关于多元幂级数

[]2005-10-06第23卷第3期大学数学Vol.23,.32007年6月COLLEGEMATHEMATICSJun.2007高朝邦(,610106)[],;.[];;;[]O173[]C[]1672-1454(2007)03-0125-051,[1-2],[3-4].,,.,;,.,.Du1(x,y),u2(x,y),,un(x,y),,n=0un(x,y)D,.(x0,y0)D,n=0un(x0,y0),(x0,y0)n=0un(x,y);n=0un(x0,y0),(x0,y0).n=0un(x,y),D0;.D0(x,y),n=0un(x,y)s(x,y),D0n=0un(x,y)s(x,y)(x,y),n=0un(x,y),s(x,y)=n=0un(x,y),(x,y)D0.2un(x,y)=nj=0anjxn-jyj,n=0nj=0anjxn-jyj,,anj(j=0,1,,n;n=0,1,2,).,n.,un(x,y)=(x+y)n=nj=0Cjnxn-jyj,n=0nj=0Cjnxn-jyj=n=0(x+y)n=1+(x+y)+(x+y)2++(x+y)n+.(1)1|x+y|1,,.(1),(1)11-x-y=n=0(x+y)n,D0|x+y|1.,,1f(x1,x2,,xn)Dn,{ak}k=0,R=limkakak+1.nn=0ak(f(x1,x2,,xn))k(i)G:|f(x1,x2,,xn)|R;(ii)G:|f(x1,x2,,xn)|R;(iii)S:|f(x1,x2,,xn)|=R.11Rk=0ak(f(x1,x2,,xn))k.1k=0ak(f(x1,x2,,xn))k.R=,D.R=0,0f(x1,x2,,xn)S:f(x1,x2,,xn)=0;0f(x1,x2,,xn).0R,m=inff(x1,x2,,xn),M=supf(x1,x2,,xn):()Rf(x1,x2,,xn),a)m-RRMD0:|f(x1,x2,,xn)|R;b)-RmRMD0:mf(x1,x2,,xn)R.()Rf(x1,x2,,xn),a)m-RMRD0:-Rf(x1,x2,,xn)M;b)-RmMRD;c)Rm-RM.m,Mf(x1,x2,,xn).1,f(x1,x2,,xn)n,k=0ak(f(x1,x2,,xn))kn,2f(x1,x2,,xn)n,{ak}k=0,R=limkakak+1.nk=0ak(f(x1,x2,,xn))k(i)G:|f(x1,x2,,xn)|R;(ii)G:|f(x1,x2,,xn)|R;(iii)S:|f(x1,x2,,xn)|=R.21.126大学数学第23卷1k=1(-1)k-1k(xy)k.2-1xy1,ak=(-1)k-1k,k=1,2,3,,R=limkakak+1=1.xy=1,xy=-1,D0:-1xy1.(2)t=xy,k=0(-1)kktk,ln(1+t)=k=1(-1)k-1ktk,,ln(1+xy)=k=1(-1)k-1k(xy)k.23-1(x2+1)(y2-1)1k=0(x2+a)k(y2+b)k,a,b.R=1,f(x,y)=(x2+a)(y2+b).a0,b0,ab1,f(x,y)1,;ab1,D0:ab(x2+a)(y2+b)1.a0b0,D0:-1(x2+a)(y2+b)1.11-(x2+a)(y2+b)=k=0(x2+a)k(y2+b)k.a=1,b=-1,(3).3n=0n+13n+22nx2ny3n.4-12x2y312n=0n+13n+22nx2ny3n=n=0n+13n+22n(x2y3)n,R=limnanan+1=limn2n(n+1)3n+23(n+1)+22n+1(n+2)=12,,f(x,y)=x2y3=-12,f(x,y)=x2y3=12,D0:-12x2y312.(4)127第3期高朝邦:关于多元幂级数5x1,y14n=1n!nn1+1xyxn(x1,y1).f(x,y)=1+1xyx,x1,y1,1f(x,y)e.R=limnanan+1=limnn!nn(n+1)n+1(n+1)!=e,D:x1,y1.(5),..3,.f(x,y)(0,0),nTaylorf(x,y)=f(0,0)+xx+yyf(0,0)+12!xx+yy2f(0,0)++1n!xx+yynf(0,0)+1(n+1)!xx+yyn+1f(1x,2y),(01,21).,f(x,y):f(0,0)+xx+yyf(0,0)+12!xx+yy2f(0,0)++1n!xx+yynf(0,0)+,f(x,y)(0,0)TaylorMaclaurin.Taylor,:3f(x,y)(0,0)U(0,0),f(x,y)U(0,0)Taylorf(x,y)TaylorRn(x,y)=1(n+1)!xx+yyn+1f(1x,2y)n0,limnRn(x,y)=0,(x,y)U(0,0)..(x,y)U(0,0)f(x,y)=sn(x,y)+Rn(x,y),sn(x,y)f(x,y)nTaylor.,f(x,y)=limnsn(x,y),,limnRn(x,y)=limn(f(x,y)-sn(x,y))=0..(x,y)U(0,0),limnRn(x,y)=0,limn(f(x,y)-sn(x,y))=limnRn(x,y)=0,f(x,y)=limnsn(x,y),(x,y)U(0,0).3,Taylor:,f(x,y),fx,fy,fxx,fxy,fyy,;,,fx(0,0),fy(0,0),fxx(0,0),fxy(0,0),fyy(0,0),;,f(x,y):f(x,y)=n=01n!xx+yynf(0,0).,.128大学数学第23卷5f(x,y)=arctan(x+y2).,arctanx=n=0(-1)n2n+1x2n+1,x[-1,1],arctan(x+y2)=n=0(-1)n2n+1(x+y2)2n+1,D:|x+y2|1.[][1],,.()[M].:,52-103.[2].()[M].:,33-73.[3].[J].,1997,(2):3-5.[4].[J].,1998,28(3):267-269.OnPowerSeriesofFunctionsinSeveralVariablesGAOChao-bang(CollegeofInformationScienceandTechnology,ChengduUniversity,ChengduSichuan,610106,China)Abstract:Theconceptofseriesoffunctionsinseveralvariablesisintroduced,anditsdomainofconvergenceandsumfunctionaredefined.Thetechniquesofcomputingthedomainofconvergenceandsumfunctionandexpandingfunctionsinseveralvariablestopowerseriesoffunctionsinseveralvariablesaremainlydiscussedbymanyexamples.Keywords:seriesoffunctionsinseveralvariables;domainofconvergence;sumfunction;powerseriesoffunctionsinseveralvariables129第3期高朝邦:关于多元幂级数