This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is known today. Each module consists of five video lectures with embedded quizzes, followed by an electronically graded homework assignment. Additionally, modules 1, 3, and 5 also contain a peer assessment.

제공자:

## Introduction to Complex Analysis

## About this Course

### 학습자 경력 결과

## 50%

## 25%

### 귀하가 습득할 기술

### 학습자 경력 결과

## 50%

## 25%

#### 100% 온라인

#### 유동적 마감일

#### 중급 단계

#### 완료하는 데 약 35시간 필요

#### 영어

### 제공자:

#### 웨슬리언 대학교

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.

## 강의 계획 - 이 강좌에서 배울 내용

**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.

**완료하는 데 5시간 필요**

**5개의 동영상**

**5개의 읽기 자료**

**1개 연습문제**

**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.

**완료하는 데 3시간 필요**

**5개의 동영상**

**5개의 읽기 자료**

**1개 연습문제**

**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!

**완료하는 데 5시간 필요**

**5개의 동영상**

**5개의 읽기 자료**

**1개 연습문제**

**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.

**완료하는 데 3시간 필요**

**5개의 동영상**

**5개의 읽기 자료**

**1개 연습문제**

### 검토

#### 4.8

##### INTRODUCTION TO COMPLEX ANALYSIS의 최상위 리뷰

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!

With this wonderful complex analysis course under your belt you will be ready for the joys of Digital Signal Processing, solving Partial Differential Equations and Quantum Mechanics.

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.

it was very challenging course , not so easy to pass the assignments but if you have gone through lectures, it will helps a lot while doing the assignments especially the final quiz

## 자주 묻는 질문

강의 및 과제를 언제 이용할 수 있게 되나요?

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.

이 수료증을 구매하면 무엇을 이용할 수 있나요?

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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