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Visual Complex Analysis

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Visual Complex Analysis

A nice example of the approach comes from one of the many fantastic exercises (Ch. 2, #22). It explores the function Pn(z) =(1 + z/n)n geometrically. The reader is asked to break down this function into a translation, a contraction, and a power mapping, and then explore the effect of these transformations on circular arcs and rays. For large n, of course, we see behavior consistent with what we should expect from the exponential map. This limit is discussed in all analysis texts, but is rarely considered as a geometric statement. The book is filled with uncommonly insightful geometric interpretations like this.

Historical and physical context play important roles in the book and are integrated into the narrative in very natural way. The author notes that many ideas in complex analysis developed from physical intuition and works to impart that intuition on the reader. This is well exemplified in his development of vector fields and harmonic flows in the context of complex integration. There are also plenty of interesting topics included as optional sections throughout the text, including an extended discussion of Möbius transformations, non-euclidean geometry, curvature, analytic continuation, and celestial mechanics.

This book develops a theory of complex analysis based on geometric ideas instead of on analytic concepts such as power series or the Cauchy-Riemann equations. Compared to many current math textbooks, this one is very concrete, and spends most of its time working with examples rather than developing and applying general theorems.

The author summarizes his intentions by saying (p. 222), "The basic philosophy of this book is that while it often takes more imagination and effort to find a picture than to do a calculation, the picture will always reward you by bringing you nearer to the Truth." The book's key concept is the "amplitwist" (p. 194), a portmanteau of amplify and twist. The amplitwist is a geometric view of the complex derivative: an analytic function will map an infinitesimal vector at a point to another infinitesimal vector that is grown or shrunk and rotated compared to the original.

I also felt that this is more of a geometry book than a complex analysis book. The geometry is relevant and interesting, but it's not what I call complex analysis. As (perhaps) a confession of bias, I'll state that my all-time favorite complex analysis book is E. C. Titchmarsh's Theory of Functions, followed not too far behind by the similar but more modern Complex Analysis of Bak and Newman (Springer, 1999).

3B1B has recommended the book Visual Complex Analysis by Needham. I read on Goodreads that this book is super good except it normally requires a traditional textbook on complex analysis as a formal introduction. These two books will be complementary to each other.

I'm comfortable with complex functions with engineering PhD (control theory to be specific) background. I'd like to learn about complex analysis (mainly because if not I "have no heart", per Grant's comments in the Riemann hypothesis video).

The study of complex analysis is important for students in engineering and thephysical sciences and is a central subject in mathematics. In addition to beingmathematically elegant, complex analysis provides powerful tools for solvingproblems that are either very difficult or virtually impossible to solve in anyother way.

This book is an interactive introduction to the theory and applications of complex functionsfrom a visual point of view. However, it does not cover all the topics of astandard course. In fact, it is a collection of selected topics and interactive appletsthat can be used as a supplementary learning resource by anyone interested in learningthis fascinating branch of mathematics.

Some of the topics covered here are basic arithmetic of complex numbers, complex functions,Riemann surfaces, limits, derivatives, domain coloring, analytic landscapes andsome applications of conformal


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