Polikrug takozh policilindr polidisk geometrichnij bagatovimirnij ob yekt sho ye dobutkom zvichajnih krugiv Polikrugi chasto vikoristovuyutsya v kompleksnomu analizi pri vivchenni funkcij bagatoh kompleksnih zminnih zokrema chasto integralni teoremi dlya bagatoh zminnih formulyuyutsya cherez integrali na granici polikruga Zmist 1 Oznachennya 2 Polikrug i bagatovimirna kulya 3 Uzagalnennya 4 LiteraturaOznachennya red Oznachennya tut bude dano dlya kompleksnogo prostoru C n displaystyle mathbb C n nbsp Analogi vsih oznachen legko sformulyuvati takozh dlya prostoru R 2 n displaystyle mathbb R 2n nbsp Yaksho poznachiti D z r displaystyle D z r nbsp vidkritij krug z centrom v tochci z i radiusom r v kompleksnij ploshini todi vidkritim polikrugom rozmirnosti n z multiradiusom r 1 r n displaystyle r 1 dots r n nbsp i centrom v tochci z 1 z n C n displaystyle z 1 dots z n in mathbb C n nbsp nazivayetsya mnozhina D z 1 r 1 D z n r n displaystyle D z 1 r 1 times dots times D z n r n nbsp Inshimi slovami vidkritim polidiskom z multiradiusom r 1 r n displaystyle r 1 dots r n nbsp i centrom v tochci z 1 z n C n displaystyle z 1 dots z n in mathbb C n nbsp ye mnozhina w w 1 w 2 w n C n z k w k lt r k k 1 n displaystyle w w 1 w 2 dots w n in mathbf C n mid vert z k w k vert lt r k forall k 1 dots n nbsp Podibnim chinom mozhna viznachiti i zamknutij polikrug w w 1 w 2 w n C n z k w k r k k 1 n displaystyle w w 1 w 2 dots w n in mathbf C n mid vert z k w k vert leqslant r k forall k 1 dots n nbsp Vin ye dekartovim dobutkom zamknutih krugiv Mezha polikruga mozhe buti zapisana tak D z r w w 1 w 2 w n C n z k w k r k k 1 n j z j w j r j displaystyle partial D z r w w 1 w 2 dots w n in mathbf C n mid vert z k w k vert leqslant r k forall k 1 dots n land exists j vert z j w j vert r j nbsp Takozh vazhlivoyu ye taka chastina mezhi yak kistyak polikruga sho viznachenij formuloyu T z r w w 1 w 2 w n C n z k w k r k k 1 n displaystyle T z r w w 1 w 2 dots w n in mathbf C n mid vert z k w k vert r k forall k 1 dots n nbsp Polikrug i bagatovimirna kulya red Okrim polikruga v C n displaystyle mathbb C n nbsp takozh viznachena standartna vidkrita kulya w C n z w lt r displaystyle w in mathbf C n mid lVert z w rVert lt r nbsp de normoyu ye evklidova norma v C n displaystyle mathbb C n nbsp Koli n gt 1 displaystyle n gt 1 nbsp vidkriti polikrugi i vidkriti kuli ne ye bigolomorfno ekvivalentnimi tobto mizh nimi ne isnuye golomorfnogo biyektivnogo vidobrazhennya z golomorfnim obernenim vidobrazhennyam Cej fakt buv dovedenij Anri Puankare v 1907 roci Puankare pokazav sho grupi avtomorfizmiv vidkritih kul i polikrugiv mayut rizni rozmirnosti yak grupi Li Uzagalnennya red Ochevidnim uzagalnennyam polikrugiv v kompleksnomu analizi ye polioblasti D 1 D n displaystyle D 1 times dots times D n nbsp sho ye dobutkom oblastej v kompleksnij ploshini Zamknuti polioblasti mezhi ta kistyaki v comu vipadku viznachayutsya analogichno do poperednogo Polikrug ye chastkovim vipadkom logarifmichno opukloyi oblasti Rejnharta Literatura red Steven G Krantz Jan 1 2002 Function Theory of Several Complex Variables American Mathematical Society ISBN 0 8218 2724 3 John P D Angelo D Angelo P D Angelo Jan 6 1993 Several Complex Variables and the Geometry of Real Hypersurfaces CRC Press ISBN 0 8493 8272 6 Otrimano z https uk wikipedia org w index php title Polikrug amp oldid 24028028
