数学论文_英文翻译

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1、毕业设计（论文）附录（翻译）课题名称-些周期性的二阶线性微分方程解的方法学生姓名万益目录1•毕业设计（论文）附录（翻译）英文2•毕业设计（论文）附录（翻译）中文2014年5月25日SomePropertiesofSolutionsofPeriodicSecondOrderLinearDifferentialEquations1・IntroductionandmainresultsInthispaper,weshallassumethatthereaderisfamiliarwiththefundamentalresultsandthestardardnotation

2、softheNevanlinnarsvaluedistributiontheoryofmeromorphicfunctions[12,14,16].Inaddition,wewillusethenotationcr(/),“(/)andA(/)todenoterespectivelytheorderofgrowth,thelowerorderofgrowthandtheexponentofconvergenceofthezerosofameromorphicfunctionf,(ye(/)([see8]),thee-typeorderoff(z),isdefined

3、tobe")=诂沁4Similarly,Ae(/)，thec-typcexponentofconvergenceofthezerosofmeromorphicfunctionf,isdefinedtobeS乔心迪r^+ocyWesaythatf(z)hasregularorderofgrowthifameromorphicfunctionf(z)satisfiesr卄0logrWeconsiderthesecondorderlineardifferentialequation•厂+AU0Where=B(eaz)isaperiodicentirefunctionwit

4、hperiodCO=2m/a.Thecomplexoscillationtheoryof(1.1)wasfirstinvestigatedbyBankandLaine[6].Studiesconcerning(1.1)haveeencarriedonandvariousoscillationtheoremshavebeenobtained[2{11,13,17{19J.When4⑵isrationaline*,BankandLaine[6]provedthefollowingtheoremTheoremALetA(z)=)beaperiodicentirefunct

5、ionwithperiodcd=2m/aandrationalineaz.Ifhaspolesofoddorderatboth：=ooandJ=0,thenforeverysolution0)of(1.1),2(/)=4-ooBank[5Jgeneralizedthisresult:Theaboveconclusionstillholdsifwejustsupposethatboth：=ooand匚=OarepolesofB($),andatleastoneisofoddorder.Inaddition,thestrongerconclusionIog+N0,l//

6、)HO(厂)(1.2)holds.When4⑵istranscendentaleaz.Gao[10]provedthefollowingtheoremwithcr(g)<1,pisanoddpositiveintegerandbpH()，LetA(^)=B(e).Thenanynon-triviasolutionfof(1.1)musthaveA(/)=+oo.Infact,thestrongerconclusion(1.2)holds.TheoremBLet〃(G=g(i/G+£=bj7‘whereg(/)isatranscendentalentirefun

7、ctionAnexamplewasgivenin[10]showingthatTheoremBdoesnotholdwhen1,butisnotapositiveinteger,whatcanwesay?ChiangandGao[8]obtainedthefollowingtheoremsTheoremCLetA(z)=B(£”),whereB(G=小(1/$)+g?O，gandareentirefunctionsg2transcendentalandcr(g2)notequal