📚 The Division Rule for Laplace Transforms of f(t)/t | 拉普拉斯变换中 f(t)/t 的除法法则
The Laplace transform is one of the most powerful tools in applied mathematics and engineering. It converts a function of time f(t) into a function of a complex or real parameter s, turning differential equations into algebraic equations. A particularly useful, yet often under-emphasised, property is the division rule: if F(s) is the Laplace transform of f(t), then the transform of f(t)/t can be obtained by integrating F(s) from s to infinity.
拉普拉斯变换是应用数学和工程中最强大的工具之一。它将时间函数 f(t) 转化为关于实数或复参数 s 的函数，从而把微分方程变成代数方程。一个特别有用、但又常常被低估的性质是除法法则：如果 F(s) 是 f(t) 的拉普拉斯变换，那么 f(t)/t 的变换可以通过将 F(s) 从 s 到无穷积分来得到。
1. The Basic Definition of the Laplace Transform | 拉普拉斯变换的基本定义
For a function f(t) defined for t ≥ 0, its Laplace transform is written as F(s) or L{f(t)}, and is defined by the improper integral
对于定义在 t ≥ 0 上的函数 f(t)，其拉普拉斯变换记为 F(s) 或 L{f(t)}，由如下反常积分定义：
L{f(t)} = F(s) = ∫0∞ e-st f(t) dt, s > c.
Here c is a real constant chosen so that the integral converges; in many elementary problems one may simply take s > 0. The transformation is linear, meaning that L{αf(t) + βg(t)} = αF(s) + βG(s).
其中 c 是使积分收敛的实常数；在许多初等问题中，可以简单地取 s > 0。该变换是线性的，即 L{αf(t) + βg(t)} = αF(s) + βG(s)。
In this article we focus on the property that handles a quotient by t. This rule is sometimes called the “division by t” rule, and it is the natural counterpart of the well-known “multiplication by t” rule.
本文我们重点讨论处理除以 t 的性质。这条规则有时称为“除以 t”法则，它是著名的“乘以 t”法则的自然对应。相应地
2. Statement of the Division Rule | 除法法则的陈述
Suppose that f(t) satisfies the conditions for the Laplace transform to exist, and that f(t)/t also has a Laplace transform. Then the division rule states:
设 f(t) 满足拉普拉斯变换存在的条件，并且 f(t)/t 也存在拉普拉斯变换。则除法法则表明：
L{f(t)/t}(s) = ∫s∞ F(u) du.
In words: to obtain the transform of f(t)/t, we integrate the ordinary transform F(u) from u = s to u = ∞. This is the inverse operation to the differentiation rule for t f(t).
也就是说：要求 f(t)/t 的变换，只需将普通变换 F(u) 从 u = s 到 u = ∞ 积分。这是 t f(t) 的微分法则的逆运算。
The following table shows the duality between the two rules.
下表展示了这两个规则之间的对偶关系。
| Property | Time domain | s-domain |
| Multiplication by t | t f(t) | -F'(s) |
| Division by t | f(t)/t | ∫s∞ F(u) du |
3. Conditions for the Division Rule | 除法法则的条件
It is tempting to apply the rule formally, but the following conditions must be checked to justify the exchange of integrals or the differentiation under the integral sign.
人们总是想形式化地套用该法则，但为了保证积分交换或积分号下求导的合理性，必须检查以下条件。
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f(t) must be piecewise continuous on every finite interval [0, T].
f(t) 在每个有限区间 [0, T] 上必须分段连续
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