Definitions of Hyperbolic Functions | 双曲函数的定义

2026-08-21 21:21 · 本文部分内容由AI辅助创作

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

Hyperbolic functions are a family of functions built from the exponential function eˣ and its reciprocal e⁻ˣ. In AQA A-Level Mathematics, the first step is to learn their precise definitions, because almost every identity, derivative and equation-solving method follows from these definitions.

双曲函数是一族由指数函数 eˣ 及其倒数 e⁻ˣ 构成的函数。在 AQA A-Level 数学中，第一步是掌握它们的精确定义，因为几乎所有恒等式、导数和方程求解方法都从这些定义推导而来。

1. What Are Hyperbolic Functions? | 什么是双曲函数

Hyperbolic functions arise naturally when eˣ and e⁻ˣ are combined by addition or subtraction. They are named 'hyperbolic' angular hyperbola x² − y² = 1, just as (cos t, sin t) lies on the unit circle x² + y² = 1.

当我们将 eˣ 与 e⁻ˣ 相加或相减时，双曲函数自然出现。它们之所以被命名为“双曲”，是因为点 (cosh t, sinh t) 落在直角双曲线 x² − y² = 1 上，正如点 (cos t, sin t) 落在单位圆 x² + y² = 1 上。

This geometric link explains why many hyperbolic identities look similar to trigonometric identities, but with important sign changes.

这种几何联系解释了为什么许多双曲恒等式看起来与三角恒等式相似，但符号上有重要变化。

2. Core Definitions of sinh x and cosh x | sinh x 与 cosh x 的核心定义

The two fundamental hyperbolic functions are the hyperbolic sine and hyperbolic cosine. They are defined for all real x by the following exponential expressions:

两个基本的双曲函数是双曲正弦和双曲余弦。它们对所有实数 x 由以下指数表达式定义：

sinh x = (eˣ − e⁻ˣ) ÷ 2

cosh x = (eˣ + e⁻ˣ) ÷ 2

Here e is Euler's number, approximately 2.71828, and e⁻ˣ means 1 ÷ eˣ.

这里 e 是欧拉数，约为 2.71828，e⁻ˣ 表示 1 ÷ eˣ。

Notice that sinh x is half the difference of eˣ and e⁻ˣ, while cosh x is half their sum. These two definitions are the foundation for every other hyperbolic function.

注意 sinh x 是 eˣ 与 e⁻ˣ 之差的一半，而 cosh x 是两者之和的一半。这两个定义是所有其他双曲函数的基础。

广告 / Sponsored

🛒

TI-Nspire CX II CAS 图形计算器 A-Level数学

¥1298

📱 去京东购买

京东精选