## Introduction to Complex Analysis

## Course Description

Complex analysis is one of the most beautiful and useful disciplines of mathematics, with applications in engineering, physics, and astronomy, as well as other branches of mathematics. This introductory course reviews the basic algebra and geometry of complex numbers; develops the theory of complex differential and integral calculus; and concludes by discussing a number of elegant theorems, including many–the fundamental theorem of algebra is one example–that are consequences of Cauchy’s integral formula. Other topics include De Moivre’s theorem, Euler’s formula, Riemann surfaces, Cauchy-Riemann equations, harmonic functions, residues, and meromorphic functions. The course should appeal to those whose work involves the application of mathematics to engineering problems as well as individuals who are interested in how complex analysis helps explain the structure and behavior of the more familiar real number system and real-variable calculus.

## Prerequisites

Basic calculus or familiarity with differentiation and integration of real-valued functions.

## Details

MATH X 451.37 – 268651 Introduction to Complex Analysis

Fall 2016

Time 7:00PM to 10:00PM

Dates Tuesdays, Sep 20, 2016 to Dec 06, 2016

Contact Hours 33.00

Location: UCLA, Math Sciences Building

Standard credit (3.9 units) $453.00

Instructor: Michael Miller

Register Now at UCLA

For many who will register, this certainly won’t be their first course with Dr. Miller — yes, he’s that good! But for the newcomers, I’ve written some thoughts and tips to help them more easily and quickly settle in and adjust:

Dr. Michael Miller Math Class Hints and Tips | UCLA Extension

I often recommend people to join in Mike’s classes and more often hear the refrain: “I’ve been away from math too long”, or “I don’t have the prerequisites to even begin to think about taking that course.” For people in those categories, you’re in luck! If you’ve even had a soupcon of calculus, you’ll be able to keep up here. In fact, it was a similar class exactly a decade ago by Mike Miller that got me back into mathematics. (Happy 10th math anniversary to me!)

I look forward to seeing everyone in the Fall!

#### Update 9/1/16

### Textbook

Dr. Miller is back from summer vacation and emailed me this morning to say that he’s chosen the textbook for the class. We’ll be using *Complex Analysis with Applications* by Richard A. Silverman. [1]

(Note that there’s another introductory complex analysis textbook from Silverman that’s offered through Dover, so be sure to choose the correct one.)

As always in Dr. Miller’s classes, the text is just *recommended* (read: not required) and in-class notes are more than adequate. To quote him directly, “We will be using as a basic guide, but, as always, supplemented by additional material and alternate ways of looking at things.”

The bonus surprise of his email: He’s doing two quarters of Complex Analysis! So we’ll be doing both the Fall and Winter Quarters to really get some depth in the subject!

### Alternate textbooks

If you’re like me, you’ll probably take a look at some of the other common (and some more advanced) textbooks in the area. Since I’ve already compiled a list, I’ll share it:

#### Undergraduate

*Complex Analysis*by Joseph Bak and Donald J. Newman [2]*Complex Analysis*by Theodore Gamelin [3]*Complex Variables and Applications*by James Brown and Ruel Churchill [4]*Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics*by Edward Saff and Arthur D. Snider (Pearson, 2014, 3rd edition) [5]

#### More advanced

*Complex Analysis*by Lars Ahlfors [6]*Complex Analysis*by Serge Lang [7]*Functions of One Complex Variable*(Graduate Texts in Mathematics by John B. Conway (Springer, 1978) [8]*Complex Analysis*(Princeton Lectures in Analysis, No. 2) by Elias M. Stein and Rami Shakarchi (Princeton University Press, 2003) [9]

### References

*Complex Analysis with Applications*, 1st ed. Dover Publications, Inc., 2010, pp. 304–304 [Online]. Available: http://amzn.to/2c7KaQy

*Complex Analysis*, 3rd ed. Springer, 2010, pp. 328–328 [Online]. Available: http://amzn.to/2bLPW89

*Complex Analysis*. Springer, 2003, pp. 478–478 [Online]. Available: http://amzn.to/2bGNQct

*Complex Variables and Applications*, 8th ed. McGraw-Hill, 2008, pp. 468–468 [Online]. Available: http://amzn.to/2bLQWcu

*Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics*, 3rd ed. Pearson, 2003, pp. 563–563 [Online]. Available: http://amzn.to/2f3Nyj6

*Complex Analysis*, 3rd ed. McGraw-Hill, 1979, pp. 336–336 [Online]. Available: http://amzn.to/2bMXrxm

*Complex Analysis*, 4th ed. Springer, 2003, pp. 489–489 [Online]. Available: http://amzn.to/2c7OaR0

*Functions of One Complex Variable*, 2nd ed. Springer, 1978, pp. 330–330 [Online]. Available: http://amzn.to/2cggbF1

*Complex Analysis*. Princeton University Press, 2003, pp. 400–400 [Online]. Available: http://amzn.to/2bGOG9c