complex variable

简明释义

复变数

英英释义

例句

1.In engineering, we often analyze systems using a complex variable 复数变量 to understand their behavior in the frequency domain.

在工程学中，我们经常使用 复数变量 complex variable 来分析系统，以理解它们在频域中的行为。

2.The function defined by a complex variable 复数变量 can exhibit unique properties not found in real numbers.

由 复数变量 complex variable 定义的函数可以表现出在实数中找不到的独特性质。

3.Many physical phenomena can be modeled using a complex variable 复数变量, allowing for easier computation of results.

许多物理现象可以使用 复数变量 complex variable 建模，从而更容易计算结果。

4.When solving differential equations, introducing a complex variable 复数变量 can simplify the problem significantly.

在解决微分方程时，引入 复数变量 complex variable 可以显著简化问题。

5.In mathematics, contour integration is often performed over a path in the complex plane, where the integrand is a complex variable 复数变量 function.

在数学中，轮廓积分通常是在复平面中的路径上进行的，其中被积函数是一个 复数变量 complex variable 函数。

作文

In the field of mathematics, particularly in complex analysis, the term complex variable refers to a variable that can take on values in the complex number system. Complex numbers are numbers that have both a real part and an imaginary part, typically expressed in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit, which satisfies the equation i² = -1. Understanding complex variable is essential for various applications in engineering, physics, and applied mathematics. The study of complex variable opens up a rich area of mathematics that allows for the analysis of functions that are defined on complex numbers. One of the most significant aspects of functions of a complex variable is the concept of analyticity. A function is said to be analytic at a point if it is differentiable in a neighborhood around that point. This property leads to powerful results, such as the Cauchy-Riemann equations, which provide necessary and sufficient conditions for a function to be analytic. Analytic functions exhibit remarkable properties, including the ability to be represented by power series. This means that any analytic function can be expressed as an infinite sum of terms calculated from the values of the function at a single point. This characteristic not only simplifies calculations but also enhances our understanding of the behavior of these functions. For instance, the exponential function, sine, and cosine functions can all be expressed in terms of their Taylor series expansions, showcasing the beauty and utility of working with complex variable. Moreover, the study of complex variable is not limited to theoretical mathematics; it has practical applications in various fields. In electrical engineering, for example, alternating current (AC) circuits can be analyzed using complex variable techniques. The use of complex numbers allows engineers to easily represent and manipulate phase differences and magnitudes of voltages and currents. This simplification is crucial for designing efficient electrical systems. In fluid dynamics, complex variable methods facilitate the modeling of potential flow, which describes the motion of ideal fluids. By using complex potentials, engineers can derive solutions to complex flow problems that would otherwise be challenging to solve using real variables alone. Furthermore, the importance of complex variable extends into the realm of quantum mechanics, where wave functions are often represented as complex-valued functions. The probabilistic interpretation of quantum mechanics relies heavily on the mathematical framework provided by complex analysis. In conclusion, the concept of complex variable is a cornerstone of modern mathematics and its applications. From providing insights into the nature of analytic functions to enabling practical solutions in engineering and physics, the study of complex variable enriches our understanding of the mathematical world. As students and professionals alike delve deeper into this fascinating area, they uncover the profound implications and utilities that arise from the interplay between real and imaginary components of numbers. Mastering the concept of complex variable is not only an academic pursuit but also a gateway to solving real-world problems across various disciplines.

在数学领域，尤其是复分析中，术语复变量指的是可以取复数系统中值的变量。复数是同时具有实部和虚部的数，通常以 a + bi 的形式表示，其中 'a' 是实部，'b' 是虚部，而 'i' 是虚数单位，满足方程 i² = -1。理解复变量对于工程、物理和应用数学的各种应用至关重要。对复变量的研究开启了一个丰富的数学领域，允许对定义在复数上的函数进行分析。复变量函数最重要的方面之一是解析性的概念。如果一个函数在某一点附近可微分，则该函数被称为在该点解析。这一特性导致了一些强大的结果，例如柯西-黎曼方程，这些方程提供了函数解析的必要和充分条件。解析函数展现出显著的特性，包括能够用幂级数表示。这意味着任何解析函数都可以表示为一个无限项的和，这些项是从函数在某一点的值计算得出的。这一特性不仅简化了计算，而且增强了我们对这些函数行为的理解。例如，指数函数、正弦和余弦函数都可以用它们的泰勒级数展开来表示，展示了使用复变量的美丽和实用性。此外，复变量的研究不限于理论数学；它在各个领域都有实际应用。例如，在电气工程中，交变电流（AC）电路可以通过复变量技术进行分析。使用复数使工程师能够轻松表示和操作电压和电流的相位差和幅度。这一简化对于设计高效的电气系统至关重要。在流体动力学中，复变量方法促进了潜流建模，描述了理想流体的运动。通过使用复势，工程师可以推导出复杂流动问题的解决方案，这些问题如果仅用实变量解决将会非常困难。此外，复变量的重要性延伸到量子力学的领域，在那里波函数通常表示为复值函数。量子力学的概率解释在很大程度上依赖于复分析提供的数学框架。总之，复变量的概念是现代数学及其应用的基石。从提供对解析函数性质的深入理解，到在工程和物理中启用实际解决方案，复变量的研究丰富了我们对数学世界的理解。当学生和专业人士深入探索这一迷人领域时，他们揭示了实数与虚数之间相互作用所产生的深远影响和实用性。掌握复变量的概念不仅是一种学术追求，也是解决各个学科现实问题的途径。