Hyperbolic Functions | 双曲函数

📚 Hyperbolic Functions | 双曲函数

Hyperbolic functions extend the analogy of trigonometric functions to the geometry of the unit hyperbola. They appear naturally in many areas of pure and applied mathematics, including calculus, differential equations, and complex analysis. In the IB AQA Mathematics course, you will learn their definitions, identities, differentiation, integration, and applications. Mastering hyperbolic functions not only prepares you for advanced topics but also deepens your understanding of the exponential function.

双曲函数将三角函数的类比推广到单位双曲线的几何当中。它们自然地出现在数学的许多领域，包括微积分、微分方程和复分析。在 IB AQA 数学课程中，你将学习双曲函数的定义、恒等式、微分、积分和应用。掌握双曲函数不仅能为你学习进阶内容做好准备，也能加深你对指数函数的理解。

1. Definitions and the Unit Hyperbola | 定义与单位双曲线

Hyperbolic sine and cosine are defined using exponential functions: sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. These definitions mirror the circular functions, which can be expressed via complex exponentials. The identity cosh² x − sinh² x = 1 shows that the point (cosh t, sinh t) lies on the unit hyperbola x² − y² = 1, just as (cos t, sin t) lies on the unit circle.

双曲正弦和双曲余弦通过指数函数定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。这些定义与可用复指数表达的圆函数类似。恒等式 cosh² x − sinh² x = 1 表明点 (cosh t, sinh t) 在单位双曲线 x² − y² = 1 上，正如 (cos t, sin t) 在单位圆上。

2. Graphing sinh x, cosh x, and tanh x | 双曲正弦、双曲余弦和双曲正切的图像

sinh x is an odd, unbounded function passing through the origin with increasing slope. cosh x is an even function, symmetric about the y-axis, with a minimum value of 1 at x = 0. tanh x = sinh x / cosh x is an odd function bounded between −1 and 1, with horizontal asymptotes y = ±1. These graphs are essential for understanding behaviour in limits and applications.

sinh x 是过原点的奇函数，无界且斜率递增。cosh x 是偶函数，关于 y 轴对称，在 x = 0 处取得最小值 1。tanh x = sinh x / cosh x 是界于 −1 与 1 之间的奇函数，水平渐近线为 y = ±1。这些图像对于理解函数的极限行为及应用至关重要。

3. Basic Hyperbolic Identities | 基本双曲恒等式

Beyond cosh² x − sinh² x = 1, you must know related identities: 1 − tanh² x = sech² x, and coth² x − 1 = csch² x. Similarly, compound-angle formulas mirror trigonometric ones with sign changes: sinh(x ± y) = sinh x cosh y ± cosh x sinh y; cosh(x ± y) = cosh x cosh y ¬ sinh x sinh y. The sign difference in the cosh formula is crucial.

除 cosh² x − sinh² x = 1 之外，你还需掌握相关恒等式：1 − tanh² x = sech² x，以及 coth² x − 1 = csch² x。同样，和角公式与三角函数类似，但要注意符号变化：sinh(x ± y) = sinh x cosh y ± cosh x sinh y；cosh(x ± y) = cosh x cosh y ¬ sinh x sinh y。cosh 公式中的符号差异至关重要。

4. Osborne’s Rule | 奥斯本法则

Osborne’s rule helps convert trigonometric identities into hyperbolic identities: replace the trigonometric function with the corresponding hyperbolic function, and change the sign of any product (or implied product) of two sines. For example, from cos(A+B) = cos A cos B − sin A sin B, we get cosh(A+B) = cosh A cosh B + sinh A sinh B (sign change for the product of two sinh terms).

奥斯本法则有助于将三角恒等式转化为双曲恒等式：将三角函数替换为对应的双曲函数，并将任何两个正弦的乘积（或隐含乘积）的符号取反。例如，由 cos(A+B) = cos A cos B − sin A sin B 可得 cosh(A+B) = cosh A cosh B + sinh A sinh B（两个 sinh 项的乘积符号改变）。

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

The derivatives of hyperbolic functions are straightforward and parallel trigonometric derivatives, without the sign changes. d/dx (sinh x) = cosh x; d/dx (cosh x) = sinh x; d/dx (tanh x) = sech² x. For reciprocal functions, d/dx (coth x) = −csch² x, d/dx (sech x) = −sech x tanh x, d/dx (csch x) = −csch x coth x. These can all be derived from the exponential definitions.

