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 analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. They appear throughout mathematics, physics, and engineering, and are particularly useful in calculus and differential equations.

双曲函数是普通三角函数的类比，但基于双曲线而非圆来定义。它们广泛出现在数学、物理和工程中，在微积分和微分方程中尤其有用。

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1. The Basic Definitions | 基本定义

The two fundamental hyperbolic functions are the hyperbolic sine, denoted \(\sinh x\) (pronounced ‘sinch’), and the hyperbolic cosine, denoted \(\cosh x\) (pronounced ‘cosh’). They are defined in terms of the exponential function \(e^x\) as follows:

两个基本的双曲函数是双曲正弦，记作 \(\sinh x\)（读作“sinch”），以及双曲余弦，记作 \(\cosh x\)（读作“cosh”）。它们通过指数函数 \(e^x\) 定义如下：

\(\sinh x = \frac{e^x – e^{-x}}{2}\)

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

From these, the other four hyperbolic functions are defined in direct analogy with trigonometric ratios:

由此，其余四个双曲函数直接类比三角比进行定义：

• \(\tanh x = \dfrac{\sinh x}{\cosh x}\) (hyperbolic tangent / 双曲正切)
• \(\coth x = \dfrac{\cosh x}{\sinh x}\) (hyperbolic cotangent / 双曲余切)
• \(\mathrm{sech}\, x = \dfrac{1}{\cosh x}\) (hyperbolic secant / 双曲正割)
• \(\mathrm{cosech}\, x = \dfrac{1}{\sinh x}\) (hyperbolic cosecant / 双曲余割)

Notice that \(\sinh x\) uses the minus sign, while \(\cosh x\) uses the plus sign. This seemingly small difference leads to very different behaviours.

注意 \(\sinh x\) 使用减号，而 \(\cosh x\) 使用加号。这看似微小的差异导致完全不同的性质。

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2. The Exponential Forms | 指数形式

Because the definitions use \(e^x\) and \(e^{-x}\), it is often useful to rewrite hyperbolic functions purely in exponential form. This is especially helpful when solving equations or proving identities.

由于定义使用了 \(e^x\) 和 \(e^{-x}\)，将双曲函数完全写成指数形式往往很有用。这在解方程或证明恒等式时特别有帮助。

\(\sinh x = \frac{e^x – e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2}\)

For example, the product-to-sum formula for \(\sinh x \cosh y\) can be derived directly by expanding the exponentials. The exponential view also explains why \(\cosh x\) is even and \(\sinh x\) is odd:

例如，\(\sinh x \cosh y\) 的和差化积公式可以直接通过展开指数来推导。指数视角也解释了为何 \(\cosh x\) 是偶函数而 \(\sinh x\) 是奇函数：

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

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

The exponential form is also the key to differentiating hyperbolic functions, since differentiation of \(e^x\) is straightforward.

指数形式也是对双曲函数求导的关键，因为对 \(e^x\) 的求导非常直接。

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

The graph of \(y = \sinh x\) passes through the origin and is symmetric about the origin. It behaves like \(e^x/2\) for large positive \(x\) and like \( -e^{-x}/2\) for large negative \(x\).

\(y = \sinh x\) 的图像经过原点，并且关于原点对称。当 \(x\) 很大时它近似于 \(e^x/2\)，当 \(x\) 为很大的负数时近似于 \(-e^{-x}/2\)。

The graph of \(y = \cosh x\) is a U-shaped curve with a minimum at \((0,1)\). It is always greater than or equal to 1, and is symmetric about the \(y\)-axis. This shape is called a catenary when transformed by scaling.

\(y = \cosh x\) 的图像是一条 U 形曲线，在 \((0,1)\) 处取得最小值。它始终大于等于 1，且关于 \(y\) 轴对称。经过缩放后，这种形状称为悬链线。

Key properties to remember for A-Level:

A-Level 需要记住的关键性质：

• \(\sinh 0 = 0\), \(\cosh 0 = 1\)
• \(\sinh x\) is odd / \(\sinh x\) 是奇函数
• \(\cosh x\) is even / \(\cosh x\) 是偶函数
• \(\cosh x \ge 1\) for all real \(x\) / 对所有实数 \(x\) 有 \(\cosh x \ge 1\)
• \(\tanh x\) lies between \(-1\) and \(1\) / \(\tanh x\) 位于 \(-1\) 和 \(1\) 之间

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4. The Relationship with Trigonometric Functions | 与三角函数的关系

Hyperbolic functions satisfy identities that closely resemble trigonometric identities, but with some sign changes. The comparison is most striking when using complex numbers, but even without complex numbers we can see the algebraic similarity.

