📚 Hyperbolic Functions | 双曲函数
Hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. They appear naturally in many areas of mathematics, including calculus, differential equations, and complex analysis. For IB Mathematics HL students, a solid command of hyperbolic functions—their definitions, identities, derivatives, integrals, and inverses—is essential for success in the exam and further study.
双曲函数是普通三角函数的类似物，但基于双曲线而非圆来定义。它们在数学的许多领域中自然出现，包括微积分、微分方程和复分析。对于 IB 数学 HL 学生来说，熟练掌握双曲函数——其定义、恒等式、导数、积分和反函数——是考试成功和进一步学习的基础。
1. What Are Hyperbolic Functions? | 什么是双曲函数？
Hyperbolic functions are defined by combining exponential functions. The two fundamental ones are the hyperbolic sine and cosine:
双曲函数通过指数函数的组合来定义。最基本的两个是双曲正弦和双曲余弦：
sinh x = (eˣ − e⁻ˣ)/2
cosh x = (eˣ + e⁻ˣ)/2
The remaining four hyperbolic functions—tanh, coth, sech, and cosech— t hyperbola x² − y² = 1, in the same way that (cos θ, sin θ) parametrises the unit circle.
其余四个双曲函数——tanh、coth、sech 和 cosech——由 sinh 和 cosh 得出，就像三角函数由正弦和余弦得出一样。它们参数化了单位双曲线 x² − y² = 1 的右支，如同 (cos θ, sin θ) 参数化单位圆一样。
2. Definitions and Basic Properties | 定义与基本性质
The six hyperbolic functions are defined as follows:
六个双曲函数的定义如下：
- sinh x = (eˣ − e⁻ˣ)/2
- 双曲正弦：sinh x = (eˣ − e⁻ˣ)/2
- cosh x = (eˣ + e⁻ˣ)/2
- 双曲余弦：cosh x = (eˣ + e⁻ˣ)/2
- tanh x = sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)
- 双曲正切：tanh x = sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)
- coth x = cosh x / sinh x = (eˣ + e⁻ˣ)/(eˣ − e⁻ˣ), x ≠ 0
- 双曲余切：coth x = cosh x / sinh x = (eˣ + e⁻ˣ)/(eˣ − e⁻ˣ), x ≠ 0
- sech x = 1 / cosh x = 2/(eˣ + e⁻ˣ)
- 双曲正割：sech x = 1 / cosh x = 2/(eˣ + e⁻ˣ)
- cosech x = 1 / sinh x = 2/(eˣ − e⁻ˣ), x ≠ 0
- 双曲余割：cosech x = 1 / sinh x = 2/(eˣ − e⁻ˣ), x ≠ 0
is [1, ∞). tanh x has horizontal asymptotes at y = 1 and y = −1, so its range is (−1, 1).
重要的基本性质：sinh x 是奇函数，cosh x 是偶函数，tanh x 是奇函数。sinh x 的值域是所有实数；cosh x 在 x = 0 处取得最小值 1，值域为 [1, ∞)。tanh x 有水平渐近线 y = 1 和 y = −1，因此其值域为 (−1, 1)。
3. Graphs of Hyperbolic Functions | 双曲函数图像
otonically without bound. It resembles a cubic curve but with exponential growth for large |x| -shaped, passing through the origin, and approaches the horizontal lines y = ±1 as x → ±∞.
y = sinh x 的图像通过原点并单调递增无界。它类似于三次曲线，但对于大的 |x| 呈指数增长。y = cosh x 的图像关于 y 轴对称，形状像悬链线，最低点在 (0,1)。y = tanh x 的图像呈 S 形，通过原点，并在 x → ±∞ 时趋近于水平线 y = ±1。
ial equations. Note that cosh x is never zero, while sinh x has exactly one zero at x = 0.
理解这些图像有助于直观地把握恒等式、反函数以及微分方程解的行为。注意 cosh x 永不为零，而 sinh x 仅在 x = 0 处有一个零点。
4. Fundamental Hyperbolic Identities | 基本双曲恒等式
The most important identity is the hyperbolic analogue of the Pythagorean identity:
最重要的恒等式是勾股恒等式的双曲类似：
cosh² x − sinh² x = 1
Dividing by cosh² x and sinh² x yields two further useful forms:
除以 cosh² x 和 sinh² x 可得到另外两种有用形式：
1 − tanh² x = sech² x
coth² x − 1 = cosech² x
Angle sum and difference formulas are similar to those for trigonometric functions but with sign changes in certain products:
角的和差公式与三角函数的公式相似，但在某些乘积的符号上有变化：
sinh(A ± B) = sinh A cosh B ± cosh A sinh B
cosh(A ± B) = cosh A cosh B ± sinh A sinh B
tanh(A ± B) = (tanh A ± tanh B) / (1 ± tanh A tanh B)
The double-angle formulas follow directly:
倍角公式可直接得出：
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x
5. Osborn's Rule | 奥斯本规则
Osborn's rule states that any trigonometric identity can be converted into the corresponding hyperbolic identity by replacing each trigonometric function with its hyperbolic equivalent, and changing the sign of every term that contains the product of two sines. This is because sin² θ terms appear with a negative sign when moving from the circular to the hyperbolic Pythagorean identity.
奥斯本规则指出，任何三角恒等式都可以通过将每个三角函数替换为其双曲等价物，并改变每个包含两个正弦乘积的项的符号，从而转换为相应的双曲恒等式。这是因为在从圆的勾股恒等式过渡到双曲勾股恒等式时，sin² θ 项会出现负号。
lic identity cosh 2x = 1 + 2 sinh² x, where the sign of the sine-squared term is reversed.
