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1、Fourier,Joseph(1768--1830)lFourierwasaFrenchmathematician,whoasayoungboyaspiredtobecomeanArmyofficer.Hewasdeniedthatopportunityformilitaryserviceand,Fourier变换therefore,switchedhispassiontomathematics.Afterbuildinghisreputationasamathematicalscholar,hebecameinvolvedin
2、thepoliticalturmoiloftheFrenchrevolution.HewasimprisonedbutthenreleasedtoattendtheÉcolenormaleandlatertoteachattheÉcolepolytechnique,asateachingaidtoMongeandLagrange.MongechoseFouriertoaccompanyNapoleon'sEgyptianexpeditionasatechnicaladvisorforengineeringandtechnical
3、research.HebecameafriendofNapoleon.FourierspentanumberofyearsdirectingpublicimprovementsfortheNapoleonicgovernmentandpublishingthefindingsoftheEgyptianexpedition.Somehow,Fouriercontinuedhismathematicalresearchandmadecontributionsinthestudyandcomputationofheatdiffusio
4、nandthesolutionofdifferentialequations.MuchofthatworkappearsinThéorieanalytiquedelachaleur(TheAnalyticalTheoryofHeat,1822),inwhichhemadeextensiveuseoftheseriesthatbearhisname.However,hecontributednothingtothemathematicaltheoryoftheseseries,whichwerewellknownmuchearli
5、ertoEuler,DanielBernoulli,andLagrange.FourierreceivedsupportforhisworkfromLaplace,despitereceivingcriticismforthatworkfromPoisson.周期函数的Fourier展开基本三角函数系的正交性质a¥+¥ìü22nnppf(t)=0++(acosntbwwsin)nt基本三角函数系íýcosxx，sin的正交性质：TånnîþTT2n=1n=0T22nmpp其中f为以周期为T的函数，满足Dirichlet条件。2c
6、osxsinxdx=0,Tò-TTT2TT2p2222Tì0,m¹n,0mn==ww=，a==f(t)dt,af()costntdt22nmppï0òòTTTnT2cosxcosxdx=íTTTT--ò-T22TT,0mn=¹2ïî2T22ì0,m¹n,0mn==bn=òTfT()sitnnwtdtT22nmppïT-2sinxsinxdx=íT2òT-TTï,0mn=¹2î2Fourier级数的复数形式的导出非周期函数的Fourier展开a¥f(t)=limftT()f(t)=0++(acosntwwbsin)ntT®¥Tån
7、n2n=1¥éùT¥inwt--inwtinwwtintf(t)=lim1êú2fe()tt-iwnntwdeita0e+-eeeåòTT=++å()abnnT®¥Tn=-¥-ëû2222n=1i2p¥D=-www=a0an-+ibninwwtainnb-intnnn-1T1¥éùT=++å()ee=Dlimêú2fe()ttw-iwnntwdeit2n=122D®wn02påò-TTnn=-¥ëû21T¥æöeeinwtTT-intw=2ft()dt++ç÷22ft()e-inwtdtft()eintwdtT¥òTTåòòTT
8、TT1éùT-2n=1èøTT--22=Dlimêú2fe()ttw-iwnntwdeitåòTTn¥¥¥TT®¥2pn=-¥-1éùëû2=c++==éùceitwncceiw-nteeiwntêú2fd()ttiwnntweitw=nw0åëûnn-åånò
