The book starts off with some standard history and a look at some remarkable, but well-known, ideas such as Euler's formula and generally the strange way complex numbers make trigonometric relationships so easy. From here we move on from algebra to functions - power series, the standard functions and the strange idea of multifunctions.

Next there is an in-depth look at the Mobius transformation. This is a topic that I have previously puzzled over - why are these rational functions so important. Having read the section now I know, but I'm still thinking about it.

Perhaps the most rewarding idea in the book is that the complex derivative has to be an "amplitwist", Needham's own word. In a standard course you restrict your attention to functions that have derivatives which are independent of how you aproach a point. This is good way to meet the complex derivative, but to discover that this is that same as the derivative being everywhere, a scaling and a rotation - an amplitwist - is a really interesting idea and worth reading the book for.

Unfortunately from here we go over the usual ground, conformal is the same as analytic, critical points, and the Cauchy-Riemann conditions, without anything that is as new or as exciting. But this is not a new theory that admits much inovation.

The section on the geometry of differentiation will help some and confuse others. If you hare happy to just work with the mechanisms of real calculus applied to analytic functions you will wonder why so much energy is being expended on exploring things in depth.

From here we have a large section on non-Euclidean geometry which is interesting, but if you are wanting to know about the topic then the best approach is differential geometry - the complex approach is a sideshow. From geometry we move to topology and the winding number and the argument principle. This is usually not covered in applied courses.

At last we reach integration and the explanation of what complex integration is really all about is the second gem in the book. From this idea we follow the conventional path (pun intended)  to end up with Cauchy's theorem. This is generally considered to be the high point of complex analysis as far as getting results go - how many integrals that seem impossible are easy once you know the calculus of residuals.

The last part of the book is a collection of applications - vector fields, flows and harmonic functions. To me these are less intesting because they are essentially limited to 2D, but they are remarkable and suprising.

This is a good book but not as good as the author's latest volume - don't let this put you off if you are interested in complex functions and how they generalize the familiar real functions. To me it is still surprising that extending the field R to C brings so much to the theory of R. For example, finding the integral of a real function by evaluating a path integral in the complex plane that involves on considerations of the "singularities" in the plane - it seems, at first to have nothing to do with the real function and yet it does.

The big problem with complex analysis and this sort of book is that unlike differential geometry it doesn't have lots of big ideas. Instead it has lots of big applications and this makes the storyline much more convoluted and branched.

Is this a good book - yes of course it is, but avoid the kindle version which is great value but low in quality - also see the note at the start of the review.

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