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This unit explores the limits and derivatives of complex functions. This is the unit where we really start to appreciate the richness of complex analysis. We'll discover in this unit that complex differentiability implies a very specific relationship between the real and imaginary parts of a complex function; this relationship is the Cauchy-Riemann equations. We'll then discuss the ramifications of these equations, and their application to fluid dynamics problems.
Lesson Preview
We’re now moving toward discussing complex differentiation. But before doing so we need to talk about complex limits.
Module I. First, we’ll define what we mean by a complex limit. We’ll see that we can reduce the definition (and calculations) to that of limits of real-valued functions of two real variables.
Module II. Then, we’ll prove two results that hold in the complex case that will remind you of similar results from real analysis.
Review
Review
Learn
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 Notes
Lesson 8.pdfReflect
Reflect
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
Practice
Work through the practice problems suggested below to see how much of this lesson you've understood.
Lesson 8 PP.pdfPage updated
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