關(guān)于復(fù)振蕩中的輻角分布

1、506200711ACTAMATHEMATICASINICA,ChineseSeriesVol.50,No.6Nov.,2007:0583-1431(2007)06-1297-08:A437100510631E-mail:wzj52510631,.BorelBorel.MR(2000)O174.52;Borel;30D35,34A20OnAngularDistributioninComplexOscillationZhaoJunWUDepartmentofMathematics,XianningUniversity,Xianning437100,P.R.ChinaSchoolofMathema

2、ticsScience,SouthChinaNormalUniversity,Guangzhou510631,P.R.ChinaE-mail:wzj52DaoChunSUNSchoolofMathematicsScience,SouthChinaNormalUniversity,Guangzhou510631,P.R.ChinaAbstractInthispaper,byusingtheinnityordertypefunctionofXiongQinglaisandasucientandnecessaryconditionforinnityorderBoreldirectionwhichwa

3、sestablishedbyChuangChitai,theangulardistributionofthesolutionsofdierentialequationswithentirecoecientsisdiscussed,theconnectionoflocationofzerosofthesolutionofsecondorderorhigherorderdierentialequationetal.theBoreldirectionisestablished,whichcanberegardedasanalternatiueversionofWuShengjianetal.Keyw

4、ordsinnityordertypefunction;Boreldirection;theexponentofconvergenceofzero-sequenceMR(2000)SubjectClassication30D35,34A20ChineseLibraryClassicationO174.52Nevanlinna12.:2005-11-25;:2006-10-23;(10471048);:2007-04-30(KZ0629,KT0623)129850,191934,JuliaJuliaIshizakiYanagihara5,20046.,68.g(z)(00aC,logn(r,g=

5、a)rlogr=,(r,g=a)nga(,+,r)=z;argz+,0|z|0)(r)(r).(2)U(r)=r(r(rr0)rlimlogU(R)logU(r)=1,R=r+rlogU(r).(3)logT(r,g)r=1.9,(r),10J:argz=g(z)(r)Borel,logn(r,g=a)r(r)logr=1,0aC,a.,6,2f1,ff +Af=0,A6.:AA(z)n1,f1,f2(1)E=f1f2=n+2,argz=EBorelA(z),:A(z)B0,1213n(r,f=0)=cr(1+o(1).,L:argz=(2)f(z)Stokes(f)=.12,An1(z)0,

6、f1,f2,.,fn(2)n,E=f1f2fnE(E)=,StokesL:argz=.f(z)Borel.L(2),(2)(14).f1,f2,.,fn3(2)An1(z)0,(2)n,E=f1f2fn.E(E)=,(r)E,argz=130050E(r)Borellogn(r,E=0)=1.0r(r)logr3,.4f1,f2,.,fnf(n)+An2(z)f(n2)+A1(z)f +A0(z)f=0,(n3,A0(z),.,An2(z)1)n.E=f1f2.fn,(r)E,argz=E(r)Borellogn(r,E=0)=1.0r(r)logr2Langley154,E(E)=,34.,

7、16,1,2.2Borel,Nevanlinna7.f(z),L:argz=0=0argz0+=,01kA(r,f)= 1r 11|bv|k sink(v),C(r,f)=C(r,f)=2vr2kB(r,f)=r tk dt+ii;|f(te)|+log|f(te)|logtrtlog+|f(rei)|sink()d;bvb f(z) :1|z|r,argzbv=|bv|eiv, C(r,a)=Cr,.(r,f).aC,1 ,C(r,)=C(r,f).faD(r,f)=A(r,f)+B(r,f),S(r,f)=S(r,f)=C(r,f)+D(r,f).S(r,f),1 Sr,=S(r,f)+O

8、(1)fa ,aC,ajC,j=1,2,.,q,6:1301(q2)S(r,f)qj=1(r,aj)+h(r),f h(r)=Dr,+f 1jq,aj=f Dr,+O(1).fajValironh(r)=O(logU(r)(7),(q2)S(r,f)Borelq j=1r,aj)+O(logU(r).(3),17f(z)f(z)(r)Borel,(r):f(z),(00,logS,+(r,E)=1r(r)logr,1.0r0.(r)E,0r,n(r,E=a)(U(r)1+.(3),logS,+(r,E)1.r(r)logr.1302500,logS,+(r,E)1.r(r)lograC,log

9、n(r,E=a)1.r(r)logr0,logS,+(r,E)0(1logS,+(r,E),rrRn=rn+nRn(n), n(rn)=nrn,E=a(U(Rn)1.3bv=|bv|eiv(v=1,2,.)v+,v=1,2,.,E=a(,+),5v+.626S(r,f),(R,a)+O(1)S,+(Rn,E)C,+(Rn,a)+O(1)Cn,+ 1 |bv|k 2sinv+O(1)(R)2vn1|bv|rn,v+2k=. 1|bv|rn,v+ 1|bv|k sinv+O(1)(R)2vn 1|bv|rn,v+ 1|bv|k +O(1),(R)vn rn11S,+(Rn,E)dn(t)+tkdn

10、(t)+O(1)tRn11 rn rnkn(rn)rnk1n(rn)kdn(t)+tk1dn(t)+O(1)trnRnRn11kkn(rn)rnn(rn)Rnn(rn)n(rn)+O(1)+O(1)rnRnrnRn 11 n(rn)+O(1),rnRn rnStieltjes,6:logS,+(r,E)logS,+(Rn,E)r(r)logrr(Rn)logRnlog(11)lognn(R+n(rn)n)logRnr(Rn)logRn1+log(Rknrkn)k(logRkn+logrkn)(Rn)logRn=1+limlog(rn+rkn)rkn(R=1.n)logRn(r,E),r0,lo

11、gS,+logn(r,2,E=a)r(r)logr1.,logn(r,E=0)0r(r)logr1,0,logn(r,E=a)r(r)logr1.logn(r,E=0)0r(r)logr0, 1 ,+,+r,E.0,logS,+(r,E)r(r)logr=1.(4)log,+(r,1)r(r)logr=1.,f=a)(,+(r,a)2n(r,7),logn(r,E=0)r(r)logr1.,logn(r,E=0)0r(r)logr1.1303(4)130450317,S,+(r,E)=C,+ 1 r,+O(1).E,0,(5)1(4)(5)C,+(r,a)2n(r,f=a).1,3.,12.H

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