The Laplace Transform

235 Pages·2012·5.56 MB·English

The Laplace Transform:
Theory and Applications
Joel L. Schiff
Springer

Contents
Preface ix
1 BasicPrinciples 1
1.1 TheLaplaceTransform . . . . . . . . . . . . . . . . . 1
1.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 ContinuityRequirements . . . . . . . . . . . . . . . . 8
1.4 ExponentialOrder . . . . . . . . . . . . . . . . . . . . 12
1.5 TheClassL . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 BasicPropertiesoftheLaplaceTransform . . . . . . 16
1.7 InverseoftheLaplaceTransform . . . . . . . . . . . 23
1.8 TranslationTheorems . . . . . . . . . . . . . . . . . . 27
1.9 Differentiation and Integration of the
LaplaceTransform . . . . . . . . . . . . . . . . . . . . 31
1.10 PartialFractions . . . . . . . . . . . . . . . . . . . . . 35
2 ApplicationsandProperties 41
2.1 GammaFunction . . . . . . . . . . . . . . . . . . . . 41
2.2 PeriodicFunctions . . . . . . . . . . . . . . . . . . . . 47
2.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4 OrdinaryDifferentialEquations . . . . . . . . . . . . 59
2.5 DiracOperator . . . . . . . . . . . . . . . . . . . . . . 74
xiii
xiv Contents
2.6 AsymptoticValues . . . . . . . . . . . . . . . . . . . . 88
2.7 Convolution. . . . . . . . . . . . . . . . . . . . . . . . 91
2.8 Steady-StateSolutions . . . . . . . . . . . . . . . . . . 103
2.9 DifferenceEquations . . . . . . . . . . . . . . . . . . 108
3 ComplexVariableTheory 115
3.1 ComplexNumbers . . . . . . . . . . . . . . . . . . . . 115
3.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . 128
3.4 PowerSeries . . . . (cid:2). . . . . . . . . . . . . . . . . . . 136
∞
3.5 IntegralsoftheType −∞f(x)dx . . . . . . . . . . . . 147
4 ComplexInversionFormula 151
5 PartialDifferentialEquations 175
Appendix 193
References 207
Tables 209
LaplaceTransformOperations . . . . . . . . . . . . . . . . 209
TableofLaplaceTransforms . . . . . . . . . . . . . . . . . . 210
AnswerstoExercises 219
Index 231
1
C H A P T E R ........................................... BPraisnicciples
Ordinaryandpartialdifferentialequationsdescribethewaycertain
quantitiesvarywithtime,suchasthecurrentinanelectricalcircuit,
theoscillationsofavibratingmembrane,ortheflowofheatthrough
aninsulatedconductor.Theseequationsaregenerallycoupledwith
initialconditionsthatdescribethestateofthesystemattimet(cid:1)0.
Averypowerfultechniqueforsolvingtheseproblemsisthatof
theLaplacetransform,whichliterallytransformstheoriginaldiffer-
entialequationintoanelementaryalgebraicexpression.Thislatter
canthensimplybetransformedonceagain,intothesolutionofthe
originalproblem.Thistechniqueisknownasthe“Laplacetransform
method.”ItwillbetreatedextensivelyinChapter2.Inthepresent
chapter we lay down the foundations of the theory and the basic
propertiesoftheLaplacetransform.
1.1 The Laplace Transform
Suppose that f is a real- or complex-valued function of the (time)
variablet > 0andsisarealorcomplexparameter.Wedefinethe
1
2 1. BasicPrinciples
Laplacetransformoff as
(cid:1)
(cid:3) (cid:4) ∞
F(s)(cid:1)L f(t) (cid:1) e−stf(t)dt
0 (cid:1)
τ
(cid:1) lim e−stf(t)dt (1.1)
τ→∞ 0
whenever the limit exists (as a finite number). When it does, the
integral(1.1)issaidtoconverge.Ifthelimitdoesnotexist,theintegral
issaidtodivergeandthereisnoLaplacetransformdefinedforf.The
notation L(f) will also be used to denote the Laplace transform of
f,andtheintegralistheordinaryRiemann(improper)integral(see
Appendix).