双曲函数满足的恒等式与三角恒等式非常相似，但某些符号有所不同。使用复数时这种对比最为显著，但即使不使用复数，我们也能看到代数上的相似性。

For example, the standard Pythagorean identity for trigonometric functions is:

例如，三角函数的标准毕达哥拉斯恒等式为：

\(\cos^2 x + \sin^2 x = 1\)

The analogous identity for hyperbolic functions is:

双曲函数的对应恒等式为：

\(\cosh^2 x – \sinh^2 x = 1\)

Notice the crucial minus sign. This identity can be verified directly from the exponential definitions:

注意这里关键的减号。这个恒等式可以直接通过指数定义验证：

\(\cosh^2 x – \sinh^2 x = \left(\frac{e^x+e^{-x}}{2}\right)^2 – \left(\frac{e^x-e^{-x}}{2}\right)^2 = \frac{4}{4} = 1\)

Because of the minus sign, hyperbolas, not circles, are described by the parametric equations \(x = a\cosh t, y = b\sinh t\).

由于这个减号，由参数方程 \(x = a\cosh t, y = b\sinh t\) 描述的是双曲线而非圆。

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5. Basic Identities | 基本恒等式

Just as with trigonometry, hyperbolic functions have a rich set of identities. The most important ones for A-Level are the same in structure as trigonometric identities, except where a product of two sines appears.

与三角学一样，双曲函数也有一系列丰富的恒等式。A-Level 最重要的恒等式在结构上与三角恒等式相同，唯一例外是出现两个正弦乘积的地方。

Pythagorean identities:

毕达哥拉斯恒等式：

\(\cosh^2 x – \sinh^2 x = 1\)

\(\tanh^2 x + \mathrm{sech}^2 x = 1\)

\(\coth^2 x – \mathrm{cosech}^2 x = 1\)

These follow from the first identity by dividing by \(\cosh^2 x\) or by \(\sinh^2 x\).

这些恒等式由第一个恒等式分别除以 \(\cosh^2 x\) 或 \(\sinh^2 x\) 得到。

Double-angle formulas:

二倍角公式：

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

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

Notice that the formula for \(\cosh 2x\) uses a plus sign, unlike \(\cos 2x = \cos^2 x – \sin^2 x\).

注意 \(\cosh 2x\) 的公式中使用加号，不同于 \(\cos 2x = \cos^2 x – \sin^2 x\)。

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6. Odd and Even Properties | 奇偶性质

Understanding the parity of hyperbolic functions helps in simplifying integrals and series expansions.

理解双曲函数的奇偶性有助于简化积分和级数展开。

• \(\sinh x\) is odd: \(\sinh(-x) = -\sinh x\)
• \(\cosh x\) is even: \(\cosh(-x) = \cosh x\)
• \(\tanh x\) is odd: \(\tanh(-x) = -\tanh x\)
• \(\coth x\) is odd: \(\coth(-x) = -\coth x\)
• \(\mathrm{sech}\, x\) is even: \(\mathrm{sech}(-x) = \mathrm{sech}\, x\)
• \(\mathrm{cosech}\, x\) is odd: \(\mathrm{cosech}(-x) = -\mathrm{cosech}\, x\)

These properties are direct consequences of the definitions. For example, \(\tanh(-x) = \frac{\sinh(-x)}{\cosh(-x)} = \frac{-\sinh x}{\cosh x} = -\tanh x\).

这些性质是定义的直接结果。例如，\(\tanh(-x) = \frac{\sinh(-x)}{\cosh(-x)} = \frac{-\sinh x}{\cosh x} = -\tanh x\)。

Because \(\sinh x\) is odd, its Maclaurin series contains only odd powers of \(x\). Because \(\cosh x\) is even, its series contains only even powers.

