Hyperbolic functions · 双曲函数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| catenary/kæˈtiːnəri/ | 悬链线 | xuán liàn xiàn |
| hyperbolic function/ˌhaɪpəˈbɒlɪk ˈfʌŋkʃn/ | 双曲函数 | shuāng qū hán shù |
| exponential/ˌekspəˈnenʃl/ | 指数 | zhǐ shù |
| identity/aɪˈdentɪti/ | 恒等式 | héng děng shì |
| derivative/dɪˈrɪvətɪv/ | 导数 | dǎo shù |
| inverse/ɪnˈvɜːs/ | 反函数 | fǎn hán shù |
The curves that hang from chains
- A hanging chain forms a catenary 悬链线 curve — not a parabola, but the graph of $\cosh x$.
- Hyperbolic functions 双曲函数 are built from exponentials 指数 and appear everywhere: suspension bridges, power lines, and even the shape of the universe in general relativity.
从链条上垂下的曲线
- 一条悬挂的链条形成一条悬链线(catenary)曲线——不是抛物线,而是 $\cosh x$ 的图。
- 双曲函数(hyperbolic functions)由指数函数构建,到处出现:悬索桥、电力线,甚至广义相对论中宇宙的形状。
Definitions
- Built from the exponential function:
Worked example. $\cosh 0 = \dfrac{e^0 + e^0}{2} = \dfrac{2}{2} = 1$. $\sinh 0 = \dfrac{1 - 1}{2} = 0$. So the catenary passes through $(0, 1)$, not the origin.
cosh and sinh are the average and half-difference of e^x and e^-x
定义
- 由指数函数构建:
算例。 $\cosh 0 = \dfrac{e^0 + e^0}{2} = \dfrac{2}{2} = 1$。$\sinh 0 = \dfrac{1 - 1}{2} = 0$。所以悬链线过 $(0, 1)$,不是原点。
cosh 和 sinh 是 e^x 和 e^-x 的平均和半差
Hyperbolic functions · 双曲函数
y = a cosh(x)
cosh is the catenary (a hanging chain) — even, with its minimum at (0, a). · cosh 是悬链线(一条悬挂的链)——偶函数,最小值在 (0, a)。
Using cosh x = (eˣ + e⁻ˣ)/2, what is cosh 0? · 用 cosh x = (eˣ + e⁻ˣ)/2,cosh 0 是多少?
cosh 0 = (e⁰ + e⁰)/2 = (1 + 1)/2 = 1. · cosh 0 = (e⁰ + e⁰)/2 = (1 + 1)/2 = 1。
Using sinh x = (eˣ − e⁻ˣ)/2, what is sinh 0? · 用 sinh x = (eˣ − e⁻ˣ)/2,sinh 0 是多少?
sinh 0 = (1 − 1)/2 = 0. · sinh 0 = (1 − 1)/2 = 0。
Key identities
- Main identity 恒等式: $\cosh^2 x - \sinh^2 x = 1$ (compare with $\cos^2 x + \sin^2 x = 1$).
- Addition: $\sinh(A + B) = \sinh A \cosh B + \cosh A \sinh B$.
Minus sign, not plus. The identity is $\cosh^2 x - \sinh^2 x = 1$ (minus), not $\cosh^2 x + \sinh^2 x = 1$. This is the key difference from trigonometric identities.
A hanging cable forms a catenary, the curve of the hyperbolic cosine y = a cosh(x/a)
关键恒等式
- 主要恒等式:$\cosh^2 x - \sinh^2 x = 1$(与 $\cos^2 x + \sin^2 x = 1$ 比较)。
- 加法:$\sinh(A + B) = \sinh A \cosh B + \cosh A \sinh B$。
是减号,不是加号。 恒等式是 $\cosh^2 x - \sinh^2 x = 1$(减),不是 $\cosh^2 x + \sinh^2 x = 1$。这是与三角恒等式的关键区别。
一根悬挂的缆绳形成一条悬链线,双曲余弦 y = a cosh(x/a) 的曲线
For any x, what does cosh²x − sinh²x equal? · 对任何 x,cosh²x − sinh²x 等于多少?
The hyperbolic identity is cosh²x − sinh²x = 1. · 双曲恒等式是 cosh²x − sinh²x = 1。
The identity cosh²x + sinh²x = 1 is correct. · 恒等式 cosh²x + sinh²x = 1 是正确的。
The correct identity is cosh²x − sinh²x = 1 (minus, not plus). · 正确的恒等式是 cosh²x − sinh²x = 1(减,不是加)。
Derivatives 导数
- $\dfrac{d}{dx}\sinh x = \cosh x$, $\quad\dfrac{d}{dx}\cosh x = \sinh x$ (no minus sign!).
- $\dfrac{d}{dx}\tanh x = \text{sech}^2 x$.
导数
- $\dfrac{d}{dx}\sinh x = \cosh x$,$\quad\dfrac{d}{dx}\cosh x = \sinh x$(没有负号!)。
- $\dfrac{d}{dx}\tanh x = \text{sech}^2 x$。
The derivative of sinh x is: · sinh x 的导数是:
d/dx(sinh x) = cosh x (no minus sign, unlike the trig derivative d/dx(sin x) = cos x). · d/dx(sinh x) = cosh x(没有负号,不像三角导数 d/dx(sin x) = cos x)。
The derivative of cosh x is: · cosh x 的导数是:
d/dx(cosh x) = sinh x (no minus sign, unlike d/dx(cos x) = −sin x). · d/dx(cosh x) = sinh x(没有负号,不像 d/dx(cos x) = −sin x)。
Inverse 反函数 hyperbolic functions
- Inverse forms are logarithmic:
- $\sinh^{-1}x = \ln(x + \sqrt{x^2 + 1})$
- $\cosh^{-1}x = \ln(x + \sqrt{x^2 - 1})$ for $x \geq 1$.
反双曲函数
- 反(inverse)形式是对数的:
- $\sinh^{-1}x = \ln(x + \sqrt{x^2 + 1})$
- $\cosh^{-1}x = \ln(x + \sqrt{x^2 - 1})$,对 $x \geq 1$。
You've got it
- $\cosh x = \dfrac{e^x + e^{-x}}{2}$, $\sinh x = \dfrac{e^x - e^{-x}}{2}$
- key identity: $\cosh^2 x - \sinh^2 x = 1$
- inverse hyperbolics have logarithmic forms
你掌握了
- $\cosh x = \dfrac{e^x + e^{-x}}{2}$,$\sinh x = \dfrac{e^x - e^{-x}}{2}$
- 关键恒等式:$\cosh^2 x - \sinh^2 x = 1$
- 反双曲函数有对数形式