Analysis of a Complex Kind
Wesleyan University via Coursera
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Overview
Class Central Tips
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. Don’t worry! We’ll help you remember facts from calculus in case you have forgotten. After this exploration we will be ready to meet the main players: analytic functions. These are functions that possess complex derivatives in lots of places, a fact which endows these functions with some of the most beautiful properties mathematics has to offer. We’ll explore these properties!
Who would want to differentiate without being able to undo it? Clearly we’ll have to learn about integration as well. But we are in the complex plane, so what are the objects we’ll integrate over? Curves! We’ll study these as well, and we’ll tie everything together via Cauchy’s beautiful and all encompassing integral theorem and formula.
Throughout this course we'll tell you about some of the major theorems in the field (even if we won't be able to go into depth about them) as well as some outstanding conjectures.
Syllabus
Week Two: The Mandelbrot set, Julia sets, a famous outstanding conjecture, history of complex numbers, sequences of complex numbers and convergence, complex functions.
Week Three: Complex differentiation and the Cauchy-Riemann equations.
Week Four: Conformal mappings, Möbius transformations and the Riemann mapping theorem.
Week Five: Complex integration, Cauchy-Goursat theorem, Cauchy integral formula, Liouville's Theorem, maximum principle, fundamental theorem of algebra.
Week Six: Power series representation of analytic functions, singularities, the Riemann zeta function, Riemann hypothesis, relation to prime numbers.
Taught by
Petra Bonfert-Taylor
Tags
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Reviews
4.7 rating, based on 7 reviews
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Anonymous completed this course.
It was a very informative class. Besides the analytics of complex numbers, we learned to communicate our reasoning and to plot graphs in the complex plane, which was rated by some of our classmates (peer grading). We also were shown how to plot Julia sets, extremely nice pictures.
Dr. Bonfert-Taylor explained all topics in an understandable and friendly way. -
Gaetano Pagani completed this course.
Really nice course. Materials, peer graded assignments and homeworks were well proposed.
Prof. Bonfert-Taylor carried out the lessons in clear and friendly way. Excellent.
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Anonymous completed this course.
Excellent!
I got to learn all about the Mandelbrot set, and so much more! -
Anonymous completed this course.
Really enjoyed it, well presented and excellent material. -
Nicolas Bernard completed this course.
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Bhuvan N completed this course, spending 3 hours a week on it and found the course difficulty to be medium.
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Francisco Javier Jiménez completed this course.