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.,复数及其指数形式,复数z可以表示为,Z = r (cosq+jsinq ) = rejq ,欧拉公式,z=x+jy,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,复变量指数函数的性质,特殊地, 有,ex+jy = exej y= ex(cos yjsin y).,.,欧拉公式,复数项级数,设有复数项级数(univn), 其中un,
2、 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(univn)的各项的模所构成的级数|univn|收敛, 则称级数(univn)绝对收敛.,绝对收敛,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,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,复变量指数函数,Appendix
3、 Lesson - Laplace Transforms,Laplace, Pierre (1749-1827),Sources: http:/ 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 w
4、as important because it translated the geometrical study of mechanics 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
5、first to publish the value of the Gaussian integral, . He studied the Laplace 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 Expositio
6、n de systme du monde (1796). He pointed out that sound travels adiabatically, 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
7、 its emphasis on a unified physical view was important. With Lavoisier, 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 grav
8、itational potential and showed it obeyed Laplaces equation in empty space. Laplace believed the universe to be completely deterministic.,The Laplace Transform of a function, f(t), is defined as;,What is the Laplace Transform?,Let f(t) be a given function that is defined for all t 0.We can transform
9、f(t) in to a new function, F(s), via:,What is the Inverse 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,The Laplace Transform,Transform Pairs:,f(
10、t) F(s),The Laplace Transform,Transform Pairs:,f(t) F(s),Yes !,The Laplace Transform,Time Differentiation:,We can extend the previous to show;,Why the transform?,A method to solve differential equations and corresponding initial and boundary value problems, particularly useful when driving forces ar
11、e discontinuous, impulsive, or a complicated periodic/aperiodic function.,Transform the subsidiaryequations solution to obtain the solution of the given problem,An important point :,The 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 uniq
12、ue 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 can go a long way.,The Laplace Transform,Building transform pairs:,A transform,pair,The Laplace Transform,Building transform pairs:,u = t dv = e-stdt
13、,A transformpair,The Laplace Transform,Building transform pairs:,A transformpair,The Laplace Transform,Time Shift,The Laplace Transform,Frequency Shift,The Laplace Transform,Example: Using Frequency Shift,Find the Le-atcos(wt),In this case, f(t) = cos(wt) so,The Laplace Transform,Time Integration:,T
14、he property is:,The Laplace Transform,Time Integration:,Making these substitutions and carrying out The integration shows that,The Laplace Transform,Time Differentiation:,If the Lf(t) = F(s), we want to show:,Integrate by parts:,The Laplace Transform,Time Differentiation:,Making the previous substit
15、utions gives,So we have shown:,The Laplace Transform,Final Value Theorem:,If the function f(t) and its first derivative are Laplace transformable and f(t) has the Laplace transform F(s), and the exists, then,Again, the utility of this theorem lies in not having to take the inverse of F(s) in order t
16、o find out the final value of f(t) in the time domain. This is particularly useful in circuits and systems.,Final Value Theorem,The Laplace Transform,Final Value,Theorem:,Example:,Given:,Find,.,The Laplace Transform,Initial Value Theorem:,If the function f(t) and its first derivative are Laplace tra
17、nsformable and f(t) Has the Laplace transform F(s), and the exists, then,The 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 isparticularly useful in circuits and systems.,Initial Value Theorem,The Laplace Tran
18、sform,Initial Value,Theorem:,Example:,Given;,Find f(0),Partial Fractions Example #1,Partial Fractions Example #2,Partial Fractions Example #3,Laplace Transform Properties,Linearity Time shifting Frequency shifting Differentiation in time,Differentiation in Time Property,Laplace Transform Properties,
19、Differentiation in frequency Integration in time Example: f(t) = d(t)Integration in frequency,Laplace Transform Properties,Scaling in time/frequency Under integration,Convolution in time Convolution in frequency,Area reduced by factor 2,Example,Compute y(t) = e a t u(t) * e b t u(t) , where a bIf a
20、= b, then we would have resonance What form would the resonant solution take?,Linear Differential Equations,Using differentiation in time property we can solve differential equations (including initial conditions) using Laplace transforms Example: (D2 + 5D + 6) y(t) = (D + 1) f(t)With y(0-) = 2, y(0
21、-) =1, and f(t) = e- 4 t u(t) So f (t) = -4 e-4 t u(t) + e-4 t d(t), f (0-) = 0 and f (0+) = 1 See Example 4.9 in Lathi and handout H,Transform Pairs,The Laplace transforms pairs in Table 13.1 are important, and the most important are repeated here.,Laplace Transform Properties,Properties,LinearityS-shiftingTime-shiftingScaling,Derivative,Integral,Convolution,Inverse LT,Basic relations of Laplace Transform,Example I-LT,Quiz I,Example II-inverse LT,Basic relations of Laplace Transform,
