复数与拉氏变换

1、 ejx=cos x+jsin x （此时z的模r=1）其中r=|z|是z的模, q =arg z是z的辐角. v复数及其指数形式 复数z可以表示为Z = r (cosq+jsinq ) = rejq ,v欧拉公式 z=x+jyv三角函数与复变量指数函数之间的联系 因为 ejx =cos x+j sin x, e-jx=cos x-j sin x, 所以 ejx+e-jx=2cos x, ex-e-jx=2jsin x. 因此v复变量指数函数的性质特殊地, 有 ex+jy = exej y = ex(cos y+jsin y). )(21cosjxjxeex-+=, )(21sinjxjxee

2、jx-=. 2121zzzzeee=+. v复数项级数 设有复数项级数(un+ivn), 其中un, vn(n=1, 2, 3, )为实常数或实函数. 如果实部所成的级数un收敛于和u, 并且虚部所成的级数vn收敛于和v, 就说复数项级数收敛且和为u+iv. 如果级(un+ivn)的各项的模所构成的级数|un+ivn|收敛, 则称级数(un+ivn)绝对收敛. v 绝对收敛v复变量指数函数 考察复数项级数 可以证明此级数在复平面上是绝对收敛的, 在x轴上它表示指数函数ex, 在复平面上我们用它来定义复变量指数函数, 记为ez . 即 !1 ! 2112 + +nznzz. !1 ! 2112

3、+ +=nzznzze. v欧拉公式 当x=0时, z=iy , =cos y+jsin y. 于是 这就是欧拉公式. 把y换成x得 eix=cos x+jsin x, v复变量指数函数 !1 ! 2112 + +=nzznzze. )(!1 )(! 2112 + +=niyiyniyiye -+-+= ! 51! 41! 31! 2115432yiyyiyiy) ! 51! 31() ! 41! 211 (5342 -+-+ -+-=yyyiyyAppendix Lesson - Laplace TransformsLaplace,Pierre(1749-1827) Sources: htt

4、p:/ http:/ physicist and mathematician who put the final capstone on mathematical astronomy by summarizing and extending the work of his predecessors in his five volume Mcanique Cleste (Celestial Mechanics) (1799-1825). This work was important because it translated the geometrical study of mechanics

5、 used by Newton to one based on calculus, known as physical mechanics. Laplace also systematized and elaborated probability theory in Essai Philosophique sur les Probabilits (Philosophical Essay on Probability, 1814). He was the first to publish the value of the Gaussian integral, . He studied the L

6、aplace transform, although Heaviside developed the techniques fully. He proposed that the solar system had formed from a rotating solar nebula with rings breaking off and forming the planets. He discussed this theory in Exposition de systme du monde (1796). He pointed out that sound travels adiabati

7、cally, accounting for Newtons too small value. Laplace formulated the mathematical theory of interparticulate forces which could be applied to mechanical, thermal, and optical phenomena. This theory was replaced in the 1820s, but its emphasis on a unified physical view was important. With Lavoisier,

8、 whose caloric theory he subscribed to, he determined specific heats for many substances using a calorimeter of his own design. Laplace borrowed the potential concept from Lagrange, but brought it to new heights. He invented gravitational potential and showed it obeyed Laplaces equation in empty spa

9、ce. Laplace believed the universe to be completely deterministic. The Laplace Transform of a function, f(t), is defined as;-=0)()()(dtetfsFtfLstWhat is the Laplace Transform? Let f(t) be a given function that is defined for all t 0. We can transform f(t) in to a new function, F(s), via:What is the I

10、nverse Laplace Transform? Let F(s) be a Laplace transform of a function f(t). We can get f(t) by inverse Laplace Transform , via:.and we can transform it back too!The Inverse Laplace Transform is defined by+-=jjtsdsesFjtfsFL )(21)()(1Why the transform?A method to solve differential equations and cor

11、responding initial and boundary value problems, particularly useful when driving forces are discontinuous, impulsive, or a complicated periodic/aperiodic function.Transform the subsidiary equations solution toobtain the solutionof the given problemGiven thehard problem!Convert it intothe subsidiary

12、equation (Simple Problem!)Solve thesubsidiary equation (Purely algebraic!)1)0(, 245=+yydtdy -000121nnnnnnffsfssFsdtfd 2514sY sY ss-+=nSolution: First, take L of both sidesRearrange,Take L-1, 15254sy tss-+=+LFrom Table :tey8 . 05 . 05 . 0) t (-+= -000121nnnnnnffsfssFsdtfd 414 62 51 2 2+=+-+-ssssYsYss

13、sYsAn important point :)()(sFtfThe above is a statement that f(t) and F(s) are transform pairs. What this means is that for each f(t) there is a unique F(s) and for each F(s) there is a unique f(t). If we can remember the Pair relationships between approximately 10 of the Laplace transform pairs we

14、can go a long way.The Laplace TransformBuilding transform pairs:eL(+-=etasstatatdtedteetueL0)(0)(asasetueLstat+ += =+ +- -= = - - -1)()(|0astueat+-1)( A transform pairThe Laplace TransformBuilding transform pairs:-=0)(dttettuLst-=000|vduuvudvu = tdv = e-stdt21)(sttuA transform pairThe Laplace Transf

