1. Complex numbers and complex functions
1. Three forms of plural:
Coordinate form: triangle form:
exponential form:
2. Complex variable function:
Each complex number z = a + b * i in the complex number set E has (uniquely determined / infinitely multiple / finite) complex numbers corresponding to it, and (single value / multivalued) complex variable functions can be determined.
3. Zero and pole
Zero point: The point where the numerator is zero, that is, when G (s) = 0, s = z1, z2 is called the zero point of G (s);
Pole: the point where the denominator is zero, that is, when G (s) = ∞, s = p1, p2 is called the pole of G (s);
Second, Laplace transform
1. Laplace transform
The Laplace transform is a function that transforms the time function f (t) into a complex variable function F (s). The Laplace transform establishes the relationship between the time domain and the complex frequency domain (s domain).
In the formula, s = a + b * i is a complex variable, f (t) is also called the original function, and F (s) is also called the image function.
2. Pull inverse transform
The Laplace transform is a function that transforms the time function F (s) into a complex variable function f (t).
In the formula, s = a + b * i is a complex variable, f (t) is also called the original function, and F (s) is also called the image function.
3. Laplace transform of typical time function
Euler's formula: e ^ iθ = cosθ + isinθ,
The deduction can be obtained: (use of the Lassian transform of the sine and cosine function)
sinθ = （e ^ iθ-e ^ -iθ） / 2i
cosθ = （e ^ iθ + e ^ -iθ） / 2i
Unit step function:
， 
Unit pulse function:
Unit ramp function:
Exponential function:
Sine function:
Cosine function:
Power function:
4. The nature of Laplace transform
Superposition theorem
Homogeneity: L [a * f (t)] = a * F (s), superposition:, the combination of homogeneity and superposition is the linear property of Laplace transform
Linear nature:
Differential Theorem
L [df (t) / dt] = s * F (s) -f (0) Similarly, when the Lagrangian transformation of each derivative of f (t) is obtained:
L[d^2f(t)/dt]=s^2*F(s)-s*f(0)-f'(0)
L[d^3f(t)/dt]=s^3*F(s)-s^2*f(0)-s*f'(0)-s''(0).......
Complex differential theorem
L[t*f(t)]=-dF(s)/d(s)、
L[t^2*f(t)]=-d^2F(s)/d(s^2)、
......、
L[t^n*f(t)]=-d^nF(s)/d(s^n)
Integral theorem
Multiple points:
Displacement theorem
L[e^(-at)*f(t)]=F(s+a)
Delay theorem
L [f (ta)] = e ^ (-a) * F (s), the function f (ta) is the function f (t) delayed by a on the time axis
Initial value theorem and final value theorem
The initial value of the original function at t = 0 is equal to s * [the final value of F (s)]
The final value of the original function at t = + ∞ is equal to s * [the initial value of F (s)]
Convolution theorem
Laplace transform of the convolution of two original functions = product of their like functions
L[f(t)*g(t)]=F(s)*G(s)
3. Inverse Laplace transform
There are three methods of inverse Laplace transform: look-up table method, residue theorem method, and partial fraction method.
1. Look-up table method: directly find out the original function f (t) corresponding to the image function F (s) from the Laplace transform table.
2. Residue theorem method: Use the residue theorem to calculate the original function of the image function.
3. Partial fraction method: first decompose the image function into partial fractions, and then inversely transform each fraction.
Here is the URL of the example of inverse Laplace transform, the reader can practice it by himself.
https://wenku.baidu.com/view/d6335d6f3968011ca300915a.html
Four, matlab code
%拉式变换
clc
clear
syms s t
y=exp(-t)*cos(t); %原函数
LY=laplace(y) %拉氏变换
L_1=ilaplace(LY) %拉氏逆变换
e=L_1-y %做差检查结果是否正确
%原函数与象函数图像绘制
subplot(1,2,1)
fplot(y);%原函数图像
title("原函数图像")
legend("原函数")
subplot(1,2,2)
fplot(LY)%象函数图像
title("象函数图像")
legend("象函数")