双曲函数的导数十分直接，且与三

角函数的导数类似，但无符号变化。d/dx (sinh x) = cosh x；d/dx (cosh x) = sinh x；d/dx (tanh x) = sech² x。对于倒数函数，d/dx (coth x) = −csch² x，d/dx (sech x) = −sech x tanh x，d/dx (csch x) = −csch x coth x。这些都可以从指数定义推导出来。

6. Inverse Hyperbolic Functions | 反双曲函数

The inverse hyperbolic functions, arsinh x, arcosh x, and artanh x, are defined as the inverses of the restricted hyperbolic functions. Since cosh is not one-to-one over all real numbers, arcosh x uses the restriction x ≥ 0 (so the range is y ≥ 0). Their logarithmic forms are vital: arsinh x = ln(x + √(x²+1)), valid for all real x; arcosh x = ln(x + √(x²−1)), valid for x ≥ 1; artanh x = ½ ln((1+x)/(1−x)), for |x| < 1.

反双曲函数 arsinh x、arcosh x 和 artanh x 被定义为对应受限双曲函数的反函数。由于 cosh 在全实轴上不是一一映射，arcosh x 采用 x ≥ 0 的限制（因此值域为 y ≥ 0）。它们的对数形式至关重要：arsinh x = ln(x + √(x²+1))，对所有实数 x 成立；arcosh x = ln(x + √(x²−1))，x ≥ 1；artanh x = ½ ln((1+x)/(1−x))，|x| < 1。

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

Derivatives of inverse hyperbolic functions are algebraic: 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 results can be obtained via implicit differentiation or by differentiating the logarithmic forms. They frequently appear in integration problems.

反双曲函数的导数为代数形式：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。这些结果可以通过隐函数求导或对对数形式求导得到。它们经常出现在积分问题中。

8. Integration Leading to Inverse Hyperbolics | 积分与反双曲函数

Standard integrals related to hyperbolic forms include ∫ 1/√(x²+a²) dx = arsinh(x/a) + C or ln(x + √(x²+a²)) + C; ∫ 1/√(x²−a²) dx = arcosh(x/a) + C (for x > a) or ln|x + √(x²−a²)| + C; and ∫ 1/(a²−x²) dx = (1/a)artanh(x/a) + C (for |x| < a) or (1/(2a))ln|(a+x)/(a−x)| + C. Recognising these forms saves time in exams.

9. Solving Equations Involving Hyperbolics | 求解含双曲函数的方程

Equations like a sinh x + b cosh x = c can be solved by expressing sinh and cosh in terms of exponentials, leading to a quadratic in eˣ. Alternatively, rewrite the left-hand side in the form R cosh(x+α) or R sinh(x+α). For example, 3 sinh x + 4 cosh x = 5 can be recast as 5 cosh(x + artanh(3/4)) = 5, leading to x = arsinh(3/4) or a logarithmic solution.

如 a sinh x + b cosh x = c 的方程可以通过将 sinh 和 cosh 用指数表示来求解，从而得到关于 eˣ 的二次方程。另一种方法是将左边改写为 R cosh(x+α) 或 R sinh(x+α) 的形式。例如，3 sinh x + 4 cosh x = 5 可转化为 5 cosh(x + artanh(3/4)) = 5，从而得到 x = arsinh(3/4) 或对数形式的解。

10. Hyperbolic Functions in Calculus and Modelling | 双曲函数在微积分和建模中的应用

Hyperbolic functions model hanging cables (catenaries), described by y = a cosh(x/a). They also appear in velocity-dependent drag, special relativity, and the solution of linear differential equations with constant coefficients. Recognising the general solution y = A cosh(kx) + B sinh(kx) for y” − k²y = 0 is essential for differential equation units.

双曲函数可模拟悬垂的缆索（悬链线），其方程为 y = a cosh(x/a)。它们也出现在与速度相关的阻力、狭义相对论以及常系数线性微分方程的求解中。对于方程 y” − k²y = 0，识别其通解 y = A cosh(kx) + B sinh(kx) 对微分方程单元至关重要。

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Hyperbolic Functions | 双曲函数