由于 \(\sinh x\) 是奇函数，其麦克劳林级数只包含 \(x\) 的奇次幂。由于 \(\cosh x\) 是偶函数，其级数只包含偶次幂。

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7. Addition Formulas | 加法公式

The addition formulas for hyperbolic functions are essential for simplifying expressions and solving equations.

双曲函数的加法公式对于简化表达式和求解方程至关重要。

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

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

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

Notice again the sign difference: in the formula for \(\cosh(x+y)\), the last term is plus, whereas the corresponding trigonometric formula has minus.

再次注意符号差异：在 \(\cosh(x+y)\) 的公式中，最后一项是加号，而对应的三角公式中是减号。

These formulas can be proved by substituting the exponential definitions and simplifying.

这些公式可以通过代入指数定义并化简来证明。

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

The derivatives of hyperbolic functions are very simple and closely resemble their trigonometric counterparts, with no sign changes for \(\sinh\) and \(\cosh\).

双曲函数的导数非常简单，与对应的三角函数导数非常相似，而且 \(\sinh\) 和 \(\cosh\) 的导数没有任何符号变化。

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

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

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

Useful derivative formulas also include:

其他有用的导数公式包括：

• \(\frac{d}{dx}(\coth x) = -\mathrm{cosech}^2 x\)
• \(\frac{d}{dx}(\mathrm{sech}\, x) = -\mathrm{sech}\, x \tanh x\)
• \(\frac{d}{dx}(\mathrm{cosech}\, x) = -\mathrm{cosech}\, x \coth x\)

These are obtained either from the exponential definitions or from the quotient rule. Because \(\frac{d}{dx}(e^x)=e^x\) and \(\frac{d}{dx}(e^{-x})=-e^{-x}\), the derivative of \(\sinh x\) is exactly \(\cosh x\).

这些公式可以通过指数定义或商法则得到。由于 \(\frac{d}{dx}(e^x)=e^x\) 且 \(\frac{d}{dx}(e^{-x})=-e^{-x}\)，因此 \(\sinh x\) 的导数正好是 \(\cosh x\)。

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9. Inverse Hyperbolic Functions | 反双曲函数

The inverse hyperbolic functions are defined in a way analogous to inverse trigonometric functions. The most common are \(\mathrm{arsinh}\, x\), \(\mathrm{arcosh}\, x\), and \(\mathrm{artanh}\, x\).

反双曲函数的定义方式与反三角函数类似。最常见的是 \(\mathrm{arsinh}\, x\)、\(\mathrm{arcosh}\, x\) 和 \(\mathrm{artanh}\, x\)。

Since \(\sinh x\) is one-to-one on \(\mathbb{R}\), its inverse \(\mathrm{arsinh}\, x\) is defined for all real \(x\). Because \(\cosh x\) is not one-to-one, we restrict its domain to \(x \ge 0\) to define \(\mathrm{arcosh}\, x\) for \(x \ge 1\).

由于 \(\sinh x\) 在 \(\mathbb{R}\) 上是一一对应的，其反函数 \(\mathrm{arsinh}\, x\) 对所有实数 \(x\) 有定义。由于 \(\cosh x\) 不是一一对应的，我们将其定义域限制为 \(x \ge 0\)，从而为 \(x \ge 1\) 定义 \(\mathrm{arcosh}\, x\)。

There are useful logarithmic forms for the inverse functions:

反函数存在有用的对数形式：

\(\mathrm{arsinh}\, x = \ln\left(x + \sqrt{x^2 + 1}\right) \quad (x \in \mathbb{R})\)

\(\mathrm{arcosh}\, x = \ln\left(x + \sqrt{x^2 – 1}\right) \quad (x \ge 1)\)

\(\mathrm{artanh}\, x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right) \quad (|x| < 1)\)

These forms are particularly useful when integrating rational functions.

这些形式在积分有理函数时特别有用。

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10. Why the ‘Hyperbolic’ Name? | 为什么叫“双曲”？

The functions are called hyperbolic because the parametric curve \(x = a\cosh t\), \(y = b\sinh t\) traces out the hyperbola

这些函数之所以称为“双曲”，是因为参数曲线 \(x = a\cosh t\)，\(y = b\sinh t\) 描绘的是双曲线

\(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\)

Just as circular functions satisfy \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) via \(\cos t\) and \(\sin t\), hyperbolic functions naturally parametrize the hyperbola.

