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Introduction to Complex Analysis

4.8 Filled star Filled star Filled star Filled star Half star
Based on 1,066 ratings
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Dr. Petra Bonfert-Taylor

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.

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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.

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What's inside

Syllabus

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.
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Traffic lights

Read about what's good
what should give you pause
and possible dealbreakers
Taught by Dr. Petra Bonfert-Taylor, who frequently contributes to Springer's journal Complex Analysis and Operator Theory and is recognized as a mathematician who works in complex analysis and is known for her work on the Riemann zeta function and other topics in number theory
Covers topics like the geometry and algebra of complex numbers, complex dynamics, and complex trigonometric functions, which are highly relevant for understanding and using complex numbers
Develops learners' understanding of complex numbers, complex functions, and their applications, which are important in fields such as engineering, physics, and mathematics
Includes homework assignments and peer assessments that provide opportunities to practice and receive feedback on the concepts learned
Requires extensive background knowledge in complex analysis and complex functions, as it is not suitable for beginners in these topics
Learners need to have a strong foundation in complex analysis and complex functions so as to fully benefit from the course

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Reviews summary

Introduction to complex analysis: rigorous and enjoyable!

Learners say this engaging course is a largely positive introduction to complex analysis that makes understanding complex numbers more manageable. The course covers diverse topics including theorems, integrals, and fractals, emphasizing applications over proofs. The engaging, knowledgeable instructor presents clear explanations with intuitive examples that make complex analysis more understandable and practical.
Engaging, thought-provoking
"I really enjoyed the course! Now I'm interested in a more proof-oriented course in Complex Analysis."
"I really liked the course. It was extremely informative, all thanks to the brilliant professor to taught the course"
"I​ have never taken a mathematics course which I want to learn a few more times."
Challenging yet rewarding
"This course is a tough course."
"This course was really awesome. Especially Course instructor’s way of teaching was nice."
"This is a tough course, but extremely rewarding."
Relevant, well-explained
"Every topic is explained clearly and basics of complex analysis can be digested easily following the exercises and the lectures. "
"This course is a great introduction for anyone who interested in complex analysis, clear explanation of the concept, not a lot of proves. Highly recommend."
"I wish Prof. Petra Bonfert Taylor teaches more courses. She has taught this course with lot of enthusiasm and has made this course a very interesting journey from the beginning till end. Thank you very much."
Wonderful, knowledgeable
"Petra is a great teacher and I learned a lot of interesting things from complex analysis."
"Petra is a wonderful lecturer - her explanations are clear and very understandable. I really enjoyed this course!"
"It's my pleasure and honour to learn math with Professor Bonfert-Taylor, she teaches in a comprehensive, rigorous (I have rarely seen her making mistakes), and sometimes homorous style. I like her and wish more courses if it is possible."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Introduction to Complex Analysis with these activities:
Read Complex Analysis by Elias M. Stein and Rami Shakarchi
This book provides an in-depth and comprehensive treatment of complex analysis concepts, helping to supplement and reinforce understanding gained from the course.
View Functional Analysis: Introduction to Further... on Amazon
Show steps
  • Purchase or borrow the book from a library.
  • Read and study the chapters that are covered in the course.
  • Work through the practice problems and exercises.
Review complex number operations
Working through examples and applying these operations will recall and solidify these key foundational concepts.
Browse courses on Complex Numbers
Show steps
  • Find practice problems online.
  • Work through several problems involving the operations of addition, subtraction, multiplication, and division of complex numbers.
  • Check answers with a calculator or online resource.
Watch video tutorials on complex analysis
Watching video tutorials can provide an alternative perspective on the concepts, reinforce understanding, and supplement the lectures.
Browse courses on Complex Analysis
Show steps
  • Find video tutorials online that are relevant to the course topics.
  • Watch the tutorials and take notes or summarize the main points.
Six other activities
Expand to see all activities and additional details
Show all nine activities
Organize notes and materials
Going through and organizing lecture materials will improve retention of the most important concepts and improve memory.
Show steps
  • Review lecture notes.
  • Create an outline or summary of the key concepts.
  • Organize materials into folders or a digital filing system.
Solve complex equations
Repeatedly solving complex equations will help to develop problem solving strategies for complex analysis and build problem solving skills.
Show steps
  • Find practice equations online.
  • Work through several equations involving complex numbers.
  • Check answers with a calculator or online resource.
Form a study group with other students
Discussing concepts and working together on problems in a group setting helps to improve understanding, identify areas of difficulty, and encourages collaboration.
Browse courses on Complex Analysis
Show steps
  • Find a group of students who are also taking the course.
  • Meet regularly to discuss lecture material, work on assignments, and prepare for exams.
Attend a workshop on complex analysis
Attending a workshop will provide the opportunity to learn from experts and ask questions, helping to clarify concepts and improve understanding.
Browse courses on Complex Analysis
Show steps
  • Research and find a relevant workshop.
  • Register and attend the workshop.
  • Actively participate in discussions and ask questions.
Create a presentation on a complex analysis topic
Creating a presentation requires synthesizing and organizing knowledge, which helps to improve understanding and retention.
Browse courses on Complex Analysis
Show steps
  • Choose a topic related to the course material.
  • Research and gather information on the topic.
  • Create a presentation using slides or other visual aids.
  • Practice presenting the material.
Tutor other students in complex analysis
Explaining concepts to other students helps to solidify understanding of complex analysis principles and improves communication skills.
Browse courses on Complex Analysis
Show steps
  • Offer tutoring services to classmates or students at a lower level.
  • Prepare materials and lesson plans for tutoring sessions.
  • Meet with students regularly to provide guidance and support.

