复数与拉氏变换

复变量指数函数
e z =1 z 1 z 2 1 z n 2! n! 欧拉公式
当x=0时 z=iy 于是
eiy =1 iy 1 (iy)2 1 (iy)n 2! n! =1 iy 1 y 2 i 1 y3 1 y 4 i 1 y5 2! 3! 4! 5! = (1 1 y 2 1 y 4 ) i( y 1 y3 1 y5 ) 2! 4! 3! 5! =cos yjsin y
From Table :
2 5 sY s 1 4Y s = s 1 5s 2 y t = L s 5 s 4 y(t ) = 0.5 0.5e 0.8t

Using differentiation in time property dn f n n 1 n2 n 1 0 s F s s f 0 s f 0 f n
因此
1 1 jx jx = = cos x (e e ) , sin x (ejx ejx ) 2 2j 复变量指数函数的性质 ez1 z2 = ez1 ez2 特殊地 有
exjy = exej y = ex(cos yjsin y)
复数项级数
欧拉公式
设有复数项级数∑(univn) 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uiv 绝对收敛 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
dn f n n 1 n2 n 1 s F s s f 0 s f 0 f 0 n dt



5
dt
4 y = 2,
y (0) = 1
Solution:
First, take L of both sides
Rearrange, Take L-1,
Given the hard problem!
Convert it into the subsidiary equation (Simple Problem!)
Transform the subsidiary equation’s solution to obtain the solution of the given problem
Laplace, Pierre (1749-1827)
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 gravitational potential and showed it obeyed Laplace's equation in empty space. Laplace believed the universe to be completely deterministic.
Solve the subsidiary equation (Purely algebraic!)
Application: Linear Differential Equations

Using differentiation in time property


we can solve differential equations (including initial conditions) using Laplace transforms dy Example:
L [ F ( s )] = f (t ) =
1
1 2
too! j ….and we can transform it back
F ( s )e j
j
ts
ds
Why the transform?
A method to solve differential equations and corresponding initial and boundary value problems, particularly useful when driving forces are discontinuous, impulsive, or a complicated periodic/aperiodic function.
复变量指数函数 考察复数项级数
1 z 1 z 2 1 z n 2! n! 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 e z =1 z 1 z 2 1 z n 2! n!
The Laplace Transform of a function, f(t), is defined as;
L[ f (t )] = F ( s) = f (t )e dt
st 0

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: The Inverse Laplace Transform is defined by

s 1 s Y s 2 s 1 5 s Y s 2 6 Y s = s4 s4
2

With y(0-) = 2, y’(0-) =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
dt
Application: Linear Differential Equations

we can solve differential equations (including initial conditions) using Laplace transforms Example: (D2 + 5D + 6) y(t) = (D + 1) f(t)
Sources:
http://scienceworld.wolfram.com/biography/Laplace.html http://mathworld.wolfram.com/LaplaceTransform.html
What 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:
An important point :
f (t ) F ( s )
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 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 can go a long way.
把y换成x得 eix=cos xjsin x 这就是欧拉公式
Appendix Lesson - Laplace Transforms
French physicist and mathematician who put the final capstone on mathematical astronomy by summarizing and extending the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799-1825). This work was 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 Probabilités" (Philosophical Essay on Probability, 1814). He was the 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 Exposition de système du monde (1796). He pointed out that sound travels adiabatically, accounting for Newton's 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.
复数及其指数形式
复数z可以表示为
Z = r (cosqjsinq ) = rejq 其中r=|z|是z的模 q =arg z是z的辐角
wenku.baidu.com
z=x+jy
欧拉公式
ejx=cos xjsin x （此时z的模r=1）
三角函数与复变量指数函数之间的联系 因为 所以 ejx+ejx=2cos x exejx=2jsin x ejx =cos xj sin x ejx=cos xj sin x