例如，三角恒等式 cos 2θ = 1 − 2 sin² θ 变为双曲恒等式 cosh 2x = 1 + 2 sinh² x ，其中正弦平方项的符号被反转。
6. Derivatives of Hyperbolic Functions | 双曲函数的导数
The derivatives of hyperbolic functions are straightforward and free from the negative signs that appear in the derivatives of co-functions in trigonometry.
双曲函数的导数简单直接，且没有三角函数余函数导数中出现的负号。
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
These derivatives follow directly from the exponential definitions. Notice that the derivative of cosh x is sinh x (positive), not −sinh x. Only coth, sech, and cosech derivatives involve minus signs, making the chain of differentiation easier to remember than for trigonometric functions.
这些导数可直接由指数定义推导出来。注意 cosh x 的导数是 sinh x（正号），而不是 −sinh x。只有 coth、sech 和 cosech 的导数带负号，因此求导链比三角函数更容易记忆。
7. Integrals Involving Hyperbolic Functions | 双曲函数的积分
The integrals of the basic hyperbolic functions are equally simple:
基本双曲函数的积分同样简单：
∫ sinh x dx = cosh x + C
∫ cosh x dx = sinh x + C
∫ tanh x dx = ln(cosh x) + C
∫ coth x dx = ln|sinh x| + C
Hyperbolic functions also provide useful substitutions for integrals involving √x²+a², √x²−a², and √a²−x². For example, substituting x = a sinh u simplifies √x²+a² to a cosh u, while x = a cosh u works for √x²−a² when x ≥ a.
双曲函数还为涉及 √x²+a²、√x²−a² 和 √a²−x² 的积分提供了有用的代换。例如，令 x = a sinh u 可将 √x²+a² 化简为 a cosh u，而对于 x ≥ a 的情况，x = a cosh u 适用于 √x²−a²。
8. Inverse Hyperbolic Functions | 反双曲函数
wn as the principal value. The inverse hyperbolic functions are often denoted by arsinh, arcosh, artanh, etc., or equivalently by sinh⁻¹, cosh⁻¹, tanh⁻¹. The IB syllabus recognises both notations, but arsinh avoids confusion with the reciprocal function.
由于 sinh x 和 tanh x 在整个定义域上都是一一映射的，因此它们的反函数对于所有实数（sinh⁻¹）以及 −1 < x < 1（tanh⁻¹）均有定义。然而，cosh x 在整个定义域上不是一一映射的，因此我们将其限制在 x ≥ 0 来定义其反函数，即主值。反双曲函数通常记作 arsinh、arcosh、artanh 等，或等价地记作 sinh⁻¹、cosh⁻¹、tanh⁻¹。IB 课程对两种符号都认可，但 arsinh 可避免与倒数函数混淆。
in the line y = x. They are invaluable in integration and in solving hyperbolic equations.
反双曲函数的图像可通过将限制后的图像关于直线 y = x 反射得到。它们在积分和求解双曲方程中非常有用。
9. Logarithmic Forms of Inverse Hyperbolic Functions | 反双曲函数的对数形式
Because hyperbolic functions are defined by exponentials, their inverses can be expressed using natural logarithms:
由于双曲函数由指数定义，它们的反函数可以用自然对数表示：
arsinh x = ln(x + √x² + 1), x ∈ ℝ
arcosh x = ln(x + √x² − 1), x ≥ 1
artanh x = ½ ln((1 + x)/(1 − x)), |x| < 1
x in terms of y and then interchanging the variables. They are particularly helpful for differentiating the inverse functions and for evaluating definite integrals without a formula booklet.
这些对数形式是通过求解 y = sinh x（举例），用 y 表示 x，然后互换变量推导出来的。它们对于求反函数的导数以及在没有公式手册的情况下计算定积分尤其有用。
10. Derivatives of Inverse Hyperbolic Functions | 反双曲函数的导数
Differentiating the logarithmic forms, or using implicit differentiation on the original hyperbolic relations, gives the following standard derivatives:
对对数形式求导，或使用原双曲关系进行隐式求导，可得到以下标准导数：
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 results closely mirror the derivatives of inverse trigonometric functions, but with sign differences. For example, the derivative of arcsin x is 1/√1 − x², while that of arsinh x is 1/√x² + 1. Recognizing these patterns helps avoid confusion in integration problems.
这些结果与反三角函数的导数非常相似，但符号有所不同。例如，arcsin x 的导数是 1/√1 − x²，而 arsinh x 的导数是 1/√x² + 1。识别这些模式有助于避免积分问题中的混淆。
11. Connection with Trigonometric Functions | 与三角函数的联系
Hyperbolic and trigonometric functions are deeply connected via complex numbers. Using Euler's formula and the definitions of hyperbolic functions, we obtain elegant relationships for purely imaginary arguments:
双曲函数与三角函数通过复数有着深刻的联系。利用欧拉公式和双曲函数的定义，我们可以得到纯虚数自变量的优美关系：
sinh(ix) = i sin x
cosh(ix) = cos x
tanh(ix) = i tan x
From these, all the similarities and sign differences between hyperbolic and trigonometric identities can be understood. They also allow complex extensions of hyperbolic functions, which appear in advanced topics such as complex integration and conformal mappings.
从这些关系中，可以理解双曲恒等式与三角恒等式之间的所有相似之处和符号差异。它们还使得双曲函数可以推广到复数，这在诸如复积分和共形映射等高级主题中出现。
12. Solving Equations and Applications | 方程求解与应用
n solved by converting to exponential form. For instance, to solve sinh x = 2, we write (eˣ − e⁻ˣ)/2 = 2, then multiply by 2 and rearrange to obtain
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图