trigonometrical series

简明释义

三角级数

英英释义

例句

1.The trigonometrical series is essential in solving many periodic functions.

三角级数在解决许多周期性函数中是必不可少的。

2.In his research, he derived a new formula based on a trigonometrical series.

在他的研究中，他基于一个三角级数推导出了一个新公式。

3.The physicist applied a trigonometrical series to analyze wave patterns.

物理学家应用了一个三角级数来分析波形模式。

4.She learned about trigonometrical series in her advanced mathematics course.

她在高级数学课程中学习了三角级数。

5.The engineer used a trigonometrical series to model the vibrations of the bridge.

工程师使用了一个三角级数来模拟桥梁的振动。

作文

In the realm of mathematics, particularly in the study of Fourier analysis, the concept of trigonometrical series plays a pivotal role. A trigonometrical series is essentially a series that expresses a function as a sum of sine and cosine functions. This mathematical tool is not only foundational for theoretical mathematics but also has practical applications in various fields such as engineering, physics, and signal processing. The ability to decompose complex periodic functions into simpler trigonometric components allows for easier analysis and understanding of these functions.The origins of trigonometrical series can be traced back to the work of mathematicians like Joseph Fourier, who introduced the idea of representing functions as infinite sums of sine and cosine terms. This was revolutionary because it provided a systematic way to analyze periodic phenomena. For instance, any periodic function can be expressed as a trigonometrical series, which makes it possible to study its properties in a more manageable form.To illustrate this further, consider the example of sound waves. Sound can be represented as a combination of different frequencies, each corresponding to a specific pitch. By using trigonometrical series, we can break down complex sounds into their constituent sine and cosine waves, allowing sound engineers to manipulate and analyze audio signals effectively.Moreover, the convergence of trigonometrical series is an important aspect to consider. Not all series converge, and understanding the conditions under which a trigonometrical series converges to a function is crucial for practical applications. The Dirichlet conditions are often referenced in this context, providing guidelines on the types of functions that can be represented by a trigonometrical series. These conditions ensure that the series will converge to the original function at most points, providing a reliable framework for analysis.In addition to its theoretical importance, the application of trigonometrical series extends to various technologies we use today. For example, in telecommunications, signals are frequently analyzed using Fourier series, which rely on the principles of trigonometrical series. This allows for efficient data transmission and compression, which are essential in our increasingly digital world.Furthermore, the study of trigonometrical series also leads to advancements in numerical methods and computational algorithms. Understanding how to effectively compute these series enables mathematicians and scientists to solve complex problems that arise in different domains, from quantum mechanics to financial modeling.In conclusion, the concept of trigonometrical series is not merely an abstract mathematical idea; it is a powerful tool that bridges the gap between theory and practical application. Its significance in analyzing periodic functions, understanding convergence, and facilitating technological advancements cannot be overstated. As we continue to explore the depths of mathematics and its applications, the role of trigonometrical series will undoubtedly remain central to our understanding of the world around us.

在数学领域，特别是在傅里叶分析的研究中，三角级数的概念起着关键作用。三角级数本质上是一个将函数表示为正弦和余弦函数之和的级数。这一数学工具不仅是理论数学的基础，而且在工程、物理和信号处理等多个领域也有实际应用。将复杂的周期函数分解为更简单的三角成分，使这些函数的分析和理解变得更加容易。三角级数的起源可以追溯到约瑟夫·傅里叶的工作，他提出了将函数表示为无穷正弦和余弦项之和的思想。这是革命性的，因为它提供了一种系统的方法来分析周期现象。例如，任何周期函数都可以表示为三角级数，这使得以更易管理的形式研究其属性成为可能。为了进一步说明这一点，考虑声音波的例子。声音可以表示为不同频率的组合，每个频率对应于特定的音高。通过使用三角级数，我们可以将复杂的声音分解为其组成的正弦和余弦波，从而使声音工程师能够有效地操控和分析音频信号。此外，三角级数的收敛性是一个重要的方面。并非所有的级数都是收敛的，理解三角级数在何种条件下收敛到一个函数对于实际应用至关重要。狄利克雷条件通常在这种情况下被提及，为可以用三角级数表示的函数类型提供了指导。这些条件确保级数将在大多数点收敛到原始函数，为分析提供了可靠的框架。除了其理论重要性外，三角级数的应用还扩展到我们今天使用的各种技术中。例如，在电信中，信号经常使用傅里叶级数进行分析，而傅里叶级数依赖于三角级数的原理。这使得数据传输和压缩变得高效，这在我们日益数字化的世界中是必不可少的。此外，三角级数的研究也推动了数值方法和计算算法的发展。理解如何有效计算这些级数使数学家和科学家能够解决在不同领域出现的复杂问题，从量子力学到金融建模。总之，三角级数的概念不仅仅是一个抽象的数学思想；它是一个强大的工具，架起了理论与实际应用之间的桥梁。它在分析周期函数、理解收敛性以及促进技术进步方面的重要性不容小觑。随着我们继续探索数学及其应用的深度，三角级数的作用无疑将继续在我们理解周围世界的过程中占据中心地位。