📚 Hyperbolic Functions | 双曲函数
Hyperbolic functions are an important family of functions in IB Mathematics, particularly in the Analysis & Approaches HL course. They are reminiscent of trigonometric functions but are defined through exponential functions, and they play a key role in integration, differential equations, and modelling phenomena such as hanging cables and special relativity. This article provides a thorough revision of hyperbolic functions, covering definitions, identities, derivatives, integrals, inverse functions, and applications.
双曲函数是IB数学，尤其是分析与方法(HL)课程中的一类重要函数。它们让人联想到三角函数，但却是通过指数函数来定义的，在积分、微分方程以及悬链线、狭义相对论等建模中扮演着关键角色。本文对双曲函数进行全面复习，涵盖定义、恒等式、导数、积分、反函数和应用。
1. Introduction to Hyperbolic Functions | 双曲函数简介
Hyperbolic functions are constructed from the exponential function eˣ ordinary trigonometric sine and cosine but arise naturally from certain combinations of exponentials. The name "hyperbolic" t) lies on the unit hyperbola x² − y² = 1, just as (cos t, sin t) lies on the unit circle.
双曲函数是由指数函数 eˣ 构造出来的。两个最基本的双曲函数是双曲正弦和双曲余弦，记作 sinh x 和 cosh x。它们与普通的三角正弦和余弦有许多类似的性质，但自然产生于指数的特定组合。“双曲”这个名字来源于点 (cosh t, sinh t) 位于单位双曲线 x² − y² = 1 上，正如 (cos t, sin t) 位于单位圆上一样。
2. Definitions of sinh, cosh, and tanh | sinh、cosh 和 tanh 的定义
The three most commonly used hyperbolic functions are defined as follows:
三个最常用的双曲函数定义如下：
sinh x = (eˣ − e⁻ˣ) / 2
cosh x = (eˣ + e⁻ˣ) / 2
tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ)
range of sinh x is all reals, while cosh x ≥ 1, and tanh x lies strictly between −1 and 1.
由于它们由指数函数构成，sinh x 和 cosh x 的定义域为全体实数。sinh x 的值域为全体实数，cosh x ≥ 1，tanh x 严格位于 −1 和 1 之间。
3. Other Hyperbolic Functions: coth, sech, csch | 其他双曲函数：coth、sech、csch
The reciprocal hyperbolic functions are defined analogously to the trigonometric ones:
倒数的双曲函数定义类似于三角函数的情况：
coth x = 1 / tanh x = cosh x / sinh x (x ≠ 0)
sech x = 1 / cosh x
csch x = 1 / sinh x (x ≠ 0)
These functions appear less frequently but can be useful in certain integration techniques and in the study of differential equations. Note the restrictions on the domain to avoid division by zero.
这些函数出现得较少，但在某些积分技巧和微分方程研究中很有用。注意定义域的限制以避免除以零。
4. Key Identities | 重要恒等式
Hyperbolic functions satisfy a rich set of identities that closely mirror trigonometric identities, often with sign differences. The fundamental identity is:
双曲函数满足一系列丰富的恒等式，它们与三角恒等式非常相似，只是符号上常常有差异。最基本的恒等式是：
cosh²x − sinh²x = 1
ion cos²x + sin²x = 1, we have a minus sign here. This identity is the reason for the name “hyperbolic” osh t, y = sinh t, then (x, y) lies on the right-hand branch of the hyperbola x² − y² = 1.
不同于三角版本 cos²x + sin²x = 1，这里是一个减号。这个恒等式正是“双曲”这个名字的由来：若令 x = cosh t, y = sinh t，则 (x, y) 位于双曲线 x² − y² = 1 的右支上。
Other important identities include:
其他重要的恒等式包括：
- sinh(2x) = 2 sinh x cosh x
- cosh(2x) = cosh²x + sinh²x = 2 cosh²x − 1 = 1 + 2 sinh²x
- tanh(2x) = 2 tanh x / (1 + tanh²x)
These double-argument formulas are derived directly from the definitions. They are indispensable when solving hyperbolic equations or integrating functions involving hyperbolic expressions.
这些倍角公式都可以直接从定义推导。在解双曲方程或积分含有双曲表达式的函数时，它们是不可或缺的。
5. Relationship to Trigonometric Functions | 与三角函数的关系
There is a deep connection between hyperbolic and trigonometric functions when complex numbers are introduced. In particular:
当引入复数后，双曲函数与三角函数之间有着深刻的联系。具体有：
sinh(ix) = i sin x
cosh(ix) = cos x
where i² = −1. These relations show that hyperbolic functions are essentially the “imaginary” ions of the trigonometric functions. They also explain why many identities are so similar.
其中 i² = −1。这些关系表明双曲函数本质上是三角函数的“虚数”版本。这也解释了为什么许多恒等式如此相似。
Furthermore, Euler's formula gives cos x = (eix + e−ix)/2 and sin x = (eix − e−ix)/(2i), which directly parallel the definitions of cosh and sinh.
