Hyperbolic Functions | 双曲函数

📚 Hyperbolic Functions | 双曲函数

Hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the exponential function. They appear throughout mathematics, physics and engineering, and are an important topic in A-Level Further Mathematics.

双曲函数是普通三角函数的类比，但基于指数函数定义。它们在数学、物理和工程中广泛出现，是 A-Level 进阶数学的重要考点。

1. Definitions and Exponential Forms | 定义与指数形式

The three main hyperbolic functions are sinh (pronounced “shine”), cosh (“cosh”) and tanh (“than” or “tansh”). They are defined as follows for all real x:

三个主要的双曲函数是 sinh（读作 shine）、cosh（读作 cosh）和 tanh（读作 than 或 tansh）。它们对所有实数 x 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)

The reciprocal functions are defined analogously: sech x = 1/cosh x, cosech x = 1/sinh x, and coth x = 1/tanh x.

倒数函数类似地定义：sech x = 1/cosh x，cosech x = 1/sinh x，coth x = 1/tanh x。

2. Graphs and Key Properties | 图像与关键性质

The graph of y = sinh x is an odd function passing through the origin, increasing monotonically. The graph of y = cosh x is an even function with a minimum value of 1 at x = 0, often called a catenary. The graph of y = tanh x has horizontal asymptotes y = 1 and y = −1.

y = sinh x 的图像是过原点的奇函数，单调递增。y = cosh x 的图像是偶函数，在 x = 0 处取得最小值 1，常称为悬链线。y = tanh x 的图像有水平渐近线 y = 1 和 y = −1。

sinh(−x) = −sinh x and cosh(−x) = cosh x.
sinh(−x) = −sinh x，cosh(−x) = cosh x。
cosh x ≥ 1 for all x; −1 < tanh x < 1 for all x.
cosh x ≥ 1 对所有 x 成立；−1 < tanh x < 1 对所有 x 成立。

3. Osborn’s Rule and Trigonometric Identities | Osborn 法则与三角恒等式

Hyperbolic identities are similar to trigonometric identities, but with a crucial difference: whenever a product of two sines appears, the sign changes.

双曲恒等式与三角恒等式类似，但有一个关键区别：每当出现两个正弦的乘积时，符号要改变。

cosh² x − sinh² x = 1

Compare this with cos² x + sin² x = 1. Osborn’s rule states: replace sin by sinh and cos by cosh, then change the sign of any term that is a product of two sines.

与 cos² x + sin² x = 1 比较。Osborn 法则指出：将 sin 替换为 sinh，将 cos 替换为 cosh，然后将任何包含两个正弦乘积的项改变符号。

For example, cos(2x) = cos² x − sin² x becomes cosh(2x) = cosh² x + sinh² x.

例如，cos(2x) = cos² x − sin² x 变为 cosh(2x) = cosh² x + sinh² x。

4. Differentiation of Hyperbolic Functions | 双曲函数的求导

The derivatives of hyperbolic functions are straightforward because they are defined in terms of exponentials.

双曲函数的导数很直接，因为它们由指数函数定义。

d/dx (sinh x) = cosh x
d/dx (cosh x) = sinh x
d/dx (tanh x) = sech² x
d/dx (coth x) = −cosech² x
d/dx (sech x) = −sech x tanh x
d/dx (cosech x) = −cosech x coth x

Notice that the derivative of cosh x is sinh x, not −sinh x. This is a key difference from trigonometric functions.

注意 cosh x 的导数是 sinh x，而不是 −sinh x。这是与三角函数的一个关键区别。

5. Integration of Hyperbolic Functions | 双曲函数的积分

Integration follows directly from differentiation. The standard integrals you need to recognise are:

积分直接由求导得出。你需要识别以下标准积分：

∫ sinh x dx = cosh x + C
∫ cosh x dx = sinh x + C
∫ sech² x dx = tanh x + C
∫ cosech² x dx = −coth x + C
∫ sech x tanh x dx = −sech x + C
∫ cosech x coth x dx = −cosech x + C

Also remember that the constant of integration must be included.

还要记得必须加上积分常数 C。

6. Inverse Hyperbolic Functions | 反双曲函数

The inverse functions are arsinh x, arcosh x and artanh x. Their logarithmic forms are essential for solving equations and integrating certain rational functions.

反函数是 arsinh x、arcosh x 和 artanh x。它们的对数形式对解方程和积分某些有理函数至关重要。

arsinh x = ln(x + √(x² + 1))

arcosh x = ln(x + √(x² − 1)), x ≥ 1

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

These are particularly useful for integrating expressions like 1/√(x² + a²) and 1/√(x² − a²).

