex analytic varieties" , Addison-Wesley  (1972)  pp. Chapt. 8</TD></TR></table>
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====Comment====
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For polar sets in convex geometry, see [[polar body]].

Latest revision as of 17:25, 23 October 2017

The polar set of analytic function $f(z)$ of the complex variable $z=(z_1,\dots,z_n)$, $n\geq1$, is the set $P$ of points in some domain $D$ of the complex space $\mathbf C^n$ for which: a) $f(z)$ is holomorphic everywhere in $D\setminus P$; b) $f(z)$ cannot be analytically continued to any point of $P$; and c) for every point $a\in P$ there are a neighbourhood $U_a$ and a function $q_a(z)\not\equiv0$, holomorphic in $U_a$, for which the function $p_a(z)=q_a(z)f(z)$, which is holomorphic in $D\cap\{U_a\setminus P\}$, can be holomorphically continued to $U_a$. At every point $a\in P$ one has $q_a(a)=0$. The polar set $P$ consists of the poles (cf. Pole (of a function)) $a\in P$ of $f(z)$, for which $p_a(a)\neq0$, and the points $a\in P$ of indeterminacy of $f(z)$, for which $p_a(a)=0$ (it is assumed that $p_a(z)$ and $q_a(z)$ have no common factors that are holomorphic and vanish at $a$). Every polar set is a complex analytic variety (by which we mean the set of common zeros of a finite set of holomorphic functions) of complex dimension $n-1$.

A polar set in potential theory is a set $E$ of points of the Euclidean space $\mathbf R^n$, $n\geq2$, for which there exists a potential $U_\mu(x)$, $x\in\mathbf R^n$, for some Borel measure $\mu$, that takes the value $+\infty$ at the points of $E$ and only at those points.

In the case of the logarithmic potential for $n=2$ and the Newton potential for $n\geq3$, for a bounded set $E$ to be a polar set it is necessary and sufficient that $E$ is a set of type $G_\delta$ 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 $\{a\}$ which consists of a single point $a\in\mathbf R^n$ is a polar set; b) a countable union of polar sets is a polar set; c) any polar set has Lebesgue measure zero in $\mathbf R^n$; 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)


Comments

A set $E$ as described under 2) is usually called a complete polar set. A (not necessarily complete) polar set is defined as a subset of a complete polar set. A bounded set is polar if and only if it has zero outer capacity.

The sets described under 1) are also called pole sets, or sets of poles, which avoids confusion, cf. [a4] and Meromorphic function.

In parabolic potential theory, a set $A$ is polar if and only if there exists an open covering $\mathcal W$ of $A$ and, for any $V\in\mathcal W$, a positive supercaloric function $u_V$ on $V$ such that $u_V=\infty$ on $A\cap V$, see [a3]. Again, points are polar and a countable union of polar sets is polar. Any polar set is totally thin but, in contrast with classical potential theory, not every totally thin set is polar. A similar theory of polarity holds in harmonic spaces, see [a2], or in the more general case of balayage spaces, cf. [a1]. In probabilistic potential theory, a Borel set is polar if its first hitting time $T_A$ satisfies $T_A=\infty$ a.s.

References

[a1] J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)
[a2] C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)
[a3] J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1983)
[a4] H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) (Translated from German)
[a5] H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) pp. Chapt. 8

Comment

For polar sets in convex geometry, see polar body.

How to Cite This Entry:
Polar set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar_set&oldid=12257
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article