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所在平台: Udemy |
课程主页: https://www.udemy.com/course/mathematical-theory-of-complex-calculus/
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课程名称：掌握复杂微积分：从导数到留数 概述：复杂微积分是必修课程，旨在为学生提供复杂函数、复变量导数、Contour积分、Laurent级数、Fourier级数和留数等方面的基础。在本课程中，您将学习复杂微积分的关键概念，并通过数学推理进行理解，而不是死记公式和练习。以下是本课程的一些主要内容： - 复杂函数的介绍：课程开始时将重点讲解复杂函数的概念。 - 复变量的导数：接下来，将复导数的概念扩展到复杂变量的函数。 - Contour积分：您将学习Contour积分，推导以下定理：Cauchy积分定理和Cauchy积分公式。 - Laurent级数：将对Laurent级数进行数学推导，并从Laurent推导出Fourier和Taylor级数。 - 留数：您将被介绍留数及其在Contour积分中的应用。 先决条件：参加本课程之前，您应完成单变量微积分，尤其是导数和积分，以及多变量微积分，特别是线积分和Stokes定理。 原创材料：该课程基于讲师的复杂微积分笔记，因此内容呈现具有独创性。 侧重理解：解释侧重于理解和数学推导关键概念，而不是通过死记公式和练习来学习。 课程好处：本课程中呈现的一些结果构成了许多科学领域的基础，包括量子力学、量子场论和工程（例如动态系统的控制理论）。掌握本课程的内容后，您将能够处理最有趣的数学和工程问题。 适合人群：本课程适合任何希望扩展数学知识的人，包括数学、物理、工程及相关领域的学生，以及希望提升复杂微积分理解的专业人士。
Complex Calculus is an essential course that provides students with a foundation in complex functions, derivatives of complex variables, contour integration, Laurent series, Fourier series, and residues. In this course, you will learn the key concepts of Complex Calculus, and the process of reasoning by using mathematics, rather than rote memorization of formulas and exercises. Here's what you need to know about this course:Introduction to Complex Functions: The course begins by focusing on the concept of complex functions.Derivatives of Complex Variables: Next, the concept of derivative is extended to functions of a complex variable.Contour Integration: You will learn about contour integration, and the following theorems will be derived: Cauchy's integral theorem and Cauchy's integral formula.Laurent Series: The Laurent series will be mathematically derived. From Laurent, the Fourier and Taylor series are also derived.Residues: You will be introduced to residues and how to use them to do contour integration.Prerequisites: To take this course, you should have completed single variable Calculus, especially derivatives and integrals, and multivariable Calculus, especially line integrals and Stokes' theorem.Original Material: This course is based on the instructor's notes on Complex Calculus, and the presentation of the results is therefore original.Focusing on Understanding: The explanations are given by focusing on understanding and mathematically deriving the key concepts, rather than learning formulas and exercises by rote.Benefits: Some of the results presented in this course constitute the foundations of many branches of science, including Quantum Mechanics, Quantum Field Theory, and Engineering (in the Control theory of dynamical systems, for instance). By mastering the contents of this course, you will be able to tackle the most interesting mathematical and engineering problems.Who this course is for: This course is suitable for anyone interested in expanding their knowledge of mathematics, including students of mathematics, physics, engineering, and related fields, as well as professionals who wish to develop their understanding of Complex Calculus.
