The Laplace Transform | 拉普拉斯变换

📚 The Laplace Transform | 拉普拉斯变换

The Laplace transform is a powerful integral transform that converts functions of a real variable t (often time) into functions of a complex variable s (complex frequency). It transforms linear differential equations into algebraic equations, providing a systematic method for solving initial value problems. This technique is widely used in engineering, physics, and applied mathematics.

拉普拉斯变换是一种强大的积分变换，它将实变量 t（通常表示时间）的函数转换为复变量 s（复频率）的函数。它把线性微分方程转化为代数方程，为求解初值问题提供了一种系统的方法。这一技巧被广泛应用于工程、物理和应用数学。

1. Introduction and Historical Context | 引言与历史背景

The Laplace transform is named after the French mathematician Pierre-Simon Laplace (1749–1827), who used a version of this transform in his work on probability theory. The modern form of the transform was later developed and applied to solve differential equations by Oliver Heaviside and others. Today it is a fundamental tool for analysing linear time-invariant systems.

拉普拉斯变换以法国数学家皮埃尔-西蒙·拉普拉斯（1749–1827）命名，他在概率论研究中使用了这种变换的一种形式。现代形式的拉普拉斯变换随后由奥利弗·赫维赛德等人发展并应用于微分方程的求解。如今它是分析线性时不变系统的基本工具。

2. Definition and Notation | 定义与符号

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

L{f(t)} = F(s) = ∫₀^∞ e^-st f(t) dt

Here s is a complex variable. The integral converges for a certain range of s, known as the region of convergence. The notation L{f(t)} or simply F(s) is used. The original function f(t) is recovered by the inverse Laplace transform, written as L^-1{F(s)}.

对于 t ≥ 0 定义的函数 f(t)，其拉普拉斯变换由以下反常积分给出：

L{f(t)} = F(s) = ∫₀^∞ e^-st f(t) dt

其中 s 是一个复变量。该积分在一定的 s 范围内收敛，这个范围称为收敛域。记号 L{f(t)} 或简写为 F(s)。原函数 f(t) 通过逆拉普拉斯变换复原，记作 L^-1{F(s)}。

3. Transforms of Elementary Functions | 初等函数的拉普拉斯变换

By direct integration, we can find the transforms of common functions. For a constant function f(t) = 1, we have L{1} = 1/s for Re(s) > 0. For f(t) = tⁿ, repeated integration by parts gives L{tⁿ} = n! / sⁿ⁺¹. The exponential function f(t) = e^at yields L{e^at} = 1/(s – a). Trigonometric functions like sin(ωt) and cos(ωt) are obtained by using Euler’s formula; L{sin(ωt)} = ω/(s² + ω²) and L{cos(ωt)} = s/(s² + ω²).

通过直接积分，我们可以求得常见函数的变换。对于常数函数 f(t) = 1，有 L{1} = 1/s，其中 Re(s) > 0。对于 f(t) = tⁿ，反复分部积分可得 L{tⁿ} = n! / sⁿ⁺¹。指数函数 f(t) = e^at 给出 L{e^at} = 1/(s – a)。三角函数如 sin(ωt) 和 cos(ωt) 可利用欧拉公式求得：L{sin(ωt)} = ω/(s² + ω²)，L{cos(ωt)} = s/(s² + ω²)。

Hyperbolic functions are equally straightforward: L{sinh(at)} = a/(s² – a²) and L{cosh(at)} = s/(s² – a²). These results form a basic table that is useful for solving more complex problems.

双曲函数同样简单：L{sinh(at)} = a/(s² – a²)，L{cosh(at)} = s/(s² – a²)。这些结果构成了一个基本表格，可用于解决更复杂的问题。

4. Linearity and the First Shift Theorem | 线性性质与第一移位定理

The Laplace transform is a linear operator. For constants a and b and functions f(t) and g(t), L{a f(t) + b g(t)} = a F(s) + b G(s). This property allows us to decompose functions into simpler parts and transform each separately.

拉普拉斯变换是一个线性算子。对于常数 a、b 以及函数 f(t) 和 g(t)，有 L{a f(t) + b g(t)} = a F(s) + b G(s)。这一性质允许我们将函数分解为更简单的部分，并分别对每一部分进行变换。

The first shift theorem (also called the s-shift property) states that multiplying a function by e^at in the time domain corresponds to a shift in the s-domain: L{e^at f(t)} = F(s – a). This is particularly useful when dealing with damped oscillations and exponential growth or decay.

