📚 Hyperbolic Functions: Definitions and Properties | IB数学：双曲函数定义与性质
Hyperbolic functions appear naturally in many areas of mathematics, from calculus to differential equations. In the IB Mathematics Analysis and Approaches HL syllabus, they are introduced as a powerful extension of the exponential function. This article provides a complete guide to their definitions, key identities, derivatives, integrals, and common exam pitfalls, with bilingual explanations to support your revision.
双曲函数在数学的许多领域自然出现，从微积分到微分方程。在 IB 数学分析与方法 HL 课程中，它们作为指数函数的重要延伸被引入。本文提供关于其定义、核心恒等式、导数、积分及常见考试陷阱的完整指南，并配以中英双语解释，帮助你的复习。
1. Definitions of sinh x and cosh x | 双曲正弦与双曲余弦的定义
The hyperbolic sine and hyperbolic cosine are defined directly from exponential functions. For any real number x, we write:
双曲正弦和双曲余弦直接由指数函数定义。对任意实数 x，我们写成：
sinh x = (eˣ − e⁻ˣ) / 2
cosh x = (eˣ + e⁻ˣ) / 2
The notation is pronounced as “sinch” and “cosh”. Unlike the circular trigonometric functions, hyperbolic functions do not involve angles; they are algebraic combinations of exponential growth and decay.
符号读作“sinch”和“cosh”。与圆三角函数不同，双曲函数不涉及角度；它们是指数增长和衰减的代数组合。
Substituting x = 0 gives sinh 0 = 0 and cosh 0 = 1. The point (cosh t, sinh t) lies on the rectangular hyperbola x² − y² = 1, which explains the name “hyperbolic”.
代入 x = 0 可得 sinh 0 = 0，cosh 0 = 1。点 (cosh t, sinh t) 落在直角双曲线 x² − y² = 1 上，这就是“双曲”名称的由来。
2. Hyperbolic Tangent and Reciprocal Functions | 双曲正切与倒数函数
The hyperbolic tangent is defined as the ratio of sinh to cosh:
双曲正切定义为 sinh 与 cosh 之比：
tanh x = sinh x / cosh x = (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ)
The reciprocal hyperbolic functions are also useful:
倒数双曲函数同样很有用：
- coth x = 1 / tanh x = cosh x / sinh x — 双曲余切，定义域为 x ≠ 0。
- sech x = 1 / cosh x — 双曲正割，值域为 (0, 1]。
- csch x = 1 / sinh x — 双曲余割，定义域为 x ≠ 0。
All of these functions can be expressed in terms of eˣ and e⁻ˣ. In IB questions, you are often expected to simplify expressions using these exponential definitions rather than memorising every formula separately.
所有这些函数都可以用 eˣ 和 e⁻ˣ 表达。在 IB 题目中，通常希望你利用这些指数定义进行化简，而不是单独记忆每一个公式。
3. Osborne’s Rule and Key Identities | 奥斯本法则与主要恒等式
Osborne’s Rule is a practical shortcut for converting standard trigonometric identities into hyperbolic identities. Compare the fundamental identity for circular functions:
奥斯本法则是将标准三角恒等式转换为双曲恒等式的实用技巧。首先比较圆三角函数的基本恒等式：
cos² x + sin² x = 1
When converting to hyperbolic form, change cos² to cosh² and sin² to −sinh². Hence the hyperbolic analogue becomes:
转换为双曲形式时，将 cos² 改为 cosh²，将 sin² 改为 −sinh²。因此双曲类似恒等式变为：
cosh² x − sinh² x = 1
This identity can be verified directly from the exponential definitions:
该恒等式可直接从指数定义验证：
cosh² x − sinh² x = ((eˣ + e⁻ˣ)² − (eˣ − e⁻ˣ)²) / 4 = 4 / 4 = 1
Osborne’s Rule also gives other standard identities:
奥斯本法则还给出其他标准恒等式：
- 1 − tanh² x = sech² x — 由 1 − tan² x = sec² x 转换得到。
- coth² x − 1 = csch² x — 由 cot² x + 1 = csc² x 转换得到。
Addition and double-angle formulas follow similarly, but never replace sin with sinh directly without changing the sign.
