Definitions of Hyperbolic Functions | 双曲函数的定义

📚 Definitions of Hyperbolic Functions | 双曲函数的定义

Hyperbolic functions are mathematical functions that share many properties with trigonometric functions, but are defined using the exponential function. They appear frequently in calculus, physics, and engineering, particularly in problems involving catenaries, special relativity, and differential equations. This article provides a rigorous introduction to the definitions of the six main hyperbolic functions, along with their domains, ranges, and fundamental identities.

双曲函数是一类与三角函数在性质上高度相似、但由指数函数定义的数学函数。它们在微积分、物理学和工程学中频繁出现，尤其在悬链线、狭义相对论和微分方程等问题中。本文系统介绍六个主要双曲函数的定义，以及它们的定义域、值域和基本恒等式。

1. The Exponential Foundation | 指数基础

Before defining hyperbolic functions, we must recall the exponential functions \( e^x \) and \( e^{-x} \). These two functions are the building blocks for all hyperbolic functions. The function \( e^x \) grows rapidly for positive \( x \) and tends to 0 for negative \( x \), while \( e^{-x} \) is its mirror image across the y-axis.

在定义双曲函数之前，我们必须回顾指数函数 \( e^x \) 和 \( e^{-x} \)。这两个函数是所有双曲函数的构建基石。函数 \( e^x \) 当 \( x \) 为正时快速增长，当 \( x \) 为负时趋于 0；而 \( e^{-x} \) 是它关于 y 轴的镜像。

The sum and difference of these two exponentials appear so often that they are given special names: hyperbolic sine and hyperbolic cosine. These names reflect a close analogy with the trigonometric sine and cosine, but instead of being based on a circle, they are based on an equilateral hyperbola \( x^2 – y^2 = 1 \).

这两个指数的和与差出现得如此频繁，以至于被赋予了特殊名称：双曲正弦和双曲余弦。这些名称反映了与三角正弦和余弦的紧密类比，但它们的几何基础不是圆，而是等轴双曲线 \( x^2 – y^2 = 1 \)。

2. Definition of sinh x | 双曲正弦的定义

The hyperbolic sine function, written as \(\sinh x\) (pronounced “shine x” or “sinch x”), is defined as:

双曲正弦函数，写作 \(\sinh x\)（读作“shine x”或“sinch x”），定义为：

\(\sinh x = \frac{e^x – e^{-x}}{2}\)

This function is defined for all real numbers \(x\). It is an odd function, meaning \(\sinh(-x) = -\sinh x\). Its graph passes through the origin and increases monotonically. As \(x \to \infty\), \(\sinh x \approx e^x/2\), and as \(x \to -\infty\), \(\sinh x \approx -e^{-x}/2\).

该函数对所有实数 \(x\) 均有定义。它是一个奇函数，即 \(\sinh(-x) = -\sinh x\)。其图像经过原点并且单调递增。当 \(x \to \infty\) 时，\(\sinh x \approx e^x/2\)；当 \(x \to -\infty\) 时，\(\sinh x \approx -e^{-x}/2\)。

3. Definition of cosh x | 双曲余弦的定义

The hyperbolic cosine function, written as \(\cosh x\) (pronounced “cosh x”), is defined as:

双曲余弦函数，写作 \(\cosh x\)（读作“cosh x”），定义为：

\(\cosh x = \frac{e^x + e^{-x}}{2}\)

This function is also defined for all real numbers \(x\). It is an even function, meaning \(\cosh(-x) = \cosh x\). Its graph is a symmetric U-shaped curve, with a minimum value of 1 at \(x = 0\). The shape of a hanging cable supported at both ends, called a catenary, is described by the graph of \(\cosh x\).

该函数同样对所有实数 \(x\) 有定义。它是一个偶函数，即 \(\cosh(-x) = \cosh x\)。其图像是一条对称的 U 形曲线，在 \(x = 0\) 处取得最小值 1。两端悬挂的电缆所形成的形状——悬链线，正是由 \(\cosh x\) 的图像描述的。

4. Definition of tanh x | 双曲正切的定义

The hyperbolic tangent function, written as \(\tanh x\) (pronounced “tanch x” or “than x”), is defined as the ratio of \(\sinh x\) to \(\cosh x\):

双曲正切函数，写作 \(\tanh x\)（读作“tanch x”或“than x”），定义为 \(\sinh x\) 与 \(\cosh x\) 之比：

\(\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x – e^{-x}}{e^x + e^{-x}}\)

Since \(\cosh x \geq 1\) for all \(x\), the denominator is never zero, so \(\tanh x\) is defined for all real numbers. It is an odd function and its range is strictly between -1 and 1. As \(x \to \infty\), \(\tanh x \to 1\); as \(x \to -\infty\), \(\tanh x \to -1\).

