您好,欢迎访问三七文档
当前位置:首页 > 商业/管理/HR > 管理学资料 > [Fourier-series傅里叶级数]例题01
Math462:HW5SolutionsDueonAugust15,2014JackyChong1JackyChongRemark:WeareworkinginthecontextofRiemannIntegrals.Problem15.1.4FindtheFouriercosineseriesofthefunctionsjsinxjintheinterval( ;).Useittondthesums1Xn=114n2 1and1Xn=1( 1)n4n2 1:Solution:Sincejsinxjisanevenfunctionon( ;),thenithasaFouriercosineseriesgivenbyjsinxj12A0+1Xn=1AncosnxwheretheAnscanbereadilycomputedbytheintegralformulaAn=1Z jsinxjcosnxdx=1Z0sinxcosnxdx 1Z0 sinxcosnxdx=2Z0sinxcosnxdxConsiderthetrigonometricidentitysin[(n+1)x] sin[(n 1)x]2=sinxcosnx;thenitfollowsAn=( 41n2 1ifniseven0ifnisoddforalln2N.Moreover,itistrivialtocheckthatA0=4.HencetheFouriercosineseriesisgivenbyjsinxj2 4Xevenn1n2 1cosnx=2 41Xn=114n2 1cos2nx:AssumetheFouriercosineseriesconvergespointwisetojsinxjon( ;),thenwehavethatjsinxj=2 41Xn=114n2 1cos2nx:Setx=0,wegetthat1Xn=114n2 1=12:Setx=2,wegetthat1=2 41Xn=1( 1)n4n2 1or1Xn=1( 1)n4n2 1=12 4:Page2of9JackyChongProblem1Problem25.2.11FindthefullFourierseriesofexon( l;l)initsrealandcomplexforms.Solution:ThecomplexformofthefullFourierseriesofexon( l;l)isgivenbyex1Xn= 1cneinx=lwherecncanbereadilycomputeviathefollowingformulawhichyieldscn=12lZl lexp1 inlxdx=el in e l+in2(l in):Henceitfollowsex1Xn= 1el in e l+in2(l in)einx=l=1Xn= 1( 1)nsinhll+inl2+n22einx=l:Therealformisgivenbyex1Xn= 1( 1)nsinhll+inl2+n22einx=l=sinhll+ 1Xn= 1( 1)nsinhll+inl2+n22einx=l+1Xn=1( 1)nsinhll+inl2+n22einx=l=sinhll+1Xn=1( 1)nsinhll inl2+n22e inx=l+1Xn=1( 1)nsinhll+inl2+n22einx=l=sinhll+1Xn=1( 1)nsinhll2+n22l(einx=l+e inx=l) neinx=l e inx=li=sinhll+21Xn=1( 1)nsinhll2+n22hlcosnxl nsinnxli:Problem35.3.10(TheGram-Schmidtorthogonalizationprocedure)IfX1;X2;:::isanysequence(niteorinnite)oflinearlyindependentvectorsinanyvectorspacewithaninnerproduct,itcanbereplacedbyasequenceoflinearcombinationsthataremutuallyorthogonal.Theideaisthatateachsteponesubtractsothecomponentsparalleltothepreviousvectors.Theprocedureisasfollows.First,weletZ1=X1=kX1k.Second,wedeneY2=X2 (X2;Z1)Z1andZ2=Y2kY2k:Third,wedeneY3=X3 (X3;Z2)Z2 (X3;Z1)Z1andZ3=Y3kY3k;andsoon.(a)ShowthatallthevectorsZ1;Z2;Z3;:::areorthogonaltoeachother.(b)Applytheproceduretothepairoffunctionscosx+cos2xand3cosx 4cos2xintheinterval(0;)toProblem3continuedonnextpage...Page3of9JackyChongProblem3(continued)getanorthogonalpair.Solution:(a)Weshallprovethestatementbystronginduction.Thebasecaseistriviallytrue.Now,supposeZ1;:::;Zk 1aremutuallyorthogonal,i.e.