Course Information
For lecture information, including instructor and TA contact information, please see
the Canvas site for this course.
You are also encouraged to use the the optional
Piazza Page for this
Course.
Catalog Description
Complex numbers and functions. Analytic functions, harmonic functions,
elementary conformal mappings. Complex integration. Power series. Cauchy's
theorem. Cauchy's formula. Residue theorem.
Prerequisites: MATH 20E or MATH
31CH, or consent of instructor.
Course Goals
The aim is to give an introduction to complex variable techniques by
covering Chapters 1-6 and parts of 7 of the book.
Course Textbooks
Complex Variables and Applications, 9th edition by Brown and Churchill.
An eBook is available as a purchasing option for this course. You can access this eBook by clicking the RedShelf tool within Canvas (once I get it up and running). If you opt-in to this eBook by clicking the Opt-in Now button your student account will be charged directly. You will also receive an email with the exact amount of this charge. Within the add/drop period you may also opt-out of this option if you decide you’d rather use an alternate format. New hardback: $184.00 while 180-day ebook: $50.00
Book description from the Amazon Website: "Complex Variables and Applications, 9e will serve, just as the earlier editions did, as a textbook for an introductory course in the theory and application of functions of a complex variable. This new edition preserves the basic content and style of the earlier editions. The text is designed to develop the theory that is prominent in applications of the subject. You will find a special emphasis given to the application of residues and conformal mappings. To accommodate the different calculus backgrounds of students, footnotes are given with references to other texts that contain proofs and discussions of the more delicate results in advanced calculus. Improvements in the text include extended explanations of theorems, greater detail in arguments, and the separation of topics into their own sections."
Course Prologue: This course begins the study of complex functions of a complex variable which are complex differentiable, i.e. the analogue of Math 20A with the real numbers replaced by the complex numbers. In writing the complex difference quotients, one necessarily uses the division of one complex number by another. It turns out that this complex notion of differentiability is very restrictive. [We refer to complex differentiable functions as being analytic or holomorphic.] In fact, the existence of complex derivatives is so restrictive that it turns out that all analytic functions are infinitely differentiable. One of the key points that makes this theory so useful is that many of the basic functions you have seen in real variable calculus (like exp(x), cos(x), sin(x), ln(x), etc.) have a unique extension to an analytic function on the complex plane. Understanding these analytic extensions greatly enhances our understanding of the basic real variable functions listed above. [In fact, we will see that all of the functions listed above are intimately related to one another.] Besides introducing and studying many concrete analytic functions, the other key goal of 120A is to develop general properties which are common to all analytic functions.