此外，欧拉公式给出 cos x = (eix + e−ix)/2 以及 sin x = (eix − e−ix)/(2i)，这直接与 cosh 和 sinh 的定义平行。
6. Derivatives of Hyperbolic Functions | 双曲函数的导数
The derivatives of the basic hyperbolic functions exhibit a simple cyclic pattern, but unlike trigonometric derivatives there are no sign changes:
基本双曲函数的导数呈现出简单的循环模式，但与三角导数不同的是没有符号变化：
- d/dx (sinh x) = cosh x
- d/dx (cosh x) = sinh x
- d/dx (tanh x) = sech² x
- d/dx (coth x) = −csch² x
- d/dx (sech x) = −sech x tanh x
- d/dx (csch x) = −csch x coth x
sinh x / cosh x. Notice that the derivative of cosh x is positive sinh x, not negative, unlike the trigonometric cosine whose derivative is −sin x.
这些公式可以通过对指数定义求导，或者对 tanh x = sinh x / cosh x 应用商法则来推导。注意 cosh x 的导数是正的 sinh x，而不是负的，不像三角余弦的导数是 −sin x。
7. Integrals of Hyperbolic Functions | 双曲函数的积分
Because differentiation of hyperbolic functions is straightforward, integration is equally simple for the basic forms:
由于双曲函数的微分很简单，基本形式的积分也同样简单：
- ∫ sinh x dx = cosh x + C
- ∫ cosh x dx = sinh x + C
- ∫ sech² x dx = tanh x + C
- ∫ csch² x dx = −coth x + C
- ∫ sech x tanh x dx = −sech x + C
- ∫ csch x coth x dx = −csch x + C
For more complicated integrals, hyperbolic substitution can be a powerful technique. For example, integrals involving √x² + a² may be handled using x = a sinh u, while those with √x² − a² often call for x = a cosh u, analogous to trigonometric substitution.
对于更复杂的积分，双曲代换是一种非常有力的技巧。例如，含有 √x² + a² 的积分可以通过设 x = a sinh u 处理，而含有 √x² − a² 的积分通常需要设 x = a cosh u，类似于三角代换。
8. Inverse Hyperbolic Functions | 反双曲函数
The inverse hyperbolic functions, also called area functions, are denoted by arsinh, arcosh, artanh, etc. They are defined as the inverses of the corresponding hyperbolic functions with suitably restricted domains:
反双曲函数，也称为面积函数，记作 arsinh、arcosh、artanh 等。它们定义为相应双曲函数的反函数，并带有适当限制的定义域：
- y = arsinh x ⇔ x = sinh y, y ∈ ℝ
- y = arcosh x ⇔ x = cosh y, y ≥ 0, x ≥ 1
- y = artanh x ⇔ x = tanh y, y ∈ ℝ, |x| < 1
The notation “arc” is sometimes used (arcsinh), but IB typically uses the “ar” prefix to emphasize the area interpretation. The derivatives of these inverse functions are standard results that must be known:
有时也用 “arc” 作为前缀 (arcsinh)，但 IB 通常使用 “ar” 前缀以强调面积含义。这些反函数的导数是必须掌握的标准结果：
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
9. Logarithmic Forms of Inverse Hyperbolic Functions | 反双曲函数的对数形式
Because hyperbolic functions are defined by exponentials, their inverses can be expressed in terms of natural logarithms. These logarithmic forms are particularly useful for evaluating inverse hyperbolic values and for integration:
因为双曲函数是由指数定义的，它们的反函数可以用自然对数来表示。这些对数形式在计算反双曲值和积分时特别有用：
arsinh x = ln(x + √x² + 1), x ∈ ℝ
arcosh x = ln(x + √x² − 1), x ≥ 1
artanh x = ½ ln((1 + x)/(1 − x)), |x| < 1
These derived formulas can be obtained by solving for y in the equation eʸ They also help in writing the answers to certain integrals in a compact logarithmic form.
这些推导出的公式可以通过解方程 eʸ = cosh y ± sinh y 等方法得到。它们也有助于将某些积分的答案写成简洁的对数形式。
For example, the integral ∫ dx / √x² + a² can be expressed as arsinh(x/a) + C, or equivalently as ln|x + √x² + a²| + C.
例如，积分 ∫ dx / √x² + a² 既可以表示为 arsinh(x/a) + C，也可以等价地写成 ln|x + √x² + a²| + C。
10. Applications of Hyperbolic Functions | 双曲函数的应用
Hyperbolic functions appear in a variety of real-world and pure mathematical contexts:
双曲函数出现在各种现实世界和纯数学的情境中：
- Catenary curve: : The shape of a hanging flexible chain or cable under gravity is described by y = a cosh(x/a).
- 悬链线： 悬挂在重力作用下的柔性链条或电缆的形状由 y = a cosh(x/a) 描述。
- Special relativity: The Lorentz transformation and velocity addition formulas often involve tanh functions.
- 狭义相对论： 洛伦兹变换和速度叠加公式常涉及 tanh 函数。
- Calculus: They simplify many integrals using hyperbolic substitution, as seen in Section 7.
- 微积分： 它们利用双曲代换简化许多积分，如第7节所示。
- Differential equations: The general solution of y'' = k² y can be expressed via cosh and sinh when k is real and positive.
- 微分方程： 当 k 为实数且为正时，y'' = k² y 的通解可以用 cosh 和 sinh 表示。
Understanding these functions not only fulfills the IB syllabus requirement but also builds a strong foundation for further studies in mathematics, physics, and engineering.
理解这些函数不仅能满足 IB 大纲的要求，还为数学、物理和工程学的进一步学习打下坚实基础。
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