由于对所有 \(x\) 都有 \(\cosh x \geq 1\)，分母永不为零，因此 \(\tanh x\) 对所有实数都有定义。它是一个奇函数，值域严格位于 -1 和 1 之间。当 \(x \to \infty\) 时，\(\tanh x \to 1\)；当 \(x \to -\infty\) 时，\(\tanh x \to -1\)。

5. Reciprocal Hyperbolic Functions | 倒数双曲函数

Just as in trigonometry, we can define three reciprocal functions: \(\operatorname{csch} x\), \(\operatorname{sech} x\), and \(\operatorname{coth} x\). They are defined as follows:

与三角函数类似，我们可以定义三个倒数函数：\(\operatorname{csch} x\)、\(\operatorname{sech} x\) 和 \(\operatorname{coth} x\)。它们的定义如下：

\(\operatorname{csch} x = \frac{1}{\sinh x} = \frac{2}{e^x – e^{-x}}\)

\(\operatorname{sech} x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}\)

\(\operatorname{coth} x = \frac{1}{\tanh x} = \frac{e^x + e^{-x}}{e^x – e^{-x}}\)

Note that \(\sinh x = 0\) when \(x = 0\), so \(\operatorname{csch} x\) and \(\operatorname{coth} x\) are undefined at \(x = 0\). Meanwhile, \(\operatorname{sech} x\) is defined for all \(x\), with range \(0 < \operatorname{sech} x \leq 1\).

注意，当 \(x = 0\) 时 \(\sinh x = 0\)，因此 \(\operatorname{csch} x\) 和 \(\operatorname{coth} x\) 在 \(x = 0\) 处无定义。而 \(\operatorname{sech} x\) 对所有 \(x\) 都有定义，其值域为 \(0 < \operatorname{sech} x \leq 1\)。

6. Domains and Ranges | 定义域与值域

It is essential to know the domain and range of each hyperbolic function. The table below summarises them for quick reference.

掌握每个双曲函数的定义域与值域至关重要。下表总结了它们，便于快速查阅。

Function	Domain	Range
\(\sinh x\)	All real numbers	All real numbers
\(\cosh x\)	All real numbers	\([1, \infty)\)
\(\tanh x\)	All real numbers	\((-1, 1)\)
\(\operatorname{csch} x\)	\(x \neq 0\)	\((-\infty,0) \cup (0,\infty)\)
\(\operatorname{sech} x\)	All real numbers	\((0, 1]\)
\(\operatorname{coth} x\)	\(x \neq 0\)	\((-\infty,-1) \cup (1,\infty)\)

Understanding these domains and ranges is crucial when solving equations or sketching graphs. For example, because \(\cosh x \geq 1\), the inverse function \(\operatorname{arcosh} x\) is only defined for \(x \geq 1\).

理解这些定义域和值域对于解方程或绘制图像至关重要。例如，由于 \(\cosh x \geq 1\)，其反函数 \(\operatorname{arcosh} x\) 仅在 \(x \geq 1\) 时有定义。

7. Fundamental Identity: cosh²x – sinh²x = 1 | 基本恒等式：cosh²x – sinh²x = 1

The most important algebraic identity for hyperbolic functions is analogous to the Pythagorean identity in trigonometry. It is derived directly from the exponential definitions:

双曲函数最重要的代数恒等式与三角学中的毕达哥拉斯恒等式类似。它可以直接从指数定义推导出来：

\(\cosh^2 x – \sinh^2 x = 1\)

Proof: \(\cosh^2 x – \sinh^2 x = \left(\frac{e^x+e^{-x}}{2}\right)^2 – \left(\frac{e^x-e^{-x}}{2}\right)^2\). Expanding both squares, the cross terms cancel, leaving \(\frac{4}{4} = 1\).

证明：\(\cosh^2 x – \sinh^2 x = \left(\frac{e^x+e^{-x}}{2}\right)^2 – \left(\frac{e^x-e^{-x}}{2}\right)^2\)。展开两个平方后，交叉项抵消，剩下 \(\frac{4}{4} = 1\)。

This identity implies that the point \((\cosh t, \sinh t)\) lies on the hyperbola \(x^2 – y^2 = 1\), just as \((\cos t, \sin t)\) lies on the unit circle \(x^2 + y^2 = 1\).

该恒等式意味着点 \((\cosh t, \sinh t)\) 位于双曲线 \(x^2 – y^2 = 1\) 上，正如 \((\cos t, \sin t)\) 位于单位圆 \(x^2 + y^2 = 1\) 上一样。

8. Other Pythagorean-type Identities | 其他类毕达哥拉斯恒等式

Dividing the fundamental identity by \(\cosh^2 x\) gives:

将基本恒等式两边除以 \(\cosh^2 x\) 得到：

\(1 – \tanh^2 x = \operatorname{sech}^2 x\)

Similarly, dividing by \(\sinh^2 x\) gives:

类似地，除以 \(\sinh^2 x\) 得到：

\(\coth^2 x – 1 = \operatorname{csch}^2 x\)

These identities are useful for simplifying expressions and solving integrals, exactly as their trigonometric counterparts are used.

