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所在平台: Udemy |
课程主页: https://www.udemy.com/course/complex-variables-and-transforms/
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**课程名称：** 复变函数与变换 (Complex Variables and Transforms) **课程概述：** 本课程是大学本科水平的课程，适合具备大学数学和微积分知识的学习者。课程分为两大部分： 1. **复数、复变量与复变函数：** 深入介绍复数系统、复变量、复变函数等概念。 2. **变换与级数：** 重点讲解傅里叶级数与变换、拉普拉斯变换、Z 变换等重要的分析工具。 **学习目标：** 成功完成本课程后，学生将能够： * 定义、理解并描述复数系统、复变函数、复变量、变换和级数。 * 分析涉及复变函数、傅里叶级数与变换、拉普拉斯变换、Z 变换、极限、连续性、可微性等问题的求解。 **主要内容：** * **复数引论：** 复变量、阿干图、复数的模与辐角、极坐标形式、棣分定理。 * **复变函数：**解析函数、调和函数与共轭调和函数、柯西-黎曼方程。 * **积分与定理：**线积分、柯西积分定理、柯西积分公式、路径无关性、两种积分方法。 * **傅里叶级数与变换：**周期函数/非周期函数的傅里叶级数/傅里叶变换、傅里叶变换性质、傅里叶逆变换、卷积定理。 * **拉普拉斯变换：** 初等函数的拉普拉斯变换、收敛域（ROC）概念与性质、拉普拉斯变换性质、拉普拉斯逆变换、卷积定理、赫维赛德展开公式、利用拉普拉斯变换求解常微分方程。 * **Z 变换：** Z 变换基础及相关数值问题。
This course is an undergrad level course designed to be studied by any one having understanding of College Level Math and Calculus.This course can be divided into two parts. First part is related to complex numbers, complex variables and functions. While the second part is related to Transforms and Series.After successfully completing this course, student is expected to be able define, understand and describe complex number system, complex functions, complex variables, transforms and series. Student is also expected to be able to analyze problems involving functions of complex variables, transforms and series (Fourier, Laplace, z Transforms and Fourier Series), limits, continuity, differentiability. Topics: Introduction to Complex Number: Complex Variable, Argand's Diagram, Modulus and Argument of a Complex Number, Polar Form, De Moivre's Theorem. Complex Functions: Analytical Functions, Harmonic and Conjugate, Harmonic Functions. Cauchy-Riemann Equations: Line Integrals, Cauchy's Theorem, Cauchy's Integral Formula, Independence of Path, Two Methods of Integration. Fourier Series and Transform: Fourier Series / Transform of periodic / non-periodic functions, Properties of Fourier Transform, Inverse Fourier Transform, Convolution Theorem. Laplace Transform: Laplace Transform of elementary functions, Concept and properties of Region of Convergence (ROC), Properties of Laplace Transform, Inverse Laplace Transform, Convolution Theorem, Heaviside Expansion Formula, Solution of Ordinary Differential Equations by Laplace Transform. z- Transforms: basics and few numerical problems.
