復數與拉氏變換課件

ejx=cos

x+jsin

x

（此時z的模r=1）其中r=|z|是z的模,q

=arg

z是z的輻角.復數及其指數形式

復數z可以表示為Z=r(cosq+jsinq

)=rejq

,歐拉公式z=x+jy三角函數與復變量指數函數之間的聯系因為ejx

=cos

x+jsinx,

e-jx=cos

x-jsinx,所以

ejx+e-jx=2cosx,

ex-e-jx=2jsinx.

因此復變量指數函數的性質特殊地,有ex+jy

=

exej

y

=ex(cos

yjsin

y).)(21cosjxjxeex-+=,

)(21sinjxjxeejx--=.

歐拉公式復數項級數

設有復數項級數∑(univn),其中un,

vn(n=1,2,3,

)為實常數或實函數.

如果實部所成的級數∑un收斂于和u,并且虛部所成的級數∑vn收斂于和v,就說復數項級數收斂且和為u+iv.

如果級∑(univn)的各項的模所構成的級數∑|univn|收斂,則稱級數∑(univn)絕對收斂.絕對收斂復變量指數函數

考察復數項級數可以證明此級數在復平面上是絕對收斂的,在x軸上它表示指數函數ex,在復平面上我們用它來定義復變量指數函數,記為ez

.即歐拉公式

當x=0時,z=iy

,

=cos

y+jsin

y.于是這就是歐拉公式.把y換成x得eix=cos

x+jsin

x,復變量指數函數AppendixLesson-LaplaceTransformsLaplace,

Pierre

(1749-1827)Sources: http://scienceworld./biography/Laplace.html

http://mathworld./LaplaceTransform.htmlFrenchphysicistandmathematicianwhoputthefinalcapstoneonmathematicalastronomybysummarizingandextendingtheworkofhispredecessorsinhisfivevolumeMécaniqueCéleste(CelestialMechanics)(1799-1825).ThisworkwasimportantbecauseittranslatedthegeometricalstudyofmechanicsusedbyNewtontoonebasedoncalculus,knownasphysicalmechanics.Laplacealsosystematizedandelaboratedprobabilitytheoryin"Essai

Philosophique

surlesProbabilités"(PhilosophicalEssayonProbability,1814).HewasthefirsttopublishthevalueoftheGaussianintegral,.HestudiedtheLaplacetransform,althoughHeavisidedevelopedthetechniquesfully.Heproposedthatthesolarsystemhadformedfromarotatingsolarnebulawithringsbreakingoffandformingtheplanets.HediscussedthistheoryinExpositiondesystèmedumonde(1796).Hepointedoutthatsoundtravelsadiabatically,accountingforNewton'stoosmallvalue.Laplaceformulatedthemathematicaltheoryofinterparticulateforceswhichcouldbeappliedtomechanical,thermal,andopticalphenomena.Thistheorywasreplacedinthe1820s,butitsemphasisonaunifiedphysicalviewwasimportant.WithLavoisier,whosecalorictheoryhesubscribedto,hedeterminedspecificheatsformanysubstancesusingacalorimeterofhisowndesign.LaplaceborrowedthepotentialconceptfromLagrange,butbroughtittonewheights.HeinventedgravitationalpotentialandshoweditobeyedLaplace'sequationinemptyspace.Laplacebelievedtheuniversetobecompletelydeterministic.TheLaplaceTransformofafunction,f(t),isdefinedas;WhatistheLaplaceTransform?Letf(t)beagivenfunctionthatisdefinedforallt0.Wecantransformf(t)intoanewfunction,F(s),via:WhatistheInverseLaplaceTransform?LetF(s)beaLaplacetransformofafunctionf(t).Wecangetf(t)byinverse

LaplaceTransform,via:….andwecantransformitbacktoo!TheInverseLaplaceTransformisdefinedbyTheLaplaceTransformTransformPairs:

f(t)F(s)TheLaplaceTransformTransformPairs:

f(t)F(s)Yes!TheLaplaceTransformTimeDifferentiation:Wecanextendtheprevioustoshow;Whythetransform?Amethodtosolvedifferentialequationsandcorrespondinginitialandboundaryvalueproblems,particularlyusefulwhendrivingforcesarediscontinuous,impulsive,oracomplicatedperiodic/aperiodicfunction.Transformthesubsidiaryequation’ssolutiontoobtainthesolutionofthegivenproblemGiventhehardproblem!Convertitintothesubsidiaryequation(SimpleProblem!)Solvethesubsidiaryequation(Purelyalgebraic!)Animportantpoint:Theaboveisastatementthatf(t)andF(s)aretransformpairs.Whatthismeansisthatforeachf(t)thereisauniqueF(s)andforeachF(s)thereisauniquef(t).IfwecanrememberthePairrelationshipsbetweenapproximately10oftheLaplacetransformpairswecangoalongway.TheLaplaceTransformBuildingtransformpairs:

Atransform

pairTheLaplaceTransformBuildingtransformpairs:u=tdv=e-stdtAtransformpairTheLaplaceTransformBuildingtransformpairs:AtransformpairTheLaplaceTransformTimeShiftTheLaplaceTransformFrequencyShiftTheLaplaceTransformExample:UsingFrequencyShiftFindtheL[e-atcos(wt)]Inthiscase,f(t)=cos(wt)so,TheLaplaceTransformTimeIntegration:Thepropertyis:TheLaplaceTransformTimeIntegration:MakingthesesubstitutionsandcarryingoutTheintegrationshowsthatTheLaplaceTransformTimeDifferentiation:IftheL[f(t)]=F(s),wewanttoshow:Integratebyparts:TheLaplaceTransformTimeDifferentiation:Makingtheprevioussubstitutionsgives,Sowehaveshown:TheLaplaceTransformFinalValueTheorem:Ifthefunctionf(t)anditsfirstderivativeareLaplacetransformableandf(t)hastheLaplacetransformF(s),andtheexists,thenAgain,theutilityofthistheoremliesinnothavingtotaketheinverseofF(s)inordertofindoutthefinalvalueoff(t)inthetimedomain.Thisisparticularlyusefulincircuitsandsystems.FinalValueTheoremTheLaplaceTransformFinalValueTheorem:Example:Given:Find.TheLaplaceTransformInitialValueTheorem:Ifthefunctionf(t)anditsfirstderivativeareLaplacetransformableandf(t)HastheLaplacetransformF(s),andtheexists,thenTheutilityofthistheoremliesinnothavingtotaketheinverseofF(s)inordertofindouttheinitialconditioninthetimedomain.Thisisparticularlyusefulincircuitsandsystems.InitialValueTheoremTheLaplaceTransformInitialValueTheorem:Example:Given;Findf(0)PartialFractionsExample#1PartialFractionsExample#2PartialFractionsExample#3LaplaceTransformPropertiesLinearity

Timeshifting

Frequencyshifting

Differentiation

intimeDifferentiationinTimePropertyLaplaceTransformPropertiesDifferentiationinfrequency

IntegrationintimeExample:f(t)=d(t)IntegrationinfrequencyLaplaceTransformPropertiesScalingintime/frequencyUnderintegration,ConvolutionintimeConvolutioninfrequencytf(t)2-2tf(2t)1-1Areareducedbyfactor2ExampleComputey(t)=eatu(t)*ebtu(t),wherea

bIfa=b,thenwewouldhaveresonanceWhatformwouldtheresonantsolutiontake?LinearDifferentialEquationsUsingdifferentiationintimeproperty

wecansolvedifferen