加法公式和倍角公式也类似地成立，但绝不可直接将 sin 换成 sinh 而不改变符号。
sinh(x ± y) = sinh x cosh y ± cosh x sinh y
cosh(x ± y) = cosh x cosh y ± sinh x sinh y
Notice that in the cosh formula, the sign in front of the product term is always positive, even for cosh(x − y).
注意在 cosh 公式中，乘积项前的符号总是正号，即使对于 cosh(x − y) 也是如此。
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x
4. Inverse Hyperbolic Functions | 反双曲函数
Because hyperbolic functions are defined in terms of exponentials, their inverses are naturally expressed using natural logarithms. In IB, the notation arsinh x, arcosh x and artanh x is preferred, though sinh⁻¹ x is also seen.
由于双曲函数由指数函数定义，它们的反函数自然用自然对数表达。在 IB 中，优先使用 arsinh x、arcosh x 和 artanh x 的记号，但 sinh⁻¹ x 也可见于题目。
arsinh x = ln(x + √(x² + 1)), 对所有实数 x 成立
arcosh x = ln(x + √(x² − 1)), 对 x ≥ 1 成立
artanh x = ½ ln((1 + x)/(1 − x)), 对 |x| < 1 成立
These logarithmic forms are essential for solving equations and evaluating integrals. You must remember the domain restrictions: arcosh requires x ≥ 1, while artanh requires −1 < x < 1.
这些对数形式对于解方程和计算积分至关重要。你必须记住定义域限制：arcosh 要求 x ≥ 1，而 artanh 要求 −1 < x < 1。
5. Derivatives | 导数
The derivatives of hyperbolic functions closely resemble those of circular functions, but without any additional coefficient changes. The key results are:
双曲函数的导数与圆三角函数的导数非常相似，但没有任何额外的系数变化。关键结果为：
| y | dy/dx |
| sinh x | cosh x |
| cosh x | sinh x |
| tanh x | sech² x |
| coth x | −csch² x |
| sech x | −sech x tanh x |
| csch x | −csch x coth x |
For inverse hyperbolic functions, the following derivatives appear frequently in Paper 3 problems:
对于反双曲函数，以下导数经常出现在 Paper 3 的题目中：
- d/dx arsinh x = 1 / √(x² + 1)
- d/dx arcosh x = 1 / √(x² − 1), for x > 1
- d/dx artanh x = 1 / (1 − x²), for |x| < 1
These are often derived using implicit differentiation. For example, if y = arsinh x, then x = sinh y, so dx/dy = cosh y = √(1 + sinh² y) = √(1 + x²).
这些公式通常通过隐函数微分推导。例如，若 y = arsinh x，则 x = sinh y，因此 dx/dy = cosh y = √(1 + sinh² y) = √(1 + x²)。
6. Integrals | 积分
Every derivative rule gives a corresponding integral. The most important integral forms are:
每一条导数法则都对应一个积分形式。最重要的积分形式为：
∫ sinh x dx = cosh x + C
∫ cosh x dx = sinh x + C
∫ sech² x dx = tanh x + C
The inverse hyperbolic derivatives provide useful standard integrals, especially when quadratic expressions appear under a square root:
反双曲函数的导数提供了有用的标准积分，特别是当根号下出现二次表达式时：
- ∫ (1 / √(x² + 1)) dx = arsinh x + C
- ∫ (1 / √(x² − 1)) dx = arcosh x + C
- ∫ (1 / (1 − x²)) dx = artanh x + C, for |x| < 1
Compare these with the circular counterparts involving arcsin and arctan. The + sign inside the square root in the arsinh integral is a distinctive feature.
将这些与涉及 arcsin 和 arctan 的圆三角对应积分比较。arsinh 积分中根号内的加号是一个典型特征。
7. Graphs, Domain and Range | 图像、定义域与值域
Understanding the graphs of hyperbolic functions helps you interpret inequalities and transformations. The table below summarises the essential characteristics.