正如圆函数通过 \(\cos t\) 和 \(\sin t\) 满足 \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)，双曲函数自然地对双曲线进行参数化。

This geometric interpretation shows why \(\cosh t\) plays the role of the \(x\)-coordinate and \(\sinh t\) the \(y\)-coordinate, with the minus sign appearing in the equation of the hyperbola.

这种几何解释说明了为什么 \(\cosh t\) 扮演 \(x\) 坐标的角色，而 \(\sinh t\) 扮演 \(y\) 坐标的角色，并且双曲线方程中出现减号。

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11. Worked Examples | 例题演示

Let us apply these definitions to a typical A-Level problem.

让我们将上述定义应用于一道典型的 A-Level 题目。

Example 1: Solve \(\cosh x = 3\) for \(x\), giving your answer as a natural logarithm.

例 1：解方程 \(\cosh x = 3\)，将答案表示为自然对数形式。

Solution: Use the definition \(\cosh x = \frac{e^x + e^{-x}}{2} = 3\). Multiply by 2: \(e^x + e^{-x} = 6\). Multiplying by \(e^x\) gives \(e^{2x} – 6e^x + 1 = 0\). This is a quadratic in \(e^x\):

解：利用定义 \(\cosh x = \frac{e^x + e^{-x}}{2} = 3\)。两边乘以 2：\(e^x + e^{-x} = 6\)。再乘以 \(e^x\) 得到 \(e^{2x} – 6e^x + 1 = 0\)。这是关于 \(e^x\) 的二次方程：

\(e^x = \frac{6 \pm \sqrt{36 – 4}}{2} = 3 \pm 2\sqrt{2}\)

Since \(e^x > 0\), both roots are positive, so

因为 \(e^x > 0\)，两个根都是正的，因此

\(x = \ln(3 + 2\sqrt{2})\) 或 \(x = \ln(3 – 2\sqrt{2}) = -\ln(3+2\sqrt{2})\)

Both are valid. Note that \(\ln(3 – 2\sqrt{2}) = -\ln(3+2\sqrt{2})\) because \(3-2\sqrt{2} = (3+2\sqrt{2})^{-1}\).

两个解都有效。注意 \(\ln(3 – 2\sqrt{2}) = -\ln(3+2\sqrt{2})\)，因为 \(3-2\sqrt{2} = (3+2\sqrt{2})^{-1}\)。

Example 2: Differentiate \(f(x) = \sinh(2x) + \cosh^2 x\).

例 2：求 \(f(x) = \sinh(2x) + \cosh^2 x\) 的导数。

Solution: Using the chain rule and the derivatives from Section 8,

解：利用链式法则和第 8 节的导数公式，

\(f'(x) = 2\cosh(2x) + 2\cosh x \sinh x = 2\cosh(2x) + \sinh(2x)\)

Here we used \(\sinh 2x = 2\sinh x \cosh x\).

这里我们使用了 \(\sinh 2x = 2\sinh x \cosh x\)。

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12. Summary | 总结

Hyperbolic functions are defined simply through exponentials: \(\sinh x = \frac{e^x – e^{-x}}{2}\) and \(\cosh x = \frac{e^x + e^{-x}}{2}\). All other hyperbolic functions derive from these two.

双曲函数通过指数简单定义：\(\sinh x = \frac{e^x – e^{-x}}{2}\) 和 \(\cosh x = \frac{e^x + e^{-x}}{2}\)。所有其他双曲函数都由这两个函数导出。

The key identity \(\cosh^2 x – \sinh^2 x = 1\) underlies many algebraic manipulations. Derivatives of hyperbolic functions are straightforward, and their curve shapes are useful in real-world modelling, such as hanging cables and velocity in special relativity.

关键恒等式 \(\cosh^2 x – \sinh^2 x = 1\) 是许多代数运算的基础。双曲函数的导数简单直接，它们的曲线形状在现实建模中很有用，例如悬索和狭义相对论中的速度。

For A-Level AQA, you need to confidently use the exponential definitions, prove identities, differentiate hyperbolic functions, and solve equations involving them.

对于 A-Level AQA，你需要熟练运用指数定义、证明恒等式、对双曲函数求导，并求解涉及它们的方程。

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