Hyperbolic Functions: Definitions and Basic Properties | 双曲函数的定义与基本性质

📚 Hyperbolic Functions: Definitions and Basic Properties | 双曲函数的定义与基本性质

Hyperbolic functions are mathematical functions that bear a close resemblance to trigonometric functions, but are defined using exponential functions. They appear frequently in physics, engineering, and advanced calculus, especially in problems involving catenaries, special relativity, and differential equations.

双曲函数是一类与三角函数形式相似、但由指数函数定义的数学函数。它们广泛出现在物理、工程和高等微积分中，尤其在悬链线、狭义相对论和微分方程等问题中频繁出现。

1. Definitions of Hyperbolic Functions | 双曲函数的定义

The two fundamental hyperbolic functions are the hyperbolic sine, denoted sinh x, and the hyperbolic cosine, denoted cosh x. They are defined in terms of the exponential function eˣ.

两个基本的双曲函数是双曲正弦（记作 sinh x）和双曲余弦（记作 cosh x）。它们由指数函数 eˣ 定义。

sinh x = (eˣ − e⁻ˣ) / 2

cosh x = (eˣ + e⁻ˣ) / 2

From these two, the remaining four hyperbolic functions are defined analogously to trigonometric functions:

由这两个函数出发，其余四个双曲函数仿照三角函数定义如下：

tanh x = sinh x / cosh x — hyperbolic tangent | 双曲正切
coth x = cosh x / sinh x — hyperbolic cotangent | 双曲余切
sech x = 1 / cosh x — hyperbolic secant | 双曲正割
csch x = 1 / sinh x — hyperbolic cosecant | 双曲余割

2. Relationship to Exponential Functions | 与指数函数的关系

The definitions show that hyperbolic functions are simply linear combinations of eˣ and e⁻ˣ. Consequently, they satisfy simple addition formulas similar to those for trigonometric functions.

从定义可看出，双曲函数不过是 eˣ 与 e⁻ˣ 的线性组合。因此，它们满足与三角函数类似的加法公式。

The identity connecting the two basic functions is especially important:

联系两个基本函数的核心恒等式尤为重要：

cosh² x − sinh² x = 1

This is the hyperbolic analogue of the Pythagorean identity cos² x + sin² x = 1, but with a crucial minus sign.

这是三角恒等式 cos² x + sin² x = 1 的双曲版本，但关键的差别在于减号。

3. Basic Identities | 基本恒等式

All standard trigonometric identities have hyperbolic counterparts. The following are the most frequently used in A-level problems.

所有标准三角恒等式都有对应的双曲形式。以下是最常出现在 A-level 考题中的几个。

sinh(x + y) = sinh x cosh y + cosh x sinh y
sinh(x − y) = sinh x cosh y − cosh x sinh y
cosh(x + y) = cosh x cosh y + sinh x sinh y
cosh(x − y) = cosh x cosh y − sinh x sinh y
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x

Notice that the signs in the cosh addition formulas are the opposite of those in the corresponding cosine formulas.

注意，cosh 的加法公式中符号与对应的余弦加法公式相反。

4. Derivatives of Hyperbolic Functions | 双曲函数的导数

The derivatives of hyperbolic functions are remarkably simple and resemble those of their trigonometric counterparts, apart from sign differences in some cases.

双曲函数的导数形式非常简洁，与对应三角函数的导数相似，只是部分符号不同。

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

Note that the derivative of cosh x is +sinh x, unlike the derivative of cos x, which is −sin x.

注意，cosh x 的导数是 +sinh x，这与 cos x 的导数是 −sin x 不同。

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

The graph of y = cosh x is the catenary curve, which is the shape taken by a hanging flexible chain. It is an even function, since cosh(−x) = cosh x, and has a minimum value of 1 at x = 0.

y = cosh x 的图像是悬链线，也就是一条自然悬挂的柔软链条所呈现的形状。它是偶函数，因为 cosh(−x) = cosh x，且在 x = 0 处有最小值 1。

The graph of y = sinh x is an odd function, since sinh(−x) = −sinh x, and it passes through the origin with no stationary points.

y = sinh x 的图像是奇函数，因为 sinh(−x) = −sinh x，并且它经过原点，没有驻点。

Domain and range: sinh x has domain ℝ and range ℝ; cosh x has domain ℝ and range [1, ∞).
定义域与值域：sinh x 的定义域为 ℝ，值域为 ℝ；cosh x 的定义域为 ℝ，值域为 [1, ∞)。
tanh x is odd and has range (−1, 1), with horizontal asymptotes y = ±1.
tanh x 是奇函数，值域为 (−1, 1)，有水平渐近线 y = ±1。

6. Inverse Hyperbolic Functions | 反双曲函数

Because hyperbolic functions are defined in terms of exponentials, their inverses can be written using logarithms. The most common inverse functions are arsinh (or sinh⁻¹), arcosh (or cosh⁻¹), and artanh (or tanh⁻¹).

由于双曲函数由指数函数定义，其反函数可以用对数表示。最常见的反双曲函数是 arsinh（或 sinh⁻¹）、arcosh（或 cosh⁻¹）和 artanh（或 tanh⁻¹）。

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 logarithmic forms are often needed for integration and solving equations.

这些对数形式在积分和求解方程时经常需要用到。

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

The derivatives of inverse hyperbolic functions are particularly useful in integration, as they give antiderivatives of common algebraic expressions.

反双曲函数的导数在积分中尤其有用，因为它们给出了常见代数表达式的原函数。

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 formulas mirror the derivative formulas for inverse trigonometric functions, but with different signs in the denominators.

这些公式与反三角函数的导数公式形式对应，但分母中的符号有所不同。

8. Relationship with Trigonometric Functions | 与三角函数的联系

Hyperbolic functions can be obtained from trigonometric functions by replacing the real variable x with ix. Euler’s formulas give:

将三角函数中的实变量 x 换成 ix，便可得到双曲函数。由欧拉公式可得：

sinh x = −i sin(ix)

cosh x = cos(ix)

This connection explains why many identities are shared but have sign changes. It also allows solutions of certain differential equations to be written using either trigonometric or hyperbolic functions.

这一联系解释了为什么许多恒等式相似但符号有所变化，也使得某些微分方程的解既可用三角函数、也可用双曲函数表示。

9. Applications in Calculus and Modelling | 在微积分与建模中的应用

Hyperbolic functions are essential for solving problems involving the shape of a hanging cable (catenary), the velocity of a falling object in a resisting medium, and the growth of populations in certain models.

双曲函数在解决悬链线形状、物体在阻力介质中的下落速度以及某些种群增长模型等问题中至关重要。

They also appear in integration, where substitutions such as x = a sinh t or x = a cosh t can simplify expressions involving √(x² ± a²).

它们在积分中也有广泛应用，例如通过代换 x = a sinh t 或 x = a cosh t 可化简含 √(x² ± a²) 的表达式。

10. Summary | 总结

Hyperbolic functions are exponential-based analogues of trigonometric functions. Their defining identities, derivatives, and inverse forms are elegant and deeply interconnected. A firm grasp of these properties is essential for advanced calculus and applications.

双曲函数是以指数函数为基础的三角函数对应形式。它们的定义恒等式、导数和反函数形式简洁优美且联系紧密。扎实掌握这些性质是深入学习微积分及其应用的基础。

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Hyperbolic Functions: Definitions and Basic Properties | 双曲函数的定义与基本性质