第一移位定理（也称为 s-域移位性质）指出：时域中乘以 e^at 对应于 s 域中的平移：L{e^at f(t)} = F(s – a)。这在处理阻尼振动和指数增长或衰减时尤为有用。

5. Differentiation in the Time Domain | 时域中的微分性质

One of the most important properties is the transform of a derivative. If f(t) is continuous and differentiable, then L{f'(t)} = s F(s) – f(0). For the second derivative, L{f”(t)} = s² F(s) – s f(0) – f'(0). In general, the nth derivative transforms to sⁿ F(s) minus a polynomial in s involving initial conditions. These formulas convert differential equations into algebraic equations in s.

最重要的性质之一是导数的变换。若 f(t) 连续且可微，则 L{f'(t)} = s F(s) – f(0)。对于二阶导数，L{f”(t)} = s² F(s) – s f(0) – f'(0)。一般地，n 阶导数变换为 sⁿ F(s) 减去一个包含初始条件的 s 的多项式。这些公式将微分方程转化为关于 s 的代数方程。

To illustrate, consider the differential equation y” + 3y’ + 2y = 0 with y(0)=1 and y'(0)=0. Taking transforms gives (s²Y – s – 0) + 3(sY – 1) + 2Y = 0, which simplifies to an algebraic expression for Y(s). Solving for Y(s) and then finding the inverse transform yields the solution in the time domain.

举例说明，考虑微分方程 y” + 3y’ + 2y = 0，初始条件为 y(0)=1, y'(0)=0。取拉普拉斯变换得到 (s²Y – s – 0) + 3(sY – 1) + 2Y = 0，化简后可求出 Y(s) 的代数表达式。解出 Y(s) 再求逆变换，即可获得时域中的解。

6. Integration and Other Key Properties | 积分性质与其他重要性质

The transform of an integral is given by L{∫₀^t f(τ) dτ} = F(s)/s. This property is the inverse of differentiation. Another useful rule is the multiplication by t: L{t f(t)} = -F'(s), and similarly L{tⁿ f(t)} = (-1)ⁿ dⁿF/dsⁿ. These are sometimes called the frequency differentiation properties.

积分的变换由 L{∫₀^t f(τ) dτ} = F(s)/s 给出。这个性质是微分性质的逆。另一个有用的法则是乘以 t：L{t f(t)} = -F'(s)，类似地 L{tⁿ f(t)} = (-1)ⁿ dⁿF/dsⁿ。这些有时被称为频域微分性质。

A summary table of the main properties is often helpful:

Property	Time domain f(t)	s-domain F(s)
Linearity	a f(t) + b g(t)	a F(s) + b G(s)
First shift	e^at f(t)	F(s – a)
Time derivative	f'(t)	s F(s) – f(0)
Integral	∫₀^t f(τ) dτ	F(s)/s
Multiplication by t	t f(t)	-F'(s)

总结主要性质的表格常常很有帮助：

性质	时域 f(t)	s域 F(s)
线性性	a f(t) + b g(t)	a F(s) + b G(s)
第一移位	e^at f(t)	F(s – a)
时域导数	f'(t)	s F(s) – f(0)
积分	∫₀^t f(τ) dτ	F(s)/s
乘以 t	t f(t)	-F'(s)

7. The Inverse Laplace Transform and Partial Fractions | 逆拉普拉斯变换与部分分式

The inverse Laplace transform is defined by a complex contour integral: f(t) = (1/(2πi)) ∫_γ-i∞^γ+i∞ e^st F(s) ds. In practice, we rarely evaluate this integral directly. Instead, we use tables of transform pairs and algebraic techniques like partial fraction decomposition to express F(s) as a sum of simpler terms whose inverses are known.

逆拉普拉斯变换由一个复围道积分定义：f(t) = (1/(2πi)) ∫_γ-i∞^γ+i∞ e^st F(s) ds。实际应用中，我们很少直接计算这个积分。而是利用变换对表格和代数技巧（如部分分式分解），将 F(s) 表达为一些简单项之和，这些项的逆变换是已知的。

For example, consider F(s) = (3s+5)/(s² + 3s + 2). The denominator factors as (s+1)(s+2). Writing F(s) = A/(s+1) + B/(s+2) and solving for A and B gives A=2, B=1. Then f(t) = 2e^-t + e^-2t. This technique is essential when solving differential equations.

例如，考虑 F(s) = (3s+5)/(s² + 3s + 2)。分母因式分解为 (s+1)(s+2)。将 F(s) 写成 A/(s+1) + B/(s+2)，解出 A=2, B=1。则 f(t) = 2e^-t + e^-2t。这一技巧在求解微分方程时必不可少。

8. Convolution Theorem | 卷积定理

The convolution of two functions f(t) and g(t) is defined as (f * g)(t) = ∫₀^t f(τ) g(t-τ) dτ. The convolution theorem states that the Laplace transform of a convolution is the product of the individual transforms: L{f * g} = F(s) G(s). Conversely, the inverse transform of a product is the convolution of the original functions.

