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Introduction to Complex Analysis
  • Level Professional
  • Duration 27 hours
  • Course by Wesleyan University
  • Offered by

About

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.

Modules

Lesson 1

    • 1 Videos
    • 1 Readings
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1 Videos

  • History of Complex Numbers

1 Readings

  • Lecture Slides

Lesson 2

    • 1 Videos
    • 1 Readings
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1 Videos

  • Algebra and Geometry in the Complex Plane

1 Readings

  • Lecture Slides

Lesson 3

    • 1 Videos
    • 1 Readings
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1 Videos

  • Polar Representation of Complex Numbers

1 Readings

  • Lecture Slides

Lesson 4

    • 1 Videos
    • 1 Readings
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1 Videos

  • Roots of Complex Numbers

1 Readings

  • Lecture Slides

Lesson 5

    • 1 Videos
    • 1 Readings
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1 Videos

  • Topology in the Plane

1 Readings

  • Lecture Slides

Review

    • 1 PeerReview
    • 1 Assignment
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1 Assignment

  • Module 1 Homework

1 Peer Review

  • Peer-Graded Assignment #1

Lesson 1

    • 1 Videos
    • 1 Readings
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1 Videos

  • Complex Functions

1 Readings

  • Lecture Slides

Lesson 2

    • 1 Videos
    • 1 Readings
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1 Videos

  • Sequences and Limits of Complex Numbers

1 Readings

  • Lecture Slides

Lesson 3

    • 1 Videos
    • 1 Readings
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1 Videos

  • Iteration of Quadratic Polynomials, Julia Sets

1 Readings

  • Lecture Slides

Lesson 4

    • 1 Videos
    • 1 Readings
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1 Videos

  • How to Find Julia Sets

1 Readings

  • Lecture Slides

Lesson 5

    • 1 Videos
    • 1 Readings
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1 Videos

  • The Mandelbrot Set

1 Readings

  • Lecture Slides

Review

    • 1 Assignment
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1 Assignment

  • Module 2 Homework

Lesson 1

    • 1 Videos
    • 1 Readings
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1 Videos

  • The Complex Derivative

1 Readings

  • Lecture Slides

Lesson 2

    • 1 Videos
    • 1 Readings
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1 Videos

  • The Cauchy-Riemann Equations

1 Readings

  • Lecture Slides

Lesson 3

    • 1 Videos
    • 1 Readings
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1 Videos

  • The Complex Exponential Function

1 Readings

  • Lecture Slides

Lesson 4

    • 1 Videos
    • 1 Readings
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1 Videos

  • Complex Trigonometric Functions

1 Readings

  • Lecture Slides

Lesson 5

    • 1 Videos
    • 1 Readings
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1 Videos

  • First Properties of Analytic Functions

1 Readings

  • Lecture Slides

Review

    • 1 PeerReview
    • 1 Assignment
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1 Assignment

  • Module 3 Homework

1 Peer Review

  • Peer Graded Assignment #2

Lesson 1

    • 1 Videos
    • 1 Readings
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1 Videos

  • Inverse Functions of Analytic Functions

1 Readings

  • Lecture Slides

Lesson 2

    • 1 Videos
    • 1 Readings
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1 Videos

  • Conformal Mappings

1 Readings

  • Lecture Slides

Lesson 3

    • 1 Videos
    • 1 Readings
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1 Videos

  • Möbius transformations, Part 1

1 Readings

  • Lecture Slides

Lesson 4

    • 1 Videos
    • 1 Readings
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1 Videos

  • Möbius Transformations, Part 2

1 Readings

  • Lecture Slides

Lesson 5

    • 1 Videos
    • 1 Readings
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1 Videos

  • The Riemann Mapping Theorem

1 Readings

  • Lecture Slides

Review

    • 1 Assignment
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1 Assignment

  • Module 4 Homework

Lesson 1

    • 1 Videos
    • 1 Readings
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1 Videos

  • Complex Integration

1 Readings

  • Lecture Slides

Lesson 2

    • 1 Videos
    • 1 Readings
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1 Videos