Theparametersbelongstosomedomainonthereallineorin
the complex plane. We will choose s appropriately so as to ensure
theconvergenceoftheLaplaceintegral(1.1).Inamathematicaland
technicalsense,thedomainofs isquiteimportant.However,ina
practicalsense,whendifferentialequationsaresolved,thedomain
ofsisroutinelyignored.Whensiscomplex,wewillalwaysusethe
notations(cid:1)x+iy.
The symbol L is the Laplace transformation, which(cid:3)act(cid:4)s on
functionsf (cid:1)f(t)andgeneratesanewfunction,F(s)(cid:1)L f(t) .
Example1.1. Iff(t)≡1fort≥0,then
(cid:1)
(cid:3) (cid:4) ∞
L f(t) (cid:1) e−st1dt
0
(cid:5) (cid:6) (cid:7)
(cid:1)τl→im∞ e−−sst(cid:6)(cid:6)(cid:6)τ0
(cid:5) (cid:7)
e−sτ 1
(cid:1) lim + (1.2)
τ→∞ −s s
1
(cid:1)
s
providedofcoursethats>0(ifsisreal).Thuswehave
1
L(1)(cid:1) (s>0). (1.3)
s
1.1. TheLaplaceTransform 3
Ifs≤0,thentheintegralwoulddivergeandtherewouldbenore-
sultingLaplacetransform.Ifwehadtakenstobeacomplexvariable,
thesamecalculation,withRe(s)>0,wouldhavegivenL(1)(cid:1)1/s.
Infact,letusjustverifythatintheabovecalculationtheintegral
canbetreatedinthesamewayevenifsisacomplexvariable.We
requirethewell-knownEulerformula(seeChapter3)
eiθ (cid:1)cosθ+i sinθ, θ real, (1.4)
andthefactthat|eiθ|(cid:1)1.Theclaimisthat(ignoringtheminussign
aswellasthelimitsofintegrationtosimplifythecalculation)
(cid:1)
est
estdt(cid:1) , (1.5)
s
fors(cid:1)x+iyanycomplexnumber(cid:7)(cid:1)0.Toseethisobservethat
(cid:1) (cid:1)
estdt(cid:1) e(x+iy)tdt
(cid:1) (cid:1)
(cid:1) extcosytdt+i extsinytdt
byEuler’sformula.Performingadoubleintegrationbypartsonboth
theseintegralsgives
(cid:1) (cid:8) (cid:9)
ext
estdt(cid:1) (xcosyt+ysinyt)+i(xsinyt−ycosyt) .
x2+y2
Nowtheright-handsideof(1.5)canbeexpressedas
est e(x+iy)t
(cid:1)
s x+iy
ext(cosyt+isinyt)(x−iy)
(cid:1)
x2+y2
(cid:8) (cid:9)
ext
(cid:1) (xcosyt+ysinyt)+i(xsinyt−ycosyt) ,
x2+y2
whichequalstheleft-handside,and(1.5)follows.
Furthermore, we obtain the result of (1.3) for s complex if we
takeRe(s)(cid:1)x>0,sincethen
lim |e−sτ|(cid:1) lim e−xτ (cid:1)0,
τ→∞ τ→∞
4 1. BasicPrinciples
killingoffthelimittermin(1.3).
Let us use the preceding to calculate L(cosωt) and L(sinωt)
(ωreal).
