Complex Analysis

I collect STEM Books that are amenable to Self-Teaching! This book is a serious addition for such efforts!

This book is helping me a lot in the mission of reconciliation with math after being taught for many years about how to use many aspects of the complex numbers framework (in physics and electrical engineering) by intelligent people that knew real analysis well but couldn't explain well the confusing aspects as they surfaced on and on as the topics were presented. Five hermitians for clarity of its content!

Having read other textbooks by Ian Stewart, i decided to buy this book on Complex Analysis to revisit a subject I had not learned properly as an undergraduate. The book covers a core 1 semester course in complex analysis at the undergraduate level and starts to tackle higher level topics at the end including analytic continuation and homology theory. The text is readable and the proofs are understandable and the problems are not approachable and so the book is usable for self study. That being said the book has typos everywhere. The typos include obvious symbol/variable errors and at times the language is misused so it the statements of theorems need to be amended to be legitimate. This is surprising for a 2nd edition.The book starts out by laying the historical framework in which complex numbers were first used and some philosophy of mathematics. As is standard, the book then goes through the algebra of complex numbers and the various representations of complex numbers, geometric as well as coordinate representations. The book uses topological ideas for many of the proofs later in the book so the relevant point set topology of the complex plane is developed with a focus on connectedness and path connectedness. The authors go through some basic real analysis concepts as they pertain to complex numbers like series and sequences to build up the framework to consider power series representations of functions. The authors then move on to complex differentiation and the Cauchy-Riemann equations. The authors start to illustrate the difference between differentiability of real and complex functions and taylor expansions. The authors then bring up key functions that are critical to understand complex function theory, namely exponentials and trigonometric functions. The extensions of these functions to the complex plane is discussed and the relationship between hyperbolic functions is analyzed. The authors discuss Euler's formula and one is asked to derive several of the well known trigonometric identities from using the Euler formula. The authors move on to integration and the relevance of path in complex analysis. The Fundamental Theorem of Contour Integration is proved. The authors then discuss the logarithm and implicitly multi-valued functions and winding numbers. This topic is treated relatively early compared to other texts but it is well described and one can follow without much difficulty. The author then moves on to one of the most important theorem's of complex analysis, namely Cauchy's Theorem. The authors prove this via the standard triangulation methods and then go on to discuss multiple versions of the Cauchy Theorem. This is a well written chapter that sets the stage for the development of tools on Residue theory that make complex analysis so powerful. The authors go on to discuss Homotopy and effectively developing the topological tools to tackle path deformation for path integration. This sort of topic make the level of the text a bit higher level than other first year complex analysis texts but are more readable that a text on topology as the applications are on material just learned. The authors then revisit power series representations of functions by discussing Taylor and then Laurent extensions. These are all well covered topics for which the text and the exercises correspond and are approachable. The author then moves on to residues which is where one sees the real applications of contour integration. This chapter is relatively weak, the examples are not comprehensive enough and the exercises dont correspond to the material fully. One needs to have supplementary reading to be able to answer all the exercises. The conformal transformation chapter is also weak and i feel one needs supplementary material to actually learn the topic.Overall Complex Analysis by Tall and Stewart is decent but riddled with typos and less than ideal in its final chapters. I definitely was able to learn quite a few things from the book and the exercises are for the most part reasonable in building an understanding. But it is a shame to see a textbook have so many mistakes and relative weakness in the later chapters as a second edition. I think this is a reasonable text but there are others which are better, namely Complex Analysis with Applications by Asmar and Grafakos.

Purchased this textbook for my complex analysis class, as recommended by the professor. Professor did not know how many typos were in the practice problems. You will come across infinite series that diverge on the first term due to TeX typos. You will find a problem "z in C : 0 <= z <= 1" (what does that even mean??). Tick marks placed in the wrong position for the chain rule. That's all that I have found and we are only at the halfway point in the semester.

Lol there are so many typos in this book. My friend and I would be unsure about homework problems and sometimes decide it was a typo instead of us being stupid, and we’d be right. Complex analysis is sick, but use a different book if you can.