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This unit begins with a study of the complex logarithm and complex trigonometric functions. We then build up the theory of integration of complex functions. Here we'll meet another set of remarkable results, including the Cauchy-Goursat Theorem and Cauchy's integral formulas. Briefly, these use information about complex functions' values on a sufficiently nice closed curve to determine the values of those functions inside those curves. We'll end the unit again discussing the applications of what we will have learned to fluid dynamics.
Lesson Preview
There is one complex function that we have thus far avoided: the complex logarithm. Today we’ll discuss the complex logarithm.
Module I. First we’ll have to discuss some additional properties of the complex exponential that we’ll need to define the complex logarithm.
Module II. Then we’ll introduce the complex logarithm. But we’ll see that its multivaluedness will necessitate a branch cut.
Module III. Finally, we’ll see that after defining an appropriate branch of the complex logarithm it will indeed turn out to be the inverse function of the complex exponential.
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 13.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 13 PP.pdfPage updated
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