Career center

Learners who complete Introduction to Complex Analysis will develop knowledge and skills that may be useful to these careers:
Physicist
Physicists use complex mathematical models to describe the physical world. They use these models to understand the behavior of matter and energy, and to make predictions about the future. Complex Analysis is a core area of mathematics, and the course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Physicists who are working on problems in this area.
See salaries and explore the career path for Physicist
Mathematician
Mathematicians develop new mathematical theories and solve complex mathematical problems. They use their knowledge of mathematics to advance our understanding of the world around us. Complex Analysis is a core area of mathematics, and the course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Mathematicians who are working on problems in this area.
See salaries and explore the career path for Mathematician
Statistician
Statisticians use complex mathematical models to analyze data and make recommendations to businesses. They use this data to help businesses understand their customers, make better decisions, and improve their operations. Complex Analysis can be a useful tool for Statisticians, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Statisticians who are working on problems that involve complex data or systems.
See salaries and explore the career path for Statistician
Data Scientist
Data Scientists use data to solve business problems. They collect data, analyze it, and build models to help businesses understand their customers, make better decisions, and improve their operations. Complex Analysis can be a useful tool for Data Scientists, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Data Scientists who are working on problems that involve complex data or systems.
See salaries and explore the career path for Data Scientist
Computer Scientist
Computer Scientists use complex mathematical models to design and develop computer systems. They use these models to ensure that computer systems are efficient, reliable, and secure. Complex Analysis can be a useful tool for Computer Scientists, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Computer Scientists who are working on problems that involve complex data or systems.
See salaries and explore the career path for Computer Scientist
Software Engineer
Software Engineers design, develop, and test software applications. They use complex mathematical models to ensure that software applications are efficient and reliable. Complex Analysis can be a useful tool for Software Engineers, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Software Engineers who are working on problems that involve complex data or systems.
See salaries and explore the career path for Software Engineer
Engineer
Engineers use complex mathematical models to design and build structures, machines, and systems. They use these models to ensure that their designs are safe, efficient, and reliable. Complex Analysis can be a useful tool for Engineers, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Engineers who are working on problems that involve complex data or systems.
See salaries and explore the career path for Engineer
Operations Research Analyst
Operations Research Analysts use advanced analytical techniques to help organizations make better decisions. They collect data, analyze it, and build models to help organizations understand their operations and make better decisions. Complex Analysis can be a useful tool for Operations Research Analysts, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Operations Research Analysts who are working on problems that involve complex data or systems.
See salaries and explore the career path for Operations Research Analyst
Actuary
Actuaries use complex mathematical models to assess risk and make recommendations to insurance companies. They use this data to help insurance companies make informed decisions about their products and services. Complex Analysis can be a useful tool for Actuaries, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Actuaries who are working on problems that involve complex financial data.
See salaries and explore the career path for Actuary
Economist
Economists use complex mathematical models to analyze economic data and make recommendations to governments and businesses. They use this data to help governments and businesses make informed decisions about their policies and strategies. Complex Analysis can be a useful tool for Economists, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Economists who are working on problems that involve complex economic data.
See salaries and explore the career path for Economist
Investment Analyst
Investment Analysts use complex mathematical models to analyze financial data and make recommendations to investors. They use this data to help investors make informed decisions about their investments. Complex Analysis can be a useful tool for Investment Analysts, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Investment Analysts who are working on problems that involve complex financial data.
See salaries and explore the career path for Investment Analyst
Risk Analyst
Risk Analysts use complex mathematical models to analyze financial data and make recommendations to investors. They use this data to help investors make informed decisions about their investments. Complex Analysis can be a useful tool for Risk Analysts, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Risk Analysts who are working on problems that involve complex financial data.
See salaries and explore the career path for Risk Analyst
Quantitative Analyst
Quantitative Analysts use complex mathematical models to analyze financial data and make recommendations to investors. They use this data to help investors make informed decisions about their investments. Complex Analysis can be a useful tool for Quantitative Analysts, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Quantitative Analysts who are working on problems that involve complex financial data.
See salaries and explore the career path for Quantitative Analyst
Financial Analyst
Financial Analysts use complex mathematical models to analyze financial data and make recommendations to investors. They use this data to help investors make informed decisions about their investments. Complex Analysis can be a useful tool for Financial Analysts, as it can help them to develop models that are more accurate and efficient. The course's emphasis on complex functions, power series, and Laurent series can be particularly helpful for Financial Analysts who are working on problems that involve complex financial data.
See salaries and explore the career path for Financial Analyst