(Zi;Zj)=(1ifi=j0otherwisefori=1;:::;k 1andj=1;:::;k 1.ConsiderZk=Xk Pk 1i=1(Xk;Zi)ZikXk Pk 1i=1(Xk;Zi)Zikthenforax1lk 1wehavethat(Zk;Zl)=(Xk;Zl) Pk 1i=1(Xk;Zi)(Zi;Zl)kXk Pk 1i=1(Xk;Zi)Zik=(Xk;Zl) Pk 1i=1(Xk;Zi)ilkXk Pk 1i=1(Xk;Zi)Zik=(Xk;Zl) (Xk;Zl)kXk Pk 1i=1(Xk;Zi)Zik=0:HenceZkisorthogonaltoallZlwherel=1;:::;k 1.Moreover,wealsohavethat(Zk;Zk)=(Xk Pk 1i=1(Xk;Zi)Zi;Xk Pk 1i=1(Xk;Zi)Zi)kXk Pk 1i=1(Xk;Zi)Zik2=1:(b)Observekcosx+cos2xk22=Z0jcosx+cos2xj2dx=Z0cos2x+2cosxcos2x+cos22xdx=Z01+cos2x2+cos3x+cosx+1+cos4x2dx=Z01+cosx+12cos2x+cos3x+12cos4xdx=;thenitfollowsZ1=cosx+cos2xkcosx+cos2xk2=1p(cosx+cos2x):Next,observe(X2;Z1)=1pZ0(cosx+cos2x)(3cosx 4cos2x)dx=1pZ03cos2x cosxcos2x 4cos22xdx=12pZ0 1 cosx+3cos2x cos3x 4cos4xdx= p2Page4of9JackyChongProblem3whichmeansY2=X2 (X2;Z1)Z1=72(cosx cos2x):ComputingthenormofY2yieldskY2k22=494Z0(cosx cos2x)2dx=494thenitfollowsZ2=Y2kY2k2=1p(cosx cos2x):Problem45.3.12ProveGreen'srstidentity:Foreverypairoffunctionsf(x);g(x)on(a;b),Zbaf00(x)g(x)dx= Zbaf0(x)g0(x)dx+f0gba:Solution:Assumef2C2[a;b]andg2C1[a;b],thenf0g2C1[a;b].Inparticular,wemayapplytheFundamentalTheoremofCalculusforRiemannIntegraltogetf0(x)g(x)ba=Zbadds(f0(s)g(s))0ds=Zbaf00(s)g(s)+f0(s)g0(s)ds:HenceitfollowsZbaf00(x)g(x)dx=f0(x)g(x)ba Zbaf0(x)g0(x)dx:Problem55.4.5Let(x)=0for0x1and(x)=1for1x3.(a)FindtherstfournonzerotermsofitsFouriercosineseriesexplicitly.(b)Foreachx(0x3),whatisthesumofthisseries?(c)Doesitconvergeto(x)intheL2sense?Why?(d)Putx=0tondthesum1+12 14 15+17+18 110 111+:Solution:(a)It'sclearthatA0=43.Next,observeAn=23Z31cosnx3dx= 2nsinn3Page5of9JackyChongProblem5wheresinn3=8:0ifn0;3mod6p32ifn1;2mod6 p32ifn4;5mod6:Hence,itfollowsAn=8:0ifn0;3mod6 p3nifn1;2mod6p3nifn4;5mod6:ThisgivesusthefollowingFouriercosineseries(x)23 p3cosx3 p32cos2x3+p34cos4x3+p35cos5x3 :::(b)ByTheorem4(ii),theFouriercosineseriesconvergespointwiseeverywhereonR.Moreover,letusextendtoanevenperiodicfunctionthenforeachxedx2[0;3]wehavethat12A0+1Xn=1Ancosnx3=12[ext(x+)+ext(x )]where12[ext(x+)+ext(x )]=8:0if0x112ifx=1;31if1x3(c)Since(x)isL2integrable,thenbyTheorem3weknowtheFouriercosineseriesdoesindeedconvergeintheL2senseto.(d)Setx=0yields0=23 p3 p32+p34+p35 p37 p38+:::or23p3=1+12 14 15+17+18 110 111+:Problem65.4.15Let(x)1for0x.Expand1=1Xn=0Bncosn+12x:(a)FindBn.(b)Let 2x2.Forwhichsuchxdoesthisseriesconverges?Foreachsuchx,whatisthesumoftheseries?(c)ApplyParseval'sequalitytothisseries.Useittocalculatethesum1+132+152+:Problem6conti
本文标题:[Fourier-series傅里叶级数]例题01
链接地址:https://www.777doc.com/doc-6214000 .html