两个函数 f(t) 和 g(t) 的卷积定义为 (f * g)(t) = ∫₀^t f(τ) g(t-τ) dτ。卷积定理指出：卷积的拉普拉斯变换等于各自变换的乘积：L{f * g} = F(s) G(s)。反之，乘积的逆变换是原函数的卷积。

This theorem is particularly useful when the forcing function in a differential equation is not a simple exponential or trig function, but can be expressed as an integral. It also provides an alternative way to find inverse transforms when F(s) appears as a product of known transforms.

这一定理在微分方程的驱动函数不是简单的指数或三角函数，但可表示为积分形式时尤为有用。当 F(s) 表现为已知变换的乘积时，它也提供了求逆变换的替代方法。

9. Solving Differential Equations: Step-by-Step Example | 解微分方程：逐步示例

Let us solve the initial value problem y” + 4y = sin(2t), with y(0)=0 and y'(0)=1. First, take the Laplace transform of both sides using the derivative rule: [s²Y – s·0 – 1] + 4Y = 2/(s²+4). Simplify to (s²+4)Y – 1 = 2/(s²+4). Solving for Y(s) yields Y(s) = 1/(s²+4) + 2/(s²+4)².

我们来求解初值问题 y” + 4y = sin(2t)，其中 y(0)=0, y'(0)=1。首先，利用导数法则对两边取拉普拉斯变换：[s²Y – s·0 – 1] + 4Y = 2/(s²+4)。化简得 (s²+4)Y – 1 = 2/(s²+4)。解出 Y(s) 得到 Y(s) = 1/(s²+4) + 2/(s²+4)²。

The first term’s inverse is (1/2) sin(2t). For the second term, we can use the convolution theorem or the frequency differentiation property; it yields (t/4) sin(2t). Thus the complete solution is y(t) = (1/2) sin(2t) + (t/4) sin(2t). This demonstrates how initial conditions are naturally incorporated.

第一项的逆变换是 (1/2) sin(2t)。对于第二项，可以利用卷积定理或频域微分性质；其结果为 (t/4) sin(2t)。因此完整解为 y(t) = (1/2) sin(2t) + (t/4) sin(2t)。这展示了初始条件如何被自然地包含进来。

10. Applications and a Summary of Common Transform Pairs | 应用与常用变换对总结

The Laplace transform is indispensable in control theory, signal processing, and circuit analysis. It allows engineers to analyse system stability using the concept of transfer functions H(s) = Y(s)/X(s). In physics, it helps solve heat conduction and wave equations under given boundary conditions.

拉普拉斯变换在控制理论、信号处理和电路分析中不可或缺。它使工程师能够利用传递函数 H(s) = Y(s)/X(s) 的概念分析系统稳定性。在物理中，它有助于求解给定边界条件下的热传导方程和波动方程。

Below is a compact table of the most frequently used transform pairs. Students are encouraged to memorise these as a starting point.

f(t) for t>0	F(s) = L{f(t)}
1	1/s
tⁿ	n! / sⁿ⁺¹
e^at	1/(s – a)
sin(ωt)	ω/(s² + ω²)
cos(ωt)	s/(s² + ω²)
sinh(at)	a/(s² – a²)
cosh(at)	s/(s² – a²)
e^at sin(ωt)	ω/[(s-a)² + ω²]
e^at cos(ωt)	(s-a)/[(s-a)² + ω²]
tⁿ e^at	n!/(s-a)ⁿ⁺¹

下面列出了最常用的变换对简表。建议学生熟记这些作为起点。

t>0 时的 f(t)	F(s) = L{f(t)}
1	1/s
tⁿ	n! / sⁿ⁺¹
e^at	1/(s – a)
sin(ωt)	ω/(s² + ω²)
cos(ωt)	s/(s² + ω²)
sinh(at)	a/(s² – a²)
cosh(at)	s/(s² – a²)
e^at sin(ωt)	ω/[(s-a)² + ω²]
e^at cos(ωt)	(s-a)/[(s-a)² + ω²]
tⁿ e^at	n!/(s-a)ⁿ⁺¹

11. Convergence and the Region of Convergence | 收敛性与收敛域

The Laplace transform integral does not converge for all s. The set of values of s for which the integral converges is called the region of convergence (ROC). For functions of exponential order, there exists a real number σ such that the integral converges for Re(s) > σ. For example, f(t) = e^2t has ROC Re(s) > 2. Understanding the ROC is crucial when dealing with inverse transforms and stability analysis.

拉普拉斯变换的积分并非对所有 s 都收敛。使积分收敛的 s 的集合称为收敛域（ROC）。对于指数阶函数，存在实数 σ，使得当 Re(s) > σ 时积分收敛。例如，f(t) = e^2t 的收敛域为 Re(s) > 2。理解收敛域在处理逆变换和稳定性分析时

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The Laplace Transform | 拉普拉斯变换