Example1.2. Webeginwith
(cid:1)
∞
L(eiωt)(cid:1) e−steiωtdt
0 (cid:6)
(cid:1)τl→im∞ eiω(iω−−s)st(cid:6)(cid:6)(cid:6)τ0
1
(cid:1) ,
s−iω
since limτ→∞|eiωτe−sτ| (cid:1) limτ→∞e−xτ (cid:1) 0, provided x (cid:1) Re(s) >
0. Similarly, L(e−iωt) (cid:1) 1/(s+iω). Therefore, using the linearity
propertyofL,whichfollowsfromthefactthatintegralsarelinear
operators(discussedinSection1.6),
(cid:5) (cid:7)
L(eiωt)+L(e−iωt) eiωt+e−iωt
(cid:1)L (cid:1)L(cosωt),
2 2
andconsequently,
(cid:5) (cid:7)
1 1 1 s
L(cosωt)(cid:1) + (cid:1) . (1.6)
2 s−iω s+iω s2+ω2
Similarly,
(cid:5) (cid:7)
1 1 1 ω (cid:3) (cid:4)
L(sinωt)(cid:1) − (cid:1) Re(s)>0 .
2i s−iω s+iω s2+ω2
(1.7)
The Laplace transform of functions defined in a piecewise
fashionisreadilyhandledasfollows.
Example1.3. Let(Figure1.1)
(cid:10)
t 0≤t≤1
f(t)(cid:1)
1 t>1.
Exercises1.1 5
f(cid:0)t(cid:1)
(cid:2)
FIGURE1.1
O (cid:2) t
Fromthedefinition,
(cid:1)
(cid:3) (cid:4) ∞
L f(t) (cid:1) e−stf(t)dt
0
(cid:1) (cid:1)
1 τ
(cid:1) te−stdt+ lim e−stdt
0 τ→∞ 1
(cid:6) (cid:1) (cid:6)
(cid:1) te−−sst(cid:6)(cid:6)(cid:6)10+1s 01e−stdt+τl→im∞e−−sst(cid:6)(cid:6)(cid:6)τ1
1−e−s (cid:3) (cid:4)
(cid:1) Re(s)>0 .
s2
Exercises 1.1
(cid:3) (cid:4)
1. From the definition of the Laplace transform, compute L f(t)
for
(a)f(t)(cid:1)4t (b)f(t)(cid:1)e2t
(c)f(t)(cid:1)2cos3t (d)f(t)(cid:1)1−cosωt
(e)f(t)(cid:1)te2t (f)f(t)(cid:1)etsint

(g)f(t)(cid:1)(cid:11)01 tt<≥aa (h)f(t)(cid:1)sin0ωt 0π<≤tt< ωπ
ω
6 1. BasicPrinciples
(cid:11)
2 t≤1
(i)f(t)(cid:1)
et t>1.
2. ComputetheLaplacetransformofthefunctionf(t)whosegraph
isgiveninthefiguresbelow.
f(cid:0)t(cid:1) f(cid:0)t(cid:1)
(cid:0)a(cid:1) (cid:0)b(cid:1)
(cid:2) (cid:2)
O FIGUREE(cid:2).1 t O FIGU(cid:2)REE.2 (cid:3) t
1.2 Convergence
Although the Laplace operator can be applied to a great many
functions, there are some for which the integral (1.1) does not
converge.
Example1.4. Forthefunctionf(t)(cid:1)e(t2),
(cid:1) (cid:1)
τ τ
lim e−stet2dt(cid:1) lim et2−stdt(cid:1)∞
τ→∞ 0 τ→∞ 0
foranychoiceofthevariables,sincetheintegrandgrowswithout
boundasτ →∞.
InordertogobeyondthesuperficialaspectsoftheLaplacetrans-
form,weneedtodistinguishtwospecialmodesofconvergenceof
theLaplaceintegral.
Theintegral(1.1)issaidtobeabsolutelyconvergentif
(cid:1)
τ
lim |e−stf(t)|dt
(cid:3) (cid:4) τ→∞ 0
exists.IfL f(t) doesconvergeabsolutely,then
(cid:6)(cid:6)(cid:6)(cid:1) τ(cid:8) (cid:6)(cid:6)(cid:6) (cid:1) τ(cid:8)
(cid:6) e−stf(t)dt(cid:6)≤ |e−stf(t)|dt→0
(cid:6) (cid:6)
τ τ