这些对于积分 1/√(x² + a²) 和 1/√(x² − a²) 这类表达式特别有用。

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

Using the logarithmic forms or implicit differentiation, the derivatives of inverse hyperbolic functions are:

利用对数形式或隐函数求导，反双曲函数的导数为：

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 enable you to integrate many algebraic functions directly.

这些导数使你能够直接积分许多代数函数。

8. Solving Equations with Hyperbolic Functions | 解双曲函数方程

You may be asked to solve equations involving hyperbolic functions. Two common methods are: using exponential definitions, or using identities.

你可能会被要求解含双曲函数的方程。两种常用方法是：使用指数定义，或使用恒等式。

Example: Solve sinh x = 3. Using sinh x = (eˣ − e⁻ˣ)/2 = 3 gives eˣ − e⁻ˣ = 6. Multiplying by eˣ: e²ˣ − 6eˣ − 1 = 0. Let y = eˣ, then y² − 6y − 1 = 0, so y = 3 ± √10. Since y > 0, y = 3 + √10, hence x = ln(3 + √10).

例：解 sinh x = 3。利用 sinh x = (eˣ − e⁻ˣ)/2 = 3 得 eˣ − e⁻ˣ = 6。乘以 eˣ：e²ˣ − 6eˣ − 1 = 0。令 y = eˣ，则 y² − 6y − 1 = 0，所以 y = 3 ± √10。因为 y > 0，取 y = 3 + √10，因此 x = ln(3 + √10)。

9. Standard Integrals Revisited | 再探标准积分

The inverse hyperbolic functions provide a neat way to integrate two important forms:

反双曲函数为积分两种重要形式提供了简洁的方法：

∫ dx/√(x² + a²) = arsinh(x/a) + C

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

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

These results are equivalent to logarithmic forms, but the arsinh/arcosh notation is often more convenient.

这些结果等价于对数形式，但 arsinh/arcosh 记号通常更方便。

10. Hyperbolic Identities for Double and Half Angles | 双曲二倍角与半角恒等式

Using Osborn’s rule, you can quickly derive double-angle identities:

使用 Osborn 法则，你可以快速推导二倍角恒等式：

sinh(2x) = 2 sinh x cosh x
cosh(2x) = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x
tanh(2x) = 2 tanh x / (1 + tanh² x)

Also useful are the addition formulas:

还有常用的加法公式：

sinh(x ± y) = sinh x cosh y ± cosh x sinh y
cosh(x ± y) = cosh x cosh y ± sinh x sinh y

Note the plus sign in cosh(x + y): cosh(x + y) = cosh x cosh y + sinh x sinh y, whereas for cos(x + y) it would be a minus.

注意 cosh(x + y) 中的加号：cosh(x + y) = cosh x cosh y + sinh x sinh y，而 cos(x + y) 中是减号。

11. Applications and Exam-style Questions | 应用与考试题型

Hyperbolic functions appear in real-world contexts such as the shape of a hanging cable (catenary), in solving differential equations, and in special relativity. On AQA A-Level papers, you may be tested on definitions, identities, differentiation, integration, and solving equations.

双曲函数出现在现实情境中，例如悬索的形状（悬链线）、解微分方程以及狭义相对论。在 AQA A-Level 试卷中，你可能会被考察定义、恒等式、求导、积分和解方程。

Common exam pitfalls include confusing sinh and cosh derivatives, forgetting Osborn’s sign change, and omitting the constant of integration. Practice converting equations into exponential form when identities do not work directly.

常见的考试陷阱包括混淆 sinh 和 cosh 的导数、忘记 Osborn 法则的符号改变、以及漏掉积分常数。当恒等式不能直接应用时，练习将方程转换为指数形式。

12. Summary of Key Results | 关键结论总结

Here is a quick reference table for the most important hyperbolic results.

以下是最重要的双曲结论速查表。

Function	Definition	Derivative	Integral (antiderivative)
sinh x	(eˣ − e⁻ˣ)/2	cosh x	cosh x + C
cosh x	(eˣ + e⁻ˣ)/2	sinh x	sinh x + C
tanh x	sinh x / cosh x	sech² x	ln(cosh x) + C
arsinh x	ln(x + √(x² + 1))	1/√(x² + 1)	—
arcosh x	ln(x + √(x² − 1))	1/√(x² − 1)	—
artanh x	½ ln((1+x)/(1−x))	1/(1 − x²)	—

Always remember the key identity cosh² x − sinh² x = 1, and apply Osborn’s rule carefully whenever converting trigonometric identities to hyperbolic ones.

始终牢记关键恒等式 cosh² x − sinh² x = 1，并且在将三角恒等式转换为双曲恒等式时小心应用 Osborn 法则。

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Hyperbolic Functions | 双曲函数