这些恒等式可用于化简表达式和求解积分，其用法与三角函数中的对应恒等式完全相同。

9. Addition Formulas | 加法公式

Hyperbolic functions satisfy addition formulas that closely resemble those for trigonometric functions:

双曲函数满足与三角函数非常相似的加法公式：

\(\sinh(x \pm y) = \sinh x \cosh y \pm \cosh x \sinh y\)

\(\cosh(x \pm y) = \cosh x \cosh y \pm \sinh x \sinh y\)

Notice a key difference from trigonometry: in the formula for \(\cosh(x+y)\), the sign is positive, whereas for \(\cos(x+y)\) it is negative. This is due to the absence of the imaginary unit \(i\).

注意与三角函数的一个关键区别：在 \(\cosh(x+y)\) 的公式中，符号为正；而在 \(\cos(x+y)\) 中符号为负。这源于没有虚数单位 \(i\) 的出现。

The corresponding formula for \(\tanh(x \pm y)\) is:

\(\tanh(x \pm y)\) 的对应公式为：

\(\tanh(x \pm y) = \frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}\)

10. Double Angle and Half Angle Formulas | 倍角与半角公式

Setting \(x = y\) in the addition formulas gives the double-angle identities:

在加法公式中令 \(x = y\) 即可得到倍角恒等式：

\(\sinh 2x = 2\sinh x \cosh x\)

\(\cosh 2x = \cosh^2 x + \sinh^2 x = 2\cosh^2 x – 1 = 1 + 2\sinh^2 x\)

These forms are especially useful in integration when substituting to simplify a quadratic expression. From the double-angle formula for \(\cosh\), we also obtain half-angle identities:

这些形式在积分中特别有用，可以通过代换简化二次表达式。由 \(\cosh\) 的倍角公式还可以得到半角恒等式：

\(\cosh^2 x = \frac{\cosh 2x + 1}{2}\)

\(\sinh^2 x = \frac{\cosh 2x – 1}{2}\)

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

A major reason for studying hyperbolic functions in A-Level mathematics is their neat differentiation rules:

在 A-Level 数学中学习双曲函数的一个重要原因是它们简洁的求导法则：

\(\frac{d}{dx}(\sinh x) = \cosh x\)

\(\frac{d}{dx}(\cosh x) = \sinh x\)

\(\frac{d}{dx}(\tanh x) = \operatorname{sech}^2 x\)

Note that the derivative of \(\cosh x\) is \(\sinh x\), with no sign change, unlike \(\frac{d}{dx}(\cos x) = -\sin x\). This makes hyperbolic functions particularly convenient in differential equations.

注意，\(\cosh x\) 的导数是 \(\sinh x\)，没有符号变化；这与 \(\frac{d}{dx}(\cos x) = -\sin x\) 不同。这使得双曲函数在微分方程中特别方便。

Using the reciprocal rule, we can also derive:

利用倒数法则，我们还可以推导出：

\(\frac{d}{dx}(\operatorname{sech} x) = -\operatorname{sech} x \tanh x\)

\(\frac{d}{dx}(\operatorname{csch} x) = -\operatorname{csch} x \coth x\)

\(\frac{d}{dx}(\operatorname{coth} x) = -\operatorname{csch}^2 x\)

12. Graphs and Key Features | 图像与关键特征

Sketching hyperbolic function graphs is a common exam question. Here are the key features to remember:

绘制双曲函数图像是常见的考题。以下是需要记住的关键特征：

The graph of \(\sinh x\) passes through the origin, is odd, and resembles \(x^3\) near the origin but grows exponentially at the ends.

\(\sinh x\) 的图像经过原点，为奇函数，在原点附近类似 \(x^3\)，但在两端呈指数增长。
The graph of \(\cosh x\) is even, has a minimum at \((0,1)\), and never goes below the x-axis.

\(\cosh x\) 的图像为偶函数，在 \((0,1)\) 处有最小值，且永远不会低于 x 轴。
The graph of \(\tanh x\) has two horizontal asymptotes: \(y = 1\) and \(y = -1\). It is strictly increasing.

\(\tanh x\) 的图像有两条水平渐近线：\(y = 1\) 和 \(y = -1\)。它严格递增。
For \(\operatorname{sech} x\), the maximum is 1 at \(x=0\), and the graph approaches 0 as \(x \to \pm\infty\).

对于 \(\operatorname{sech} x\)，最大值在 \(x=0\) 处为 1，且当 \(x \to \pm\infty\) 时图像趋近于 0。

Because hyperbolic functions are defined in terms of exponentials, their graphs are always smooth and continuous across their domains. In AQA A-Level Mathematics, you may be asked to sketch these graphs, find asymptotes, or use the definitions to prove identities. Mastering the definitions is therefore the foundation for all related exam questions.

由于双曲函数由指数函数定义，它们的图像在其定义域上始终光滑连续。在 AQA A-Level 数学中，你可能需要绘制这些图像、求渐近线，或利用定义证明恒等式。因此，掌握定义是解答所有相关考题的基础。

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Definitions of Hyperbolic Functions | 双曲函数的定义