📚 Laplace Transform: Basic Concepts | 拉普拉斯变换基本概念
The Laplace transform is a powerful integral transform that maps a function of a real variable t (often interpreted as time) into a function of a complex variable s. By converting calculus operations into algebraic operations, it provides an elegant route for solving linear differential equations and analyzing systems in engineering and physics.
拉普拉斯变换是一种强大的积分变换，它将实变量 t（通常理解为时间）的函数映射为复变量 s 的函数。通过将微积分运算转化为代数运算，它为求解线性微分方程以及分析工程和物理中的系统提供了一条优雅的途径。
1. Definition and Basic Idea | 定义与基本思想
Let f(t) be a function defined for t ≥ 0. Its Laplace transform is denoted by L{f(t)} or F(s) and is defined as the improper integral
设 f(t) 是定义在 t ≥ 0 上的函数。其拉普拉斯变换记作 L{f(t)} 或 F(s)，定义为如下反常积分：
F(s) = ∫0∞ f(t) e-st dt
The variable s is a complex number, usually written as s = σ + iω. The transform exists when the integral converges; this typically requires f(t) to grow no faster than an exponential as t → ∞.
变量 s 是一个复数，通常写成 s = σ + iω。当积分收敛时变换存在；这通常要求 f(t) 在 t → ∞ 时增长速率不超过某个指数函数。
2. Existence and Convergence Conditions | 存在性与收敛条件
For the Laplace integral to converge, f(t) must be piecewise continuous on [0, ∞) and of exponential order. That is, there exist constants M > 0, α, and T such that |f(t)| ≤ M eαt for all t > T.
要使拉普拉斯积分收敛，f(t) 必须在 [0, ∞) 上分段连续，并且具有指数阶。也就是说，存在常数 M > 0、α 和 T，使得对所有 t > T 都有 |f(t)| ≤ M eαt。
Under these conditions, the integral converges for Re(s) > α, and the region Re(s) > α is called the region of convergence (ROC).
在这些条件下，积分在 Re(s) > α 时收敛，区域 Re(s) > α 称为收敛域（ROC）。
3. Linearity | 线性性质
The Laplace transform is a linear operator. If L{f(t)} = F(s) and L{g(t)} = G(s), then for any constants a and b,
拉普拉斯变换是线性算子。若 L{f(t)} = F(s) 且 L{g(t)} = G(s)，则对任意常数 a 和 b，有
L{a f(t) + b g(t)} = a F(s) + b G(s)
This property follows directly from the linearity of integration. For example, L{3 + 2t} = 3/s + 2/s2.
该性质直接由积分的线性性推出。例如，L{3 + 2t} = 3/s + 2/s2。
4. Laplace Transforms of Common Functions | 常见函数的拉普拉斯变换
The following table lists some frequently used transforms. Here n is a non-negative integer, a is a real constant, and the transforms are valid for Re(s) > 0 unless otherwise stated.
下表列出一些常用变换。其中 n 是非负整数，a 是实常数，除非另加说明，变换均在 Re(s) > 0 时成立。
| f(t) | F(s) = L{f(t)} |
|---|---|
| 1 | 1/s, Re(s) > 0 |
| tn | n! / sn+1, Re(s) > 0 |
| eat | 1/(s – a), Re(s) > a |
| sin(at) | a/(s2 + a2), Re(s) > 0 |
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