Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!umich!samsung!brutus.cs.uiuc.edu!psuvax1!psuvm!auvm!BROWNVM!ST401843 From: ST401843@BROWNVM.BITNET (thanasis kehagias) Newsgroups: bit.listserv.hellas Subject: Re: Quiz Message-ID: Date: 10 Feb 90 02:32:35 GMT Sender: The Hellenic Discussion List Reply-To: The Hellenic Discussion List Lines: 27 Approved: NETNEWS@AUVM Gateway In-Reply-To: Message of Fri, 9 Feb 90 17:45:00 EST from opoiosdhpote migadikos ariqmos exei n rizes niostes tachs. auto sumbainei dioti an to z einai niosth riza tou b, tote z**n=b => z**n -b=0. alla auth einai mia poluwnumikh exiswsh . to b mporei na einai migadikos ariqmos. genika opoiadhpote eciswsh z**n+a(n-1)*z**(n-1)+...+a(1)*z+a(0)=0, exei n migadikes rizes. auto einai to qemeliwdes qewrhma ths algebras (pou ston Ntziwra, Algebra Pempths Gumnasiou, parousiazotan ws aciwma ?). h apodeich den einai topologikh. basizetai sthn qewria migadikwn sunarthsewn. sugkekrimena, to qewrhma tou liouville mas leei oti mia migadikh sunarthsh ANALUTIKH kai FRAGMENH sto migadiko epipedo einai staqerh (apo auto eukola apodeiknuetai to Qemeliwdes Qewrhma ...). to qewrhma liouville meta seira tou basizetai sto QEIKO qewrhma tou Cauchy, pou leei oti oi times mias analutikhs sunarthshs mesa se ena xwrio, einai plhrws prosdiorismenes apo tis times sto sunoro tou xwriou!!! otan milame gia niostes rizes enos migadikou ariqmou, ceroume oti an h r*e**(i*theta) einai mia apo autes tis rizes, tote rizes einai kai oi r*e**(i*[theta+2*k*pi]/n), k=0,1,2,... . autos o tupos dinei mia apeiria rizwn, alla ousiastika pairnoume mono n diaforetikes rizes, dioti otan k=m kai otan k=m+n pairnoume thn idia timh. oi rizes autes einai summetrika topoqethmenes se ena kuklo me aktina r, kai apexoun metacu tous 2*k*pi/n aktinia. ola auta uparxoun kai pali sto biblio tou Ntziwra. epishs ola auta uparxoun kai sto biblio tou Schaum Outline Series , Complex Variables, by M. Spiegel. thanasis