Inverse Hyperbolic Functions | 反双曲函数

📚 Inverse Hyperbolic Functions | 反双曲函数

Inverse hyperbolic functions are the inverse operations of the hyperbolic functions sinh, cosh and tanh. They are used extensively in integration, differential equations and physical applications, and they have elegant logarithmic forms that make calculations much simpler.

反双曲函数是双曲函数 sinh、cosh 和 tanh 的反函数运算。它们在积分、微分方程和物理应用中广泛使用，并且具有简洁的对数形式，使计算大为简化。

1. Review of Hyperbolic Functions | 双曲函数回顾

Before introducing the inverse functions, we recall the standard hyperbolic functions defined from exponential functions:

在介绍反函数之前，我们先回顾由指数函数定义的标准双曲函数：

sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x

The hyperbolic sine is one-to-one over the whole real line, so it has an inverse with domain ℝ. The hyperbolic cosine is even, so to make it invertible we restrict the domain to x ≥ 0, giving an inverse with domain [1, ∞). The hyperbolic tangent is one-to-one on ℝ and has range (−1, 1), so its inverse has domain (−1, 1).

双曲正弦在整个实数轴上是单射，因此其反函数定义域为 ℝ。双曲余弦是偶函数，为使其可逆，我们将定义域限制为 x ≥ 0，于是反函数定义域为 [1, ∞)。双曲正切在 ℝ 上单射，值域为 (−1, 1)，因此其反函数定义域为 (−1, 1)。

2. Defining the Inverse Hyperbolic Functions | 反双曲函数的定义

The inverse hyperbolic functions are usually denoted by arsinh, arcosh and artanh. This notation emphasises that they are inverse functions, not reciprocals. They are defined as:

反双曲函数通常记作 arsinh、arcosh 和 artanh。这种记法强调它们是反函数，而不是倒数。它们的定义为：

arsinh x = y such that sinh y = x, with domain ℝ.

arsinh x = y，满足 sinh y = x，定义域 ℝ。
arcosh x = y such that cosh y = x, with y ≥ 0, domain [1, ∞).

arcosh x = y，满足 cosh y = x，且 y ≥ 0，定义域 [1, ∞)。
artanh x = y such that tanh y = x, domain (−1, 1).

artanh x = y，满足 tanh y = x，定义域 (−1, 1)。

For completeness, the inverse functions of sech, cosech and coth also exist, but they are not required for AQA A-Level Further Mathematics.

为完整起见，sech、cosech 和 coth 的反函数也存在，但 AQA A-Level 进阶数学不要求。

3. Logarithmic Forms | 对数形式

Each inverse hyperbolic function can be expressed in terms of natural logarithms. These logarithmic forms are essential for solving equations, simplifying expressions and evaluating derivatives and integrals.

每个反双曲函数都可以表示为自然对数的形式。这些对数形式对于解方程、化简表达式以及求导和积分至关重要。

arsinh x = ln(x + √(x² + 1)), for all x ∈ ℝ

arcosh x = ln(x + √(x² − 1)), for x ≥ 1

artanh x = ½ ln((1 + x)/(1 − x)), for |x| < 1

These are obtained by solving sinh y = x, cosh y = x or tanh y = x for y in terms of x, using the definitions of sinh and cosh as exponentials and the quadratic formula.

这些公式通过分别解 sinh y = x、cosh y = x 或 tanh y = x，并将 sinh 和 cosh 写成指数形式，再利用二次公式求得 y 关于 x 的表达式。

4. Graphs and Properties | 图像与性质

The graph of y = arsinh x resembles a logarithmic curve that passes through the origin and is symmetric about the origin. As x → ±∞, the curve approaches the line y = ±ln(2x). For negative x, arsinh x is negative because sinh is an odd function.

y = arsinh x 的图像类似对数曲线，经过原点，且关于原点对称。当 x → ±∞ 时，曲线逼近直线 y = ±ln(2x)。由于 sinh 是奇函数，所以 x 为负时 arsinh x 也为负。

The graph of y = arcosh x has domain x ≥ 1, starts at the point (1, 0) and increases slowly, similar to a logarithmic curve. There is no portion for x < 1.

y = arcosh x 的定义域为 x ≥ 1，起点为 (1, 0)，随后缓慢增加，类似对数曲线。x < 1 时无图像。

The graph of y = artanh x has vertical asymptotes at x = 1 and x = −1. As x approaches 1 from the left, artanh x tends to +∞; as x approaches −1 from the right, it tends to −∞.

y = artanh x 的图像在 x = 1 和 x = −1 处有竖直渐近线。当 x 从左侧接近 1 时，artanh x 趋于 +∞；当 x 从右侧接近 −1 时，趋于 −∞。

5. Derivatives of Inverse Hyperbolic Functions | 反双曲函数的导数

Using the logarithmic forms, we obtain the following standard derivatives:

利用对数形式，我们得到以下标准导数：

d/dx (arsinh x) = 1/√(x² + 1)

d/dx (arcosh x) = 1/√(x² − 1), x > 1

d/dx (artanh x) = 1/(1 − x²), |x| < 1

These derivatives are particularly useful in integration, and they appear directly in many standard integral results. Notice that the derivative of artanh x has the same form as the derivative of the natural logarithm of a product, which is why artanh often appears in partial fraction integration.

