Seeing is believing

Visual Complex Analysis
March 13, 1998

Real analysis comes as a shock to most mathematics undergraduates. Accompanied by a hefty slug of abstract algebra, it enforces rigour and precision. But there is a price. The familiar comforting functions of everyday mathematics - polynomials, exponentials, trigonometric functions - seem somehow illicit and specialised. Above all, the demands of abstraction and the fear of lurking pathologies mean that geometrical intuition is mistrusted and undervalued.

Complex analysis is not like that. Even a basic course covers a suite of spectacular theorems of great elegance and power. Complex analytic methods have made an enormous impact in a wide range of applications across the whole of mathematics and physics, and are still absolutely central today.

Tristan Needham's approach to the subject is unusual but deeply rooted in the evolution of the subject at the hands of giants such as Riemann. He asserts that one can - should - learn complex analysis from a geometric, or visual, point of view. More "traditional" calculation-based methods with roots in real analysis are, where possible, sidelined in favour of geometric presentations, interpretations and proofs. Every statement of any importance is illustrated with a diagram (there is an average of one per page). Newton would have approved.

The book proceeds at a leisurely pace. There are 12 chapters, each around 50 pages long. The first third of the book is devoted to the complex plane and its (Euclidean) geometry, to complex functions regarded as transformations, and (a whole chapter) to Mobius maps. The complex derivative is introduced as an "amplitwist': a local ampli(fication) by f+(*), accompanied by a twist through arg f+(*). This leads to a discussion of conformality, to "visual differentiation" of standard functions and to analytic continuation expressed via the "rigidity" of a conformal transformation. The "differential" half of the book ends with a long chapter on non-Euclidean geometry.

Integration is introduced with a chapter on winding numbers and topology, including a delightful interpretation of Rouche's theorem (walk a dog round a tree; if the lead is always shorter than your distance from the tree, you both circle the tree the same number of times). The chapters on integration cover the usual topics from Cauchy's theorem to Laurent expansions, but with some unusual geometric interpretations. The book concludes with three chapters on vector fields and physics, with a number of applications to fluid flow, heat flow and electrostatics.

One of the most fascinating and successful aspects of the book is the way in which geometry pervades the exposition. The intimate link between complex analysis and Euclidean, projective and non-Euclidean geometry is exploited at every opportunity, with many examples, greatly enriching our view of both. (It is a pity that there is no discussion of Riemann surfaces.) The theory is explained in geometrical terms as a matter of course. Anybody who already knows some complex analysis is bound to find new visual insights into every aspect of what seems familiar territory.

Would we prescribe Needham for the novice? In favour are the clarity of the writing; the general approach, informal yet tightly controlled; the copious illustrations and invitations to computer experiment; the many links to other subjects, far more than mentioned above; the excellent and interesting exercises; and the historical insights that lard the text. There is a deliberate lack of rigour, which greatly improves the readability of the book.

Rigour and calculation do, however, have a part to play in a balanced view of any area of mathematics. The overriding geometrical emphasis has perhaps pushed the analysis too far into the background. This bias neglects valuable links to what students already know; for example, the derivative is introduced without any mention of limh ' 0(f(*+h)-f(*))/h.

The treatment of contour integrals is rather cursory in view of the large number of important applications arising from asymptotic analysis and transform methods (neither of which is mentioned). The "physical" sections do not attempt to derive the models for fluid flow or electrostatics; they are easier to appreciate if the reader has experience of continuum mechanics.

Visual Complex Analysis might have been called Complex Analysis, Geometry and Physics. It is a fascinating and refreshing look at a familiar subject. It is not the ideal introduction to complex analysis - that may never exist - and it may not be the best introduction to use on its own, but rather a mandatory adjunct to almost any other text. It is essential reading for anybody with any interest at all in this absorbing area of mathematics.

Sam Howison is a lecturer in mathematics, University of Oxford. James Lawry and Linda Cummings are postdoctoral researchers at the Centre for Industrial and Applied Mathematics, Oxford.

Visual Complex Analysis

Author - Tristan Needham
ISBN - 0 19 853447 7
Publisher - Oxford University Press
Price - £32.95
Pages - 592

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