理解双曲函数的图像有助于解释不等式和变换。下表总结了基本特征。
| Function | Domain | Range | Symmetry |
| y = sinh x | ℝ | ℝ | 奇函数（原点对称） |
| y = cosh x | ℝ | [1, ∞) | 偶函数（y 轴对称） |
| y = tanh x | ℝ | (−1, 1) | 奇函数（原点对称） |
The graph of y = cosh x is sometimes called a catenary; it has a minimum point at (0, 1). The graph of y = tanh x has horizontal asymptotes y = 1 and y = −1 because (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ) tends to 1 as x → ∞ and to −1 as x → −∞.
y = cosh x 的图像有时被称为悬链线；它在 (0, 1) 处有最小值。y = tanh x 的图像具有水平渐近线 y = 1 和 y = −1，因为当 x → ∞ 时 (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ) 趋于 1，当 x → −∞ 时趋于 −1。
8. Applications | 应用
Hyperbolic functions appear in many real-world and mathematical contexts. In IB, you may encounter them in mechanics, differential equations, or calculus problems.
双曲函数出现在许多现实世界和数学背景中。在 IB 中，你可能会在力学、微分方程或微积分问题中遇到它们。
- Catenary curves: A freely hanging chain under uniform gravity takes the shape of y = a cosh(x/a). — 自由悬挂且受均匀重力的链条呈 y = a cosh(x/a) 的形状。
- Solving differential equations: The second-order equation y” − y = 0 has general solution y = A sinh x + B cosh x. — 二阶方程 y” − y = 0 的通解为 y = A sinh x + B cosh x。
- Integration substitutions: Substituting x = sinh t can simplify integrals involving √(x² + 1). — 代入 x = sinh t 可以化简含 √(x² + 1) 的积分。
- Relativity and engineering: Lorentz transformations and signal processing use hyperbolic functions. — 洛伦兹变换和信号处理中使用双曲函数。
In exam questions, applications often require combining exponential forms with limits or derivatives. Practising these connections is valuable.
在考试题目中，应用问题通常需要结合指数形式与极限或导数。练习这些联系非常有价值。
9. Common Exam Mistakes | 常见考试错误
Students frequently lose marks on hyperbolic functions due to small sign errors or misuse of Osborne’s Rule. Here are the most common pitfalls.
学生在双曲函数问题上经常因符号错误或误用奥斯本法则而失分。以下是最常见的陷阱。
- Confusing cosh² x − sinh² x with 1: For circular functions, cos² x + sin² x = 1. For hyperbolic functions, the plus becomes a minus. — 将 cosh² x − sinh² x 误认为等于 1 以外的其他值。圆三角函数中 cos² x + sin² x = 1；双曲函数中加号变减号。
- Forgetting domain restrictions for inverse functions: arcosh x is only real for x ≥ 1, and artanh x only for |x| < 1. — 忘记反函数定义域限制：arcosh x 仅对 x ≥ 1 为实数，artanh x 仅对 |x| < 1 成立。
- Using sin² x = 1 − cos² x instead of the hyperbolic analogue: When integrating, converting a circular trig identity directly without changing the sign of the sinh² term leads to wrong answers. — 使用 sin² x = 1 − cos² x 而不是双曲类似公式：当积分时，若不改变 sinh² 项的符号则会导致错误答案。
- Miscomputing d/dx coth x: The derivative is −csch² x, not +csch² x. — 错误计算 d/dx coth x：其导数为 −csch² x，不是 +csch² x。
- Neglecting the constant of integration: Always include +C for indefinite integrals. — 忽略积分常数：不定积分务必添加 +C。
Reviewing these errors before the exam can save valuable marks. Always check whether a result makes sense at x = 0 or as x → ∞.
考试前复习这些错误可以节省宝贵的分数。务必检查结果在 x = 0 或 x → ∞ 时是否合理。
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