課程信息
4.8
510 個評分
161 個審閱
100% 在線

100% 在線

立即開始,按照自己的計劃學習。
可靈活調整截止日期

可靈活調整截止日期

根據您的日程表重置截止日期。
中級

中級

完成時間(小時)

完成時間大約為35 小時

建議:8 weeks of study, 6-12 hours/week...
可選語言

英語(English)

字幕:英語(English)

您將獲得的技能

Power SeriesComplex AnalysisMappingOptimizing Compiler
100% 在線

100% 在線

立即開始,按照自己的計劃學習。
可靈活調整截止日期

可靈活調整截止日期

根據您的日程表重置截止日期。
中級

中級

完成時間(小時)

完成時間大約為35 小時

建議:8 weeks of study, 6-12 hours/week...
可選語言

英語(English)

字幕:英語(English)

教學大綱 - 您將從這門課程中學到什麼

1
完成時間(小時)
完成時間為 5 小時

Introduction to Complex Numbers

We’ll begin this module by briefly learning about the history of complex numbers: When and why were they invented? In particular, we’ll look at the rather surprising fact that the original need for complex numbers did not arise from the study of quadratic equations (such as solving the equation z^2+1 = 0), but rather from the study of cubic equations! Next we’ll cover some algebra and geometry in the complex plane to learn how to compute with and visualize complex numbers. To that end we’ll also learn about the polar representation of complex numbers, which will lend itself nicely to finding roots of complex numbers. We’ll finish this module by looking at some topology in the complex plane....
Reading
5 個視頻 (總計 119 分鐘), 5 個閱讀材料, 2 個測驗
Video5 個視頻
Algebra and Geometry in the Complex Plane30分鐘
Polar Representation of Complex Numbers32分鐘
Roots of Complex Numbers14分鐘
Topology in the Plane21分鐘
Reading5 個閱讀材料
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Quiz1 個練習
Module 1 Homework10分鐘
2
完成時間(小時)
完成時間為 3 小時

Complex Functions and Iteration

Complex analysis is the study of functions that live in the complex plane, that is, functions that have complex arguments and complex outputs. The main goal of this module is to familiarize ourselves with such functions. Ultimately we’ll want to study their smoothness properties (that is, we’ll want to differentiate complex functions of complex variables), and we therefore need to understand sequences of complex numbers as well as limits in the complex plane. We’ll use quadratic polynomials as an example in the study of complex functions and take an excursion into the beautiful field of complex dynamics by looking at the iterates of certain quadratic polynomials. This allows us to learn about the basics of the construction of Julia sets of quadratic polynomials. You'll learn everything you need to know to create your own beautiful fractal images, if you so desire. We’ll finish this module by defining and looking at the Mandelbrot set and one of the biggest outstanding conjectures in the field of complex dynamics....
Reading
5 個視頻 (總計 123 分鐘), 5 個閱讀材料, 1 個測驗
Video5 個視頻
Sequences and Limits of Complex Numbers30分鐘
Iteration of Quadratic Polynomials, Julia Sets25分鐘
How to Find Julia Sets20分鐘
The Mandelbrot Set18分鐘
Reading5 個閱讀材料
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Quiz1 個練習
Module 2 Homework10分鐘
3
完成時間(小時)
完成時間為 5 小時

Analytic Functions

When studying functions we are often interested in their local behavior, more specifically, in how functions change as their argument changes. This leads us to studying complex differentiation – a more powerful concept than that which we learned in calculus. We’ll begin this module by reviewing some facts from calculus and then learn about complex differentiation and the Cauchy-Riemann equations in order to meet the main players: analytic functions. These are functions that possess complex derivatives in lots of places; a fact, which endows them with some of the most beautiful properties mathematics has to offer. We’ll finish this module with the study of some functions that are complex differentiable, such as the complex exponential function and complex trigonometric functions. These functions agree with their well-known real-valued counterparts on the real axis!...
Reading
5 個視頻 (總計 135 分鐘), 5 個閱讀材料, 2 個測驗
Video5 個視頻
The Cauchy-Riemann Equations29分鐘
The Complex Exponential Function24分鐘
Complex Trigonometric Functions21分鐘
First Properties of Analytic Functions25分鐘
Reading5 個閱讀材料
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Quiz1 個練習
Module 3 Homework10分鐘
4
完成時間(小時)
完成時間為 3 小時

Conformal Mappings

We’ll begin this module by studying inverse functions of analytic functions such as the complex logarithm (inverse of the exponential) and complex roots (inverses of power) functions. In order to possess a (local) inverse, an analytic function needs to have a non-zero derivative, and we’ll discover the powerful fact that at any such place an analytic function preserves angles between curves and is therefore a conformal mapping! We'll spend two lectures talking about very special conformal mappings, namely Möbius transformations; these are some of the most fundamental mappings in geometric analysis. We'll finish this module with the famous and stunning Riemann mapping theorem. This theorem allows us to study arbitrary simply connected sub-regions of the complex plane by transporting geometry and complex analysis from the unit disk to those domains via conformal mappings, the existence of which is guaranteed via the Riemann Mapping Theorem....
Reading
5 個視頻 (總計 113 分鐘), 5 個閱讀材料, 1 個測驗
Video5 個視頻
Conformal Mappings26分鐘
Möbius transformations, Part 127分鐘
Möbius Transformations, Part 217分鐘
The Riemann Mapping Theorem15分鐘
Reading5 個閱讀材料
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Lecture Slides10分鐘
Quiz1 個練習
Module 4 Homework10分鐘
4.8
161 個審閱Chevron Right
工作福利

29%

通過此課程獲得實實在在的工作福利

熱門審閱

創建者 RKApr 6th 2018

The lectures were very easy to follow and the exercises fitted these lectures well. This course was not always very rigorous, but a great introduction to complex analysis nevertheless. Thank you!

創建者 NSJun 25th 2018

The prof makes it easy to understand yet fascinating. I enjoyed video checkpoints, quizzes and peer reviewed assignments. This course encourages you to think and discover new things.

講師

Avatar

Dr. Petra Bonfert-Taylor

Former Professor of Mathematics at Wesleyan University / Professor of Engineering at Thayer School of Engineering at Dartmouth

關於 卫斯连大学

At Wesleyan, distinguished scholar-teachers work closely with students, taking advantage of fluidity among disciplines to explore the world with a variety of tools. The university seeks to build a diverse, energetic community of students, faculty, and staff who think critically and creatively and who value independence of mind and generosity of spirit. ...

常見問題

  • 注册以便获得证书后,您将有权访问所有视频、测验和编程作业(如果适用)。只有在您的班次开课之后,才可以提交和审阅同学互评作业。如果您选择在不购买的情况下浏览课程,可能无法访问某些作业。

  • 您购买证书后,将有权访问所有课程材料,包括评分作业。完成课程后,您的电子课程证书将添加到您的成就页中,您可以通过该页打印您的课程证书或将其添加到您的领英档案中。如果您只想阅读和查看课程内容,可以免费旁听课程。

還有其他問題嗎?請訪問 學生幫助中心