Explore

Introduction to Complex Analysis

Approx. 27 hours to complete

Save Course

Go to Course

Course Summary

This course is an introduction to complex analysis which is a branch of mathematics that studies functions of complex numbers. It covers topics such as complex functions, differentiation, integration, and series.

Key Learning Points

Learn about the basics of complex analysis
Understand the theory behind complex functions and their properties
Apply complex analysis to real-world problems
Explore the relationship between complex analysis and other branches of mathematics

Learning Outcomes

Understand the theory and applications of complex analysis
Develop analytical skills for solving complex problems
Apply complex analysis to real-world scenarios

Prerequisites or good to have knowledge before taking this course

Knowledge of calculus and basic algebra
Familiarity with complex numbers

Course Difficulty Level

Intermediate

Course Format

Online self-paced course
Video lectures and quizzes

Similar Courses

Real Analysis
Partial Differential Equations
Abstract Algebra

Notable People in This Field

Terence Tao
John Baez
Timothy Gowers

Related Books

Description

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.

The homework assignments will require time to think through and practice the concepts discussed in the lectures. In fact, a significant amount of your learning will happen while completing the homework assignments. These assignments are not meant to be completed quickly; rather you'll need paper and pen with you to work through the questions. In total, we expect that the course will take 6-12 hours of work per module, depending on your background.

Outline

Introduction to Complex Numbers
History of Complex Numbers
Algebra and Geometry in the Complex Plane
Polar Representation of Complex Numbers
Roots of Complex Numbers
Topology in the Plane
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Module 1 Homework

Complex Functions and Iteration
Complex Functions
Sequences and Limits of Complex Numbers
Iteration of Quadratic Polynomials, Julia Sets
How to Find Julia Sets
The Mandelbrot Set
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Module 2 Homework

Analytic Functions
The Complex Derivative
The Cauchy-Riemann Equations
The Complex Exponential Function
Complex Trigonometric Functions
First Properties of Analytic Functions
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Module 3 Homework

Conformal Mappings
Inverse Functions of Analytic Functions
Conformal Mappings
Möbius transformations, Part 1
Möbius Transformations, Part 2
The Riemann Mapping Theorem
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Module 4 Homework

Complex Integration
Complex Integration
Complex Integration - Examples and First Facts
The Fundamental Theorem of Calculus for Analytic Functions
Cauchy’s Theorem and Integral Formula
Consequences of Cauchy’s Theorem and Integral Formula
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Module 5 Homework

Power Series
Infinite Series of Complex Numbers
Power Series
The Radius of Convergence of a Power Series
The Riemann Zeta Function And The Riemann Hypothesis
The Prime Number Theorem
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Module 6 Homework

Laurent Series and the Residue Theorem
Laurent Series
Isolated Singularities of Analytic Functions
The Residue Theorem
Finding Residues
Evaluating Integrals via the Residue Theorem
Bonus: Evaluating an Improper Integral via the Residue Theorem
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Lecture Slides
Module 7 Homework

Final Exam
Final Exam

Summary of User Reviews

Discover the world of complex analysis with this highly recommended course on Coursera. Users have praised the course for its comprehensive approach to the subject matter, with a focus on practical applications and real-world examples.

Key Aspect Users Liked About This Course

The course is highly comprehensive and covers a wide range of topics in complex analysis.

Pros from User Reviews

The course is well-structured and easy to follow.
The instructors are knowledgeable and engaging.
The course is suitable for both beginners and advanced learners.
The course offers plenty of opportunities for hands-on practice and application.

Cons from User Reviews

Some users have complained about the pace of the course, finding it too slow or too fast.
The course may require prior knowledge in calculus or other mathematical concepts.
Some users have found the course to be too theoretical and lacking in practical application.
The course may not be suitable for those looking for a quick, superficial overview of the subject matter.

Recommended for you

Complete Data Science Training with Python for Data Analysis

Beginners python data analytics : Data science introduction : Learn data science : Python data analysis methods tutorial THIS IS A COMPLETE DATA SCIENCE TRAINING WITH PYTHON FOR DATA ANALYSIS: ...

Save Course

Complete Introduction to Microsoft Power BI [2021 Edition]

A comprehensive guide on how to import, transform & visualise data with Power BI, with practical exercises & case study.  ...

Save Course

Leading the Life You Want

Pursue a meaningful life and improve your performance as a leader. You'll explore the core principles of leadership and learn the skills you need to bring them to life....

Save Course