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Visual complex analysis (25th anniversary ed.)
Needham T., Penrose R., OXFORD UNIVERISTY PRESS, Oxford, UK, 2023. 675 pp. Type: Book (9780192868923)
Date Reviewed: Apr 1 2024

Complex analysis has always enhanced the learning of mathematics. The application of complex numbers in other fields has also been widely studied, independent of theoretical studies in mathematics and related subjects. This book looks at the geometrical aspects of complex analysis. Its 12 chapters comprise concepts of visual complex analysis. Chapter 1 introduces Bombelli’s “wild thought” about particles and power series, the three reflections theorem, and applications to trigonometry, geometry, and so on.

Chapter 2 conceptualizes cubics, Cassinian curves, power series, and fractional powers, and presents considerations regarding exponential and inverse exponential functions. Chapter 3 discusses the Riemann sphere; Möbius transformations, including basic results as well as matrices; the preservation of circles, angles, and symmetry; elliptic, hyperbolic, and parabolic cases; and the symmetry principle. Chapter 4 highlights a puzzling phenomenon, the amplitwist concept, conformality, and Cauchy-Riemann equations. Chapter 5 explains rules of differentiation such as composition, inverse functions, addition, and multiplication. It also theorizes complex curvature, celestial mechanics, and analytics like rigidity, uniqueness, identities, and reflections.

Chapter 6 illustrates two types of geometries: spherical geometry containing spatial rotations, and hyperbolic geometry containing hyperbolic lines and reflections. Chapter 7 details winding numbers, loops, a topological argument principle, Rouché’s theorem, and the Schwarz-Pick lemma. Chapter 8 covers real and complex integrals, complex inversion, and conjugation, and provides explanations of Cauchy’s theorem. Chapter 9 goes on to discuss two explanations of Cauchy’s formula and its applications to real integrals and summations of series.

Chapter 10 discusses physical vector fields, flows and force fields, and sources and sinks, and describes the Poincaré–Hopf theorem along with its formulation and explanation. Chapter 11 goes on to describe local flux and work, divergence- and curl-free vector fields, Cauchy’s theorem in terms of area and winding number as flux, multipoles, and the potential function. Chapter 12 presents dual flows, conformal invariance, harmonic equipotentials, complex curvature, interior and exterior mappings, and Green’s formula.

The exercises following each chapter allow readers to brush up on their problem-solving skills. This 25th anniversary edition claims to facilitate graduate students and engineers in their research. It would also be an interesting read for academics and researchers working in the area of geometrical complex analysis. The authors leave room for future developments in the proposed methods, algorithms, and applications.

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Reviewer:  Lalit Saxena Review #: CR147735
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