Analysis of a Complex Kind
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Complex analysis is the study of functions that live in the complex plane, i.e. functions that have complex arguments and complex outputs. In order to study the behavior of such functions we’ll need to first understand the basic objects involved, namely the complex numbers. We’ll begin with some history: When and why were complex numbers invented? Was it the need for a solution of the equation x^2 = -1 that brought the field of complex analysis into being, or were there other reasons? Once we’ve answered these questions we’ll devote some time to learn about basic properties of complex numbers that will make it possible for us to use them in more advanced settings later on. We will learn how to do basic algebra with these numbers, how they behave in limiting processes, etc. These facts enable us to begin the study of complex functions, and at this point we can already understand the basics about the construction of the Mandelbrot set and Julia sets (if you have never heard of
these that’s quite alright, but do look at
http://en.wikipedia.org/wiki/Mandelbrot_set for example to see some
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.
Week One: Introduction to complex numbers, their geometry and algebra, working with complex numbers.
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.
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Disclaimer: the review is more or less subjective. I have absolutely no intention to make it objective. You have been informed.
这门课马上就要结课了，先谈谈这个讲师 Dr. Petra Bonfert-Taylor，德裔名字不好记，讲话语速适中，发音极为清楚，我不开字幕基本就能一遍听懂。这个讲师不太在讨论版上回复学生（我只看到过一次），TA 也不太能看到，可能是老师比较忙也可以理解，但是老师对于一些学生的诉求会通过邮件和我们交流，所以不是完全放任不管的。
课程的内容是 intro 性质的复分析。课程的时长只有6周，限定了课程的内容和深度。所以这门课的主要还是让你对复函数有一个直观的了解，对像我这种之前没接触过这方面内容的学生来说作为 first exposure 还是很适合的。课程的 syllabus 我就不多说了，课程介绍里有，虽然深度不够，不过讲师将的很清楚，对我以后更深入的学习肯定有帮助。
这门课作业的形式是每周10到左右的测试题，外加每两周一个 peer-graded assignments ，里面有几个让你找某函数在不同 domain 里的 image 的题目很有启发性，循循善诱让你自己找到函数在整个 C 上的 image。另外前面忘记说了 intro to mathematical thinking 和 uw 的 PL 也有很好的 peer-graded。