Laplace Transforms: The Division-by-s Rule and Its Link to Integration | 拉普拉斯变换中除以s与积分的关系

📚 Laplace Transforms: The Division-by-s Rule and Its Link to Integration | 拉普拉斯变换中除以s与积分的关系

In this article, we explore one of the most useful operational properties of the Laplace transform: the division-by-s rule. This rule states that integrating a time-domain function from 0 to t corresponds to dividing its Laplace transform by s. We will state the rule, prove it, apply it to several examples, and discuss its role in solving differential equations.

在本文中，我们探讨拉普拉斯变换最实用的运算性质之一：除以s法则。该法则指出，对时域函数从0到t进行积分，等价于将其拉普拉斯变换除以s。我们将阐述该法则、给出证明、应用于多个例子，并讨论它在求解微分方程中的作用。

1. The Laplace Transform: A Quick Recap | 拉普拉斯变换快速回顾

The Laplace transform of a function f(t), defined for t ≥ 0, is given by the improper integral:

函数 f(t)（定义在 t ≥ 0 上）的拉普拉斯变换由下述反常积分给出：

F(s) = L{f(t)} = ∫₀^∞ e^(-st) f(t) dt

Here s is a complex variable, and the transform exists for values of s for which the integral converges. The original function f(t) can be recovered from F(s) using the inverse Laplace transform, denoted L⁻¹{F(s)}.

其中 s 为复变量，变换在积分收敛的 s 值范围内存在。原函数 f(t) 可通过逆拉普拉斯变换从 F(s) 中恢复，记作 L⁻¹{F(s)}。

Key properties that we will use include linearity and the derivative rule:

我们将使用到的重要性质包括线性性和微分法则：

L{f'(t)} = sF(s) – f(0)

2. The Division-by-s Rule: Statement | 除以s法则：陈述

The division-by-s rule states: if L{f(t)} = F(s), then

除以s法则指出：若 L{f(t)} = F(s)，则

L{∫₀