Lesson 3: Solving Quadratic Equations in z; Topology of C
Preview
Unit Preview
In this unit we explore complex functions. We'll quickly discover that visualizing them requires 4 dimensions. To circumvent this we'll come to think of complex functions as maps from one region of a plane to another region of another plane. We'll then move on to studying the canonical mappings -- linear functions, power functions, etc. -- and conclude with a discussion of the extended complex plane (which formally adds the point "infinity" to C).
Lesson Preview
Today we’ll continue solving complex equations and discussing their geometry. This will lead us into a more general discussion of the topology of C.
Module I. We’ll learn how to solve quadratic equations with complex coefficients, making use of Theorem 2.2.
Module II. We’ll discuss the basic topology of C, including circles, annuli, and open sets.
Review
Learn
The lesson notes below contain a learning plan with three stages -- Learn, Reflect, and Practice -- and guidance for what to do within each stage. Some tips for you as you work through this resource, and those that it points to:
I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.
Lesson Notes
Lesson 3.pdfReflect
If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:
Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.
Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.
If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:
Please summarize the main mathematical takeaways from the lesson notes.
What was the most interesting part of what you learned, and why?
What, if anything, do you still find confusing?
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood.
Lesson 3 PP.pdf