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지금 바로 시작해 나만의 일정에 따라 학습을 진행하세요.

탄력적인 마감일

일정에 따라 마감일을 재설정합니다.

중급 단계

완료하는 데 약 34시간 필요

권장: 8 weeks of study, 6-12 hours/week...

영어

자막: 영어

귀하가 습득할 기술

Power SeriesComplex AnalysisMappingOptimizing Compiler

100% 온라인

지금 바로 시작해 나만의 일정에 따라 학습을 진행하세요.

탄력적인 마감일

일정에 따라 마감일을 재설정합니다.

중급 단계

완료하는 데 약 34시간 필요

권장: 8 weeks of study, 6-12 hours/week...

영어

자막: 영어

강의 계획 - 이 강좌에서 배울 내용

1
완료하는 데 5시간 필요

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....
5 videos (Total 119 min), 5 readings, 2 quizzes
5개의 동영상
Algebra and Geometry in the Complex Plane30m
Polar Representation of Complex Numbers32m
Roots of Complex Numbers14m
Topology in the Plane21m
5개의 읽기 자료
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
1개 연습문제
Module 1 Homework10m
2
완료하는 데 3시간 필요

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....
5 videos (Total 123 min), 5 readings, 1 quiz
5개의 동영상
Sequences and Limits of Complex Numbers30m
Iteration of Quadratic Polynomials, Julia Sets25m
How to Find Julia Sets20m
The Mandelbrot Set18m
5개의 읽기 자료
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
1개 연습문제
Module 2 Homework10m
3
완료하는 데 5시간 필요

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!...
5 videos (Total 135 min), 5 readings, 2 quizzes
5개의 동영상
The Cauchy-Riemann Equations29m
The Complex Exponential Function24m
Complex Trigonometric Functions21m
First Properties of Analytic Functions25m
5개의 읽기 자료
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
1개 연습문제
Module 3 Homework10m
4
완료하는 데 3시간 필요

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....
5 videos (Total 113 min), 5 readings, 1 quiz
5개의 동영상
Conformal Mappings26m
Möbius transformations, Part 127m
Möbius Transformations, Part 217m
The Riemann Mapping Theorem15m
5개의 읽기 자료
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
Lecture Slides10m
1개 연습문제
Module 4 Homework10m
4.8
181개의 리뷰Chevron Right

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최상위 리뷰

대학: 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!

대학: 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.

강사

Avatar

Dr. Petra Bonfert-Taylor

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

웨슬리언 대학교 정보

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. ...

자주 묻는 질문

  • 강좌에 등록하면 바로 모든 비디오, 테스트 및 프로그래밍 과제(해당하는 경우)에 접근할 수 있습니다. 상호 첨삭 과제는 이 세션이 시작된 경우에만 제출하고 검토할 수 있습니다. 강좌를 구매하지 않고 살펴보기만 하면 특정 과제에 접근하지 못할 수 있습니다.

  • 수료증을 구매하면 성적 평가 과제를 포함한 모든 강좌 자료에 접근할 수 있습니다. 강좌를 완료하면 전자 수료증이 성취도 페이지에 추가되며, 해당 페이지에서 수료증을 인쇄하거나 LinkedIn 프로필에 수료증을 추가할 수 있습니다. 강좌 콘텐츠만 읽고 살펴보려면 해당 강좌를 무료로 청강할 수 있습니다.

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