Large-time Behaviour of Wave Packets
Large-time Behaviour of Wave Packets
复制标题
波包的大时间行为
DOI：
10.1007/978-3-030-06194-4_3
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
A. Barletta
中科院分区：
文献类型：
--
作者：
A. Barletta
本章的目的是说明鞍点近似作为一种工具来检测在大的时间波包的行为。这一目标可以通过基于复变全纯函数理论的显示技术来实现。读者是引导沿着这种方式的简要说明的主要特点，复变量和全纯函数。讨论了复平面上路径上积分的性质。提供了洛朗级数展开式的元素、奇点、留数和柯西留数定理的陈述。综述了拉普拉斯变换的主要特点。然后，在大时间的渐近制度的波包的行为进行了研究，并提出了鞍点近似。同伦的核心作用，即连续变形的可能性，在复杂的平面上的路径，进行了讨论。
This chapter aims to illustrate the saddle-point approximation as a tool to detect the behaviour at large times of wave packets. This objective can be achieved by displaying techniques based on the theory of holomorphic functions of a complex variable. The reader is guided along this way by a brief illustration of the main features of complex variables and holomorphic functions. Properties of integrals over paths in the complex plane are discussed. Elements of Laurent series expansions, singular points, residues and a statement of Cauchy’s residue theorem are provided. The main features of the Laplace transform are surveyed. Then, the behaviour of a wave packet in the asymptotic regime of large time is studied and the saddle-point approximation is presented. The central role of homotopy, that is the possibility of deforming continuously a path in the complex plane, is discussed.