analytical continuation

简明释义

解析延拓;

英英释义

例句

1.Mathematicians often use analytical continuation 解析延拓 to find values of functions that are not defined at certain points.

数学家们经常使用 analytical continuation 解析延拓 来寻找在某些点未定义的函数值。

2.In complex analysis, the concept of analytical continuation 解析延拓 allows us to extend the domain of a given function beyond its initial definition.

在复分析中，analytical continuation 解析延拓 的概念使我们能够将给定函数的定义域扩展到其初始定义之外。

3.Researchers applied analytical continuation 解析延拓 techniques to solve problems in quantum field theory.

研究人员应用 analytical continuation 解析延拓 技术来解决量子场论中的问题。

4.Understanding analytical continuation 解析延拓 helps physicists derive results from known solutions.

理解 analytical continuation 解析延拓 有助于物理学家从已知解中推导结果。

5.The Riemann zeta function is a classic example of analytical continuation 解析延拓, which is initially defined for real numbers greater than 1.

黎曼ζ函数是 analytical continuation 解析延拓 的经典例子，它最初定义于大于1的实数上。

作文

In the realm of complex analysis, the concept of analytical continuation plays a pivotal role in extending the domain of functions beyond their initial definitions. This mathematical technique allows us to take a function that is defined in a limited region and expand it to a larger domain where the function remains valid. The idea is particularly useful when dealing with functions that have singularities or discontinuities in their original form. By employing analytical continuation, mathematicians can uncover deeper insights into the behavior of these functions and their relationships with other mathematical entities.To understand analytical continuation, consider the example of the Riemann zeta function, which is initially defined for complex numbers with a real part greater than one. However, through the process of analytical continuation, we can extend its definition to all complex numbers except for a simple pole at one. This extension reveals fascinating properties of the zeta function, including its connection to prime numbers through the famous Riemann Hypothesis. Such connections are not only intriguing but also essential for various applications in number theory and mathematical physics.The method of analytical continuation often involves finding a series representation of the function within its original domain and then using this representation to define the function in a broader context. This process typically requires a deep understanding of the function's behavior near its singularities and the ability to manipulate complex series and integrals. As such, analytical continuation is a powerful tool that bridges gaps in our understanding of complex functions and enables mathematicians to explore new territories within the field.Moreover, analytical continuation is not just limited to theoretical mathematics; it has practical implications in fields such as physics and engineering. For instance, in quantum mechanics, wave functions often need to be extended beyond their initial conditions to solve problems involving potential barriers or other complex scenarios. Here, analytical continuation provides a pathway to derive solutions that would otherwise remain inaccessible.In conclusion, the significance of analytical continuation cannot be overstated. It serves as a bridge that connects localized mathematical phenomena with global properties, allowing for a richer understanding of functions and their applications. By mastering the concept of analytical continuation, one gains access to a powerful set of tools that can unlock new possibilities in both pure and applied mathematics. As we continue to explore the complexities of mathematical functions, the role of analytical continuation will undoubtedly remain a cornerstone of our analytical toolkit, guiding us toward deeper discoveries and innovations in the field.

在复分析领域，解析延续的概念在将函数的定义域扩展到其初始定义之外时发挥着重要作用。这种数学技术使我们能够将一个在有限区域内定义的函数扩展到更大的有效域。这个想法在处理具有奇点或不连续性的函数时尤为有用。通过使用解析延续，数学家可以揭示这些函数及其与其他数学实体之间关系的更深层次见解。为了理解解析延续，考虑黎曼ζ函数的例子，该函数最初定义在实部大于1的复数上。然而，通过解析延续的过程，我们可以将其定义扩展到除了1以外的所有复数。这个扩展揭示了ζ函数的迷人特性，包括它与素数之间的联系，这一联系通过著名的黎曼假设得以体现。这些联系不仅引人入胜，而且对于数论和数学物理中的各种应用至关重要。解析延续的方法通常涉及在原始定义域内找到函数的级数表示，然后利用这个表示在更广泛的背景中定义函数。这个过程通常需要对函数在其奇点附近的行为有深入的理解，以及操纵复杂级数和积分的能力。因此，解析延续是一种强大的工具，它弥合了我们对复杂函数理解的空白，使数学家能够探索该领域的新领域。此外，解析延续不仅限于理论数学；它在物理学和工程等领域也具有实际意义。例如，在量子力学中，波函数通常需要超出其初始条件进行扩展，以解决涉及潜在障碍或其他复杂场景的问题。在这里，解析延续提供了一条获取否则无法接触的解决方案的路径。总之，解析延续的重要性不容小觑。它作为桥梁，将局部数学现象与全局属性连接起来，使我们能够更丰富地理解函数及其应用。通过掌握解析延续的概念，人们可以获得一套强大的工具，开启纯数学和应用数学的新可能性。当我们继续探索数学函数的复杂性时，解析延续的角色无疑将继续成为我们分析工具箱的基石，引导我们向更深层次的发现和创新迈进。