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1、SINGULARINTEGRALEQUATIONSWITHCOSECANTKERNELINSOLUTIONSWITHSINGULARITIESOFORDERONENo.3Dm&Du:SINGULARINTEGRALEQUATIONSWITHCOSECANTKERNEL411TheinvestigationonSIEofCauchykernelwithsolutionhavingsingularitiesoforderoneandhighorderhasbeencompletedin『4-6].Inf2-3]and[8],thei
2、nversionandmodifiedinver—sionproblemsofSIEwithcosecantkernelwerediscussed.InSection2ofthisarticle,wefirstrecallsomeresultsfortheperiodicRiemannboundaryvalueproblem(PRHproblem)alongclosedsmoothcontoursandthesolutionof(1.1)(cf.f1,2】),andthengiveoutthesolutionandthesolvable
3、conditionsofSIEwithcosecantkernel,whicharethefoundationoftheconsideredproblem.InSection3,wediscusstheperiodicitiesoforderone(PRHproblem)andobtainbytransformingittoPRHproblem.2Riemannboundaryvalueproblemwithsingular—thesolutionandthesolvableconditionsof(1.1)PRHProblemandt
4、heSolutioninH(r0)ofProblem(1.1)ThePRHproblemis+(t)=G(t)一(t)+9@),t∈L(2.1)whereLisasDefinition1,c(t)≠o,c(t+2at)=G(t),andg(t+2at)=9(t)∈日().Thesolutionfz)issectionallyanalyticandsatisfies(z+2at)=(z),orsimply(z)EPAH,andf土∞i)isbounded.ThecorrespondingPRHproblemislikethePRHprob
5、lemexceptthat口㈤∈PHandthesolution(z)∈PAH.Byconformalmap<=exp(iz/a),(2.1)istransformedtoRiemannboundaryvalueproblemof(一plane,thenreturntoz—plane,weget,when0,thePRHproblem(2.1)hassolution:(z).(cot+)dt+X㈤/n)(2.2)wherefz)isacanonicalfunction,=IndL0a(t)istheindexof(2.1),isp
6、olynomialoforderatmost.When=一1,(2.1)hasuniquesolution(2.2)(—=0);when<一1(2.1)hasuniquesolution(2.2)(=0)iffthesolvableconditions.[g(t)/x+(t)]e出=0,J=1.….一一1hold.NextweconsiderthesolutioninH(r0)ofproblem(1.1).TheassumptionsareasinSIE(1.1)exceptthatf(t)∈H(r0)andtheunknownf
7、unction(t)∈H(r0).LetFbethecontourinstrjD一血7r<Rez<0extendedsymmetricallywithrespecttothey-axisF0.Denote0:F0+r,ListhesetofcontoursextendedbyL0withperiod2at.Weextendthefunctionstoas(t+07r)=(t),B(t+a'K)=B(t),f(t+a'K)=一,(t),and(t+aTr)一(t)?Let(z)=(2ari)一(t)csc[(t—z)/a]dt
8、.If(t)isthesolutioninH(r0)ofproblem(1.1),then(z)issectionallyanalyticwiththejumpcontourLandsatisfies:+(