这些导数在积分中特别有用，并且直接出现在许多标准积分结果中。注意 artanh x 的导数与某些对数导数的形式相同，这也是 artanh 常出现在部分分式积分中的原因。

6. Integrals Involving Inverse Hyperbolic Functions | 涉及反双曲函数的积分

Because differentiation and integration are inverse operations, the derivative results provide the following standard integrals:

由于微分与积分互为逆运算，上述导数结果可直接给出以下标准积分：

∫ 1/√(x² + a²) dx = arsinh(x/a) + C

∫ 1/√(x² − a²) dx = arcosh(x/a) + C, x > a

∫ 1/(a² − x²) dx = (1/a) artanh(x/a) + C, |x| < a

More generally, by completing the square, many quadratic expressions in the denominator can be transformed into these forms. For example, ∫ dx/√(x² + 2x + 5) can be simplified by writing x² + 2x + 5 = (x + 1)² + 4 and then using the substitution u = x + 1.

更一般地，通过配方，许多分母中的二次表达式都可以转化为上述形式。例如，∫ dx/√(x² + 2x + 5) 可先把 x² + 2x + 5 写成 (x + 1)² + 4，再令 u = x + 1 进行化简。

7. Solving Equations with Inverse Hyperbolic Functions | 利用反双曲函数解方程

Equations involving sinh or tanh can often be solved by applying the inverse function. For example, to solve sinh x = 3, we take arsinh on both sides and use the logarithmic form:

涉及 sinh 或 tanh 的方程通常可以两边取反函数来求解。例如，解 sinh x = 3 时，两边取 arsinh，再利用对数形式得到：

x = arsinh 3 = ln(3 + √(3² + 1)) = ln(3 + √10)

For equations such as cosh x = c with c ≥ 1, there are two solutions because cosh is even. Using the restricted inverse arcosh gives only the non-negative solution; the negative solution is simply −arcosh c.

对于 cosh x = c（其中 c ≥ 1）这样的方程，由于 cosh 是偶函数，会有两个解。使用限制后的反函数 arcosh 只能得到非负解，负解即为 −arcosh c。

Usually you should check that a solution falls in the correct domain. For example, if an equation involves tanh x, any real solution must satisfy |tanh x| < 1, which is automatically true for real x.

通常需要检查解是否落在正确的定义域内。例如，若方程涉及 tanh x，任何实数解都必须满足 |tanh x| < 1，而这对实数 x 自动成立。

8. Useful Identities and Relationships | 常用恒等式与关系

Inverse hyperbolic functions satisfy many identities analogous to those of inverse trigonometric functions. Often these are used to simplify expressions or to differentiate more complicated composite functions.

反双曲函数满足许多与反三角函数类似恒等式。它们常用于化简表达式或对更复杂的复合函数求导。

arsinh x = ln(x + √(x² + 1)), tanh(arsinh x) = x / √(x² + 1)

cosh(arsinh x) = √(x² + 1), sinh(arcosh x) = √(x² − 1)

These identities are derived directly from the definitions of sinh and cosh. They are particularly helpful when differentiating composite functions such as y = tanh(arsinh x) or when evaluating definite integrals with hyperbolic substitutions.

这些恒等式直接由 sinh 和 cosh 的定义推出。它们对诸如 y = tanh(arsinh x) 这类复合函数求导，或者使用双曲代换计算定积分时特别有用。

9. Common Exam Questions and Tips | 常见考题与技巧

In AQA A-Level Further Mathematics, questions on inverse hyperbolic functions often appear in the calculus and algebra papers. The most common types are:

在 AQA A-Level 进阶数学中，反双曲函数的问题常出现在微积分和代数试卷中。最常见的题型包括：

Write an inverse hyperbolic function in logarithmic form.

将反双曲函数写成对数形式。
Differentiate a function that contains arsinh, arcosh or artanh, using the chain rule.

对包含 arsinh、arcosh 或 artanh 的函数使用链式法则求导。
Evaluate an integral that leads to an inverse hyperbolic function by completing the square.

通过配方计算可化为反双曲函数的积分。
Solve an equation involving hyperbolic functions using the definition of the inverse function.

利用反函数的定义求解含双曲函数的方程。

A standard technique is to remember the derivatives and integrals listed in Sections 5 and 6, and to be comfortable with the logarithmic forms in Section 3. When completing the square, always check that the resulting integral matches one of the three standard forms exactly.

一个标准技巧是熟记第 5、6 节中的导数和积分公式，并且熟悉第 3 节的对数形式。配方后，务必检查所得积分是否正好符合三种标准形式之一。

10. Summary and Revision Checklist | 总结与复习清单

Inverse hyperbolic functions are an elegant and useful extension of the hyperbolic functions. The key points to remember are:

反双曲函数是双曲函数的一个优美且实用的扩展。需要记住的关键点包括：

The definitions of arsinh, arcosh and artanh, including their domains and ranges.

arsinh、arcosh 和 artanh 的定义，包括定义域和值域。
The logarithmic forms, which allow us to evaluate them on a calculator or by hand.

对数形式，使我们能够手算或用计算器求值。
The standard derivatives and integrals, which are frequently tested.

标准导数与积分，这是考试常考内容。
The shape of their graphs, especially asymptotes for artanh.

它们的图像形状，尤其是 artanh 的渐近线。
How to complete the square to integrate expressions involving quadratic denominators.

如何通过配方来积分含二次分母的表达式。

Revise these essentials and practise with past paper questions. The more fluent you are with the logarithmic forms and standard derivatives, the faster you will solve these problems in the exam.

请复习这些要点并结合历年真题练习。你越熟练地掌握对数形式和标准导数，在考试中解决这些问题的速度就会越快。

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Inverse Hyperbolic Functions | 反双曲函数