hyperbolic function

简明释义

双曲线函数

英英释义

例句

1.The hyperbolic sine 双曲正弦 function is defined as (e^x - e^(-x))/2.

双曲正弦函数定义为(e^x - e^(-x))/2。

2.In calculus, the derivative of a hyperbolic function 双曲函数 is often used to solve complex problems.

在微积分中，双曲函数的导数常用于解决复杂问题。

3.Engineers often use hyperbolic functions 双曲函数 in calculations related to structures and materials.

工程师在与结构和材料相关的计算中经常使用双曲函数。

4.The area under the curve of a hyperbolic function 双曲函数 can be calculated using integration techniques.

可以使用积分技术计算双曲函数下的曲线面积。

5.When modeling certain physical phenomena, hyperbolic functions 双曲函数 provide more accurate results than trigonometric functions.

在建模某些物理现象时，双曲函数比三角函数提供更准确的结果。

作文

In the realm of mathematics, functions play a crucial role in understanding various phenomena. Among these functions, the hyperbolic function stands out due to its unique properties and applications. The hyperbolic function is defined using exponential functions, which can be quite fascinating for those delving into advanced mathematics. These functions include the hyperbolic sine (sinh), hyperbolic cosine (cosh), and their inverses, which are instrumental in many fields, including engineering, physics, and computer science.The hyperbolic function can be thought of as analogous to trigonometric functions but is based on hyperbolas rather than circles. For instance, while the sine and cosine functions describe relationships in circular geometry, the hyperbolic function relates to hyperbolic geometry. The definitions of the hyperbolic sine and cosine are as follows:- sinh(x) = (e^x - e^{-x})/2- cosh(x) = (e^x + e^{-x})/2These equations highlight how the hyperbolic function is derived from the exponential function, showcasing the deep connection between different areas of mathematics.One of the most intriguing aspects of the hyperbolic function is its relationship with the geometry of hyperbolas. Just as the unit circle is fundamental to the study of trigonometric functions, the unit hyperbola plays a similar role for hyperbolic functions. The equation of the unit hyperbola is given by x² - y² = 1, which leads to the definition of the hyperbolic function in terms of coordinates on this curve. This geometric interpretation allows mathematicians and scientists to visualize and better understand the behavior of these functions.In practical applications, the hyperbolic function finds its use in various fields. For example, in physics, the shape of a hanging cable or chain, known as a catenary, can be described using the hyperbolic cosine function. This is crucial in engineering when designing bridges and other structures that require an understanding of load distribution and structural integrity.Moreover, the hyperbolic function also appears in the solutions to certain differential equations, making it essential in mathematical modeling. In calculus, the derivatives and integrals of hyperbolic functions are often encountered, providing valuable tools for solving complex problems. The derivative of the hyperbolic sine function, for instance, is the hyperbolic cosine function: (d/dx)(sinh(x)) = cosh(x), which mirrors the relationship found in trigonometry where the derivative of sine is cosine.In conclusion, the hyperbolic function is a fascinating area of study within mathematics that extends beyond theoretical concepts into practical applications across various disciplines. Understanding the hyperbolic function not only enriches one’s mathematical knowledge but also provides essential tools for tackling real-world problems. Whether in engineering, physics, or computer science, the implications of hyperbolic functions are vast and significant, making them a vital part of the mathematical landscape.

在数学领域，函数在理解各种现象中起着至关重要的作用。在这些函数中，超越函数因其独特的性质和应用而脱颖而出。超越函数是通过指数函数定义的，这对那些深入研究高级数学的人来说是相当迷人的。这些函数包括超越正弦（sinh）、超越余弦（cosh）及其反函数，在工程、物理和计算机科学等多个领域中都具有重要意义。超越函数可以被视为与三角函数类似，但它是基于双曲线而非圆形。例如，虽然正弦和余弦函数描述的是圆形几何中的关系，但超越函数则与双曲几何相关。超越正弦和超越余弦的定义如下：- sinh(x) = (e^x - e^{-x})/2- cosh(x) = (e^x + e^{-x})/2这些方程突出了超越函数是如何从指数函数中导出的，展示了不同数学领域之间的深刻联系。超越函数最引人入胜的方面之一是它与双曲线几何的关系。就像单位圆对三角函数的研究至关重要一样，单位双曲线在超越函数中也扮演着类似的角色。单位双曲线的方程为x² - y² = 1，这导致了超越函数的定义，该定义涉及到该曲线上的坐标。这个几何解释使得数学家和科学家能够可视化并更好地理解这些函数的行为。在实际应用中，超越函数在多个领域中找到了它的用武之地。例如，在物理学中，悬挂电缆或链条的形状，即悬链线，可以用超越余弦函数来描述。这在工程设计桥梁和其他需要理解负载分布和结构完整性的结构时至关重要。此外，超越函数还出现在某些微分方程的解中，使其在数学建模中变得不可或缺。在微积分中，超越函数的导数和积分常常出现，为解决复杂问题提供了有价值的工具。例如，超越正弦函数的导数恰好是超越余弦函数：(d/dx)(sinh(x)) = cosh(x)，这与三角学中正弦的导数是余弦的关系相似。总之，超越函数是数学中一个迷人的研究领域，它不仅超越理论概念，还延伸到各个学科的实际应用中。理解超越函数不仅丰富了一个人的数学知识，也为解决现实世界的问题提供了必要的工具。无论是在工程、物理还是计算机科学中，超越函数的影响都是广泛而重要的，使它们成为数学领域的重要组成部分。