Reading list

We've selected 15 books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Introduction to Complex Analysis.
Cover image
Functional Analysis
Save
Provides a comprehensive introduction to complex analysis, covering all the topics discussed in the course. It is written in a clear and concise style, with plenty of examples and exercises to help students understand the material.
Functional Analysis: Introduction to Further Topics...
Hardcover
$$$
Functional Analysis: Introduction to Further Topics...
Kindle Edition
$$$
Cover image
Analytic Functions
Save
Classic textbook on complex analysis. It is written in a clear and concise style, with plenty of examples and exercises to help students understand the material.
Analytic Functions (Princeton Legacy Library, 2107)
Hardcover
$$$$
Cover image
Complex Analysis in Banach Spaces
Save
Provides an introduction to complex analysis in Banach spaces.
Complex Analysis in Banach Spaces (Dover Books on...
Paperback
$$
Cover image
Complex Analysis
Save
Provides a thorough introduction to complex analysis, with a focus on applications to differential equations and physics.
Complex Analysis (Cambridge Mathematical Textbooks)
Kindle Edition
$$$
Cover image
Complex Manifolds
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Provides an introduction to geometric function theory, which branch of complex analysis that studies the geometric properties of complex functions.
Complex Manifolds
Paperback
$$$
Introduction to Complex Variables and Applications
Save
Provides an introduction to complex variables, with a focus on applications to differential equations and mathematical physics.
Introduction to Complex Variables and Applications...
Hardcover
$$$$
Introduction to Complex Variables and Applications...
Kindle Edition
$$$
Fundamentals of Complex Analysis with Applications...
Save
Provides an introduction to complex analysis, with a focus on applications to differential equations.
Fundamentals of Complex Analysis | Applications to...
Paperback
$$$
Cover image
Functions of One Complex Variable I
Save
Classic textbook on complex analysis. It covers a wide range of topics, from the basics of complex numbers to more advanced topics such as conformal mappings and the Riemann mapping theorem.
Functions of One Complex Variable I (Graduate Texts...
Paperback
$$
Cover image
Ebook: Complex Variables and Applications
Save
Popular textbook for undergraduate complex analysis courses. It covers all the topics discussed in the course, with a focus on applications to engineering and physics.
Ebook: Complex Variables and Applications
Paperback
Check
Ebook: Complex Variables and Applications
Kindle Edition
Check
Cover image
Random Heterogeneous Materials
Save
Provides an introduction to complex analysis, with a focus on applications to the physical sciences.
Random Heterogeneous Materials: Microstructure and...
Hardcover
$$$$
Cover image
Language and Society in South Asia
Save
Covers the basics of complex analysis, with a focus on geometric methods.
Language and Society in South Asia (Publications in...
Hardcover
$$$$
Language and Society in South Asia
Hardcover
$$
Cover image
Elementary Differential Equations and Boundary...
Save
Good choice for students who are interested in learning complex analysis for applications in mathematics and engineering.
Elementary Differential Equations and Boundary...
Paperback
$$$
Elementary Differential Equations and Boundary...
Loose Leaf
$$$$
Elementary Differential Equations and Boundary...
Loose Leaf
$$$$
Elementary Differential Equations and Boundary...
Paperback
$$$
Elementary Differential Equations and Boundary...
Loose Leaf
$$$$
Elementary Differential Equations and Boundary...
Loose Leaf
$$$$
Elementary Differential Equations and Boundary...
Kindle Edition
$$$$
Elementary Differential Equations and Boundary...
Kindle Edition
$$$$
Cover image
Mathematical Methods for Engineers and Scientists 1
Save
Provides an introduction to complex analysis, with a focus on applications to engineering and science.
Mathematical Methods for Engineers and Scientists 1...
Hardcover
$$
Mathematical Methods for Engineers and Scientists 1...
Hardcover
$$$
Mathematical Methods for Engineers and Scientists 1...
Kindle Edition
$$$
Cover image
Invitation to Complex Analysis
Save
Provides an introduction to complex analysis, with a focus on applications to differential equations and physics.
Invitation to Complex Analysis (Mathematical...
Hardcover
$$$
Cover image
Schaum's Outline of Complex Variables, 2ed
Save
Good resource for students who are looking for a quick review of complex analysis.
Schaum's Outline of Complex Variables, 2ed...
Paperback
$$
Schaum's Outline of Complex Variables, 2ed...
Kindle Edition
$$

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Rating
4.8 Filled star Filled star Filled star Filled star Half star
Effort
Approx. 27 hours
Level
Intermediate
Via
Coursera
Institution
Wesleyan University
Instructor
Dr. Petra Bonfert-Taylor
Language
English
Transcripts
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Traffic lights
Read about what's good
what should give you pause
and possible dealbreakers
Taught by Dr. Petra Bonfert-Taylor, who frequently contributes to Springer's journal Complex Analysis and Operator Theory and is recognized as a mathematician who works in complex analysis and is known for her work on the Riemann zeta function and other topics in number theory
Covers topics like the geometry and algebra of complex numbers, complex dynamics, and complex trigonometric functions, which are highly relevant for understanding and using complex numbers
Develops learners' understanding of complex numbers, complex functions, and their applications, which are important in fields such as engineering, physics, and mathematics
Includes homework assignments and peer assessments that provide opportunities to practice and receive feedback on the concepts learned
Requires extensive background knowledge in complex analysis and complex functions, as it is not suitable for beginners in these topics
Learners need to have a strong foundation in complex analysis and complex functions so as to fully benefit from the course
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