  • Complex Integration - Examples and First Facts

1 Readings

  • Lecture Slides

Lesson 3

    • 1 Videos
    • 1 Readings
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1 Videos

  • The Fundamental Theorem of Calculus for Analytic Functions

1 Readings

  • Lecture Slides

Lesson 4

    • 1 Videos
    • 1 Readings
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1 Videos

  • Cauchy’s Theorem and Integral Formula

1 Readings

  • Lecture Slides

Lesson 5

    • 1 Videos
    • 1 Readings
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1 Videos

  • Consequences of Cauchy’s Theorem and Integral Formula

1 Readings

  • Lecture Slides

Review

    • 1 PeerReview
    • 1 Assignment
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1 Assignment

  • Module 5 Homework

1 Peer Review

  • Peer-Graded Assignment #3

Lesson 1

    • 1 Videos
    • 1 Readings
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1 Videos

  • Infinite Series of Complex Numbers

1 Readings

  • Lecture Slides

Lesson 2

    • 1 Videos
    • 1 Readings
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1 Videos

  • Power Series

1 Readings

  • Lecture Slides

Lesson 3

    • 1 Videos
    • 1 Readings
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1 Videos

  • The Radius of Convergence of a Power Series

1 Readings

  • Lecture Slides

Lesson 4

    • 1 Videos
    • 1 Readings
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1 Videos

  • The Riemann Zeta Function And The Riemann Hypothesis

1 Readings

  • Lecture Slides

Lesson 5

    • 1 Videos
    • 1 Readings
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1 Videos

  • The Prime Number Theorem

1 Readings

  • Lecture Slides

Review

    • 1 Assignment
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1 Assignment

  • Module 6 Homework

Lesson 1

    • 1 Videos
    • 1 Readings
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1 Videos

  • Laurent Series

1 Readings

  • Lecture Slides

Lesson 2

    • 1 Videos
    • 1 Readings
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1 Videos

  • Isolated Singularities of Analytic Functions

1 Readings

  • Lecture Slides

Lesson 3

    • 1 Videos
    • 1 Readings
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1 Videos

  • The Residue Theorem

1 Readings

  • Lecture Slides

Lesson 4

    • 1 Videos
    • 1 Readings
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1 Videos

  • Finding Residues

1 Readings

  • Lecture Slides

Lesson 5

    • 1 Videos
    • 1 Readings
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1 Videos

  • Evaluating Integrals via the Residue Theorem

1 Readings

  • Lecture Slides

Lesson 6

    • 1 Videos
    • 1 Readings
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1 Videos

  • Bonus: Evaluating an Improper Integral via the Residue Theorem

1 Readings

  • Lecture Slides

Review

    • 1 Assignment
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1 Assignment

  • Module 7 Homework

Final Exam

    • 1 Assignment
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1 Assignment

  • Final Exam

Auto Summary

Dive into the fascinating world of complex analysis with this comprehensive course designed for Data Science and AI enthusiasts. Under the expert guidance of Coursera, you’ll embark on a journey through the intricate theory of complex functions of a complex variable. This course begins with the fundamental concepts of the complex plane, algebra, and geometry of complex numbers, and progresses through differentiation, integration, complex dynamics, and power series representation, culminating in the advanced study of Laurent series. The course is structured into modules, each featuring five engaging video lectures with embedded quizzes to reinforce your learning. You'll also tackle electronically graded homework assignments and peer assessments in selected modules, ensuring a thorough understanding of each topic. Expect to invest 6-12 hours per module to truly grasp the material, with assignments designed to challenge your thinking and application skills. With a duration of approximately 1620 minutes, this professional-level course offers flexible subscription options including Starter, Professional, and Paid plans, catering to different learning needs. Ideal for professionals and advanced learners in the field, this course promises to deepen your understanding of complex analysis and equip you with valuable skills for your career in Data Science and AI.

Dr. Petra Bonfert-Taylor
Instructor

Dr. Petra Bonfert-Taylor