15、ormBuilding transform pairs:22011212)()cos(wssjwsjwsdteeewtLstjwtjwt+=+-=+=-22)()cos(wsstuwt+A transform pairThe Laplace TransformTime Shift-+-=+=-=-=-00)()()(,.,0,)()()(dxexfedxexfSoxtasandxatAsaxtanddtdxthenatxLeteatfatuatfLsxasaxsast)()()(sFeatuatfLas-=-The Laplace TransformFrequency Shift+-+=0)(

16、0)()()()(asFdtetfdtetfetfeLtasstatat)()(asFtfeLat+=-The Laplace TransformExample: Using Frequency ShiftFind the Le-atcos(wt)In this case, f(t) = cos(wt) so,2222)()()()(wasasasFandwsssF+=+=22)()()()cos(wasaswteLat+=-The Laplace TransformTime Integration:The property is:ststtsttesvdtedvanddttfdudxxfuL

17、etpartsbyIntegratedtedxxfdttfL-= 1,)(,)(:)()(0000The Laplace TransformTime Integration:Making these substitutions and carrying outThe integration shows that)(1)(1)(00sFsdtetfsdttfLst=-The Laplace TransformTime Differentiation:If the Lf(t) = F(s), we want to show:)0()()(fssFdttdfL-=Integrate by parts

18、:)(),()(,tfvsotdfdtdttdfdvanddtsedueustst= = = =- -= = =- - -The Laplace TransformTime Differentiation:Making the previous substitutions gives,-+-=-=000)()0(0)()(|dtetfsfdtsetfetfdtdfLstststSo we have shown:)0()()(fssFdttdfL-=The Laplace TransformFinal Value Theorem:If the function f(t) and its firs

19、t derivative are Laplace transformable and f(t)has the Laplace transform F(s), and the exists, then)(limssF s)()(lim)(lim = = =ftfssF0s tAgain, the utility of this theorem lies in not having to take the inverseof F(s) in order to find out the final value of f(t) in the time domain. This is particula

20、rly useful in circuits and systems.Final Value TheoremThe Laplace TransformFinal Value Theorem:Example:Given: ttesFnotesssFt3cos)(3)2(3)2()(212222- -= =+ + +- -+ += =- -Find )( f. 03)2(3)2(lim)(lim)(2222= =+ + +- -+ += = = sssssFf0s0sThe Laplace TransformInitial Value Theorem:If the function f(t) an

21、d its first derivative are Laplace transformable and f(t)Has the Laplace transform F(s), and the exists, then)(limssF0)0()(lim)(lim = = =tsftfssFThe utility of this theorem lies in not having to take the inverse of F(s) in order to find out the initial condition in the time domain. This is particula

22、rly useful in circuits and systems. sInitial Value TheoremThe Laplace TransformInitial ValueTheorem:Example:Given;225)1()2()(+ + + += =sssFFind f(0)1)26(22lim25122lim5)1()2(lim)(lim)0(22222222222= =+ + + += = + + + + += =+ + + += = =ssssssssssssssssssFf s s s s=-00010)(tttttt1t0(t)t0 The Laplace tra

23、nsform of a unit impulse:Pictorially, the unit impulse appears as follows:0t0f(t) (t t0)Mathematically: (t t0) = 0 t 0*note01)(000=-+- dttttt=-0000)(0)()(tttftttttff(t)t0t0f(t) (t-t0) The Laplace transform of a unit impulse:An important property of the unit impulse is a sifting or sampling property.

24、 The following is an important.=-21201020100,0)()()(ttttttttttfdttttf =0100)(tttu1t0u(t)The Laplace TransformLaplace Transform of the unit step.*notes|0011)(-=ststesdtetuLstuL1)(=The Laplace Transform of a unit step is:s1=-atatatu10)(1t0a1t0+Tu(t-) - u(t- -T) The Laplace transform of a unit impulse:

25、In particular, if we let f(t) = (t) and take the Laplace1)()(00=-sstedtettL 22)(wss+22)(wsw+22wsw+22wss+1tntate-s121s1!+nsn)(1as +)cos(wt)sin(wt)cos(wteat-)sin(wteat-)(tf)(sF)(tf)(sFT1s1teT421-ses-1(t)F(s)(t)1u(t) a constants1e-atas +1t21st e-at21as +)()()()()()(),()(sGsFtgtfLsGtgLsFtfL+=+=)()()()(-

26、=sFtfeLsFtfLt)()(00sFettfLst-=-)(1)(csFcctfL=)() 1()()0(.)0()0()()()()()()1()1(21)(sFtftLffsfssFstfLsFtfLnnnnnnnn-=-=-wDerivativessFtfLsdfLt)()(1)(0=wIntegralwConvolution)()()()()()(),()(0sHsFdthfLhfLsHthLsFtfLt=-=TheoremProperty(t)F(s)1ScalingA (t)A F(s)2Linearity1(t) 2(t)F1(s) F2(s)3Time Scaling(a

27、t)01aasaF F4Time Shifting(t-t0) u(t-t0)e-st0 F(s) t006Frequency Shiftinge-at (t)F(s+a)9Time DomainDifferentiationdttfd)(s F(s) - (0)7Frequency DomainDifferentiationt (t)dssd)(F F-10Time DomainIntegrationtdf0)()(1ssF F11Convolution-tdtff021)()(F1(s) F2(s) tuttueetfjsjssFBABABAsBjBsAjAsjsBjsAsFjssssFtjtj cos 212121212101:numerators two theEquating :(poles)r denominatoin Roots0002020000020200