Visual Complex Analysis could almost have been titled Geometry with Complex Numbers, so much does it emphasize the visual and geometric. Needham makes extensive use of diagrams, which he gives as much prominence as the equations, and focuses on "those aspects of complex analysis for which some visual representation or interpretation is possible". Many topics are explored at length but without rigorous proofs: of Riemann's Mapping Theorem, for example, he writes "[though one of the proofs] is constructive in nature we have not yet found a way to present it in a manner consistent with the aims of this book".

Needham begins with the links between complex numbers and geometry, and a view of complex functions as transformations, focusing on Moebius transformations and inversions and ways of visualising them. Differentiation is approached using the idea of an "amplitwist", a transformation of infinitesimal vectors, and "visual differentiation" investigated for a range of different functions.

The longest chapter is devoted entirely to non-Euclidean geometry, though this is "starred" as not necessary for the rest of the book (as are sections of other chapters).

Winding numbers lend themselves to a topological perspective. Needham follows that with a visual approach to complex integration, in which he uses Cauchy's Theorem before proving it, and Cauchy's Formula, relating the integral of an analytic function around a simple loop with its values at interior points. A series of chapters then look at vector fields, flows, and harmonic functions, concluding with a lovely geometric perspective on Dirichlet's Problem, concerning heat flows in a metal disc. (Physical models and intuitions are used throughout, but not given nearly the same prominence as geometric ones.)

Needham's approach is, as mentioned, informal and not rigorous, and curiosity-driven: the quite extensive exercises at the end of each chapter are designed to grab the reader and make them want to solve them. Visual Complex Analysis could be used by itself, as an introduction to the subject for those of us — a majority, I believe — whose visual intuitions are stronger than our formal and symbolic ones, and who are perhaps driven more by curiosity than the need to solve engineering problems. It could also be used alongside more traditional complex analysis textbooks, perhaps to help when intuition or motivation fail.

July 2020

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