Enhanced descriptions from Syndetics:

This radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Aimed at undergraduate students in mathematics, physics, and engineering, the book's intuitive explanations, lack of advanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is the book's use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicated with the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to a classical topic, this work will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields.

Table of contents provided by Syndetics

• 1 Geometry and Complex Arithmetic (p. 1)
• I Introduction (p. 1)
• II Euler's Formula (p. 10)
• III Some Applications (p. 14)
• IV Transformations and Euclidean Geometry* (p. 30)
• V Exercises (p. 45)
• 2 Complex Functions as Transformations (p. 55)
• I Introduction (p. 55)
• II Polynomials (p. 57)
• III Power Series (p. 64)
• IV The Exponential Function (p. 79)
• V Cosine and Sine (p. 84)
• VI Multifunctions (p. 90)
• VII The Logarithm Function (p. 98)
• VIII Averaging over Circles* (p. 102)
• IX Exercises (p. 111)
• 3 Mobius Transformations and Inversion (p. 122)
• I Introduction (p. 122)
• II Inversion (p. 124)
• III Three Illustrative Applications of Inversion (p. 136)
• IV The Riemann Sphere (p. 139)
• V Mobius Transformations: Basic Results (p. 148)
• VI Mobius Transformations as Matrices* (p. 156)
• VII Visualization and Classification* (p. 162)
• VIII Decomposition into 2 or 4 Reflections* (p. 172)
• IX Automorphisms of the Unit Disc* (p. 176)
• X Exercises (p. 181)
• 4 Differentiation: The Amplitwist Concept (p. 189)
• I Introduction (p. 189)
• II A Puzzling Phenomenon (p. 189)
• III Local Description of Mappings in the Plane (p. 191)
• IV The Complex Derivative as Amplitwist (p. 194)
• V Some Simple Examples (p. 199)
• VI Conformal = Analytic (p. 200)
• VII Critical Points (p. 204)
• VIII The Cauchy-Riemann Equations (p. 207)
• IX Exercises (p. 211)
• 5 Further Geometry of Differentiation (p. 216)
• I Cauchy-Riemann Revealed (p. 216)
• II An Intimation of Rigidity (p. 219)
• III Visual Differentiation of log(z) (p. 222)
• IV Rules of Differentiation (p. 223)
• V Polynomials, Power Series, and Rational Functions (p. 226)
• VI Visual Differentiation of the Power Function (p. 229)
• VII Visual Differentiation of exp(z) (p. 231)
• VIII Geometric Solution of E' = E (p. 232)
• IX An Application of Higher Derivatives: Curvature* (p. 234)
• X Celestial Mechanics* (p. 241)
• XI Analytic Continuation* (p. 247)
• XII Exercises (p. 258)
• 6 Non-Euclidean Geometry* (p. 267)
• I Introduction
• II Spherical Geometry (p. 278)
• III Hyperbolic Geometry (p. 293)
• IV Exercises (p. 328)
• 7 Winding Numbers and Topology (p. 338)
• I Winding Number (p. 338)
• II Hopf's Degree Theorem (p. 341)
• III Polynomials and the Argument Principle (p. 344)
• IV A Topological Argument Principle* (p. 346)
• V Rouche's Theorem (p. 353)
• VI Maxima and Minima (p. 355)
• VII The Schwarz-Pick Lemma* (p. 357)
• VIII The Generalized Argument Principle (p. 363)
• IX Exercises (p. 369)
• 8 Complex Integration: Cauchy's Theorem (p. 377)
• I Introduction (p. 377)
• II The Real Integral (p. 378)
• III The Complex Integral (p. 383)
• IV Complex Inversion (p. 388)
• V Conjugation (p. 392)
• VI Power Functions (p. 395)
• VII The Exponential Mapping (p. 401)
• VIII The Fundamental Theorem (p. 402)
• IX Parametric Evaluation (p. 409)
• X Cauchy's Theorem (p. 410)
• XI The General Cauchy Theorem (p. 414)
• XII The General Formula of Contour Integration (p. 418)
• XIII Exercises (p. 420)
• 9 Cauchy's Formula and Its Applications (p. 427)
• I Cauchy's Formula (p. 427)
• II Infinite Differentiability and Taylor Series (p. 431)
• III Calculus of Residues (p. 434)
• IV Annular Laurent Series (p. 442)
• V Exercises (p. 446)
• 10 Vector Fields: Physics and Topology (p. 450)
• I Vector Fields (p. 450)
• II Winding Numbers and Vector Fields* (p. 456)
• III Flows on Closed Surfaces* (p. 462)
• IV Exercises (p. 468)
• 11 Vector Fields and Complex Integration (p. 472)
• I Flux and Work (p. 472)
• II Complex Integration in Terms of Vector Fields (p. 481)
• III The Complex Potential (p. 494)
• IV Exercises (p. 505)
• 12 Flows and Harmonic Functions (p. 508)
• I Harmonic Duals (p. 508)
• II Conformal Invariance (p. 513)
• III A Powerful Computational Tool (p. 517)
• IV The Complex Curvature Revisited* (p. 520)
• V Flow Around an Obstacle (p. 527)
• VI The Physics of Riemann's Mapping Theorem (p. 540)
• VII Dirichlet's Problem (p. 554)
• VIII Exercises (p. 570)
• References (p. 573)
• Index (p. 579)

Reviews provided by Syndetics

CHOICE Review

One variable calculus pedagogy centers around the representation of functions by graphs and the concomitant graphical interpretation of fundamental concepts: the derivative as tangential slope, the integral as area. Following the same strategy with complex functions of a complex variable requires contemplating "graphs" with the form not of curves in a plane but surfaces extended through four dimensional space. With visual aids so difficult to visualize, it might seem better to elucidate the analytical ideas directly, and many complex analysis texts accordingly proceed almost exclusively along formal lines with just the occasional diagram. Needham appeals to various alternative pictorial representations to offer perhaps the first thoroughgoingly visual exposition of complex analysis. Many components of his approach seem familiar, but the novelty consists of having them all in one place. He emphasizes general principles and says little concerning the geometry of special functions, an important aspect of complex analysis well suited to such an approach. He includes an unusually substantial chapter on non-Euclidean geometry. For all the appeal to pictures, the book's notation can seem surprisingly heavy at times. From the start, Needham intentionally forswears rigor. This unfortunately seems to mean more than simply omitting certain proofs or details thereof, and too many sentences do not mean exactly what they say. A special topic in complex analysis, the theory of elliptic (doubly-periodic meromorphic) functions extends the more familiar theory of trigonometric (singly-periodic meromorphic) functions. Any development of elliptic functions generalizes some corresponding approach to the theory of trigonometric functions. The most popular approach to elliptic functions originates with Weierstrass and Eisenstein, but its trigonometric analogue remains underexposed. Walker starts by carefully redoing trigonometry along the lines that best prepare readers to approach elliptic functions the right way. As the source for this approach to elliptic functions, he cites A. Weil's Elliptic Functions according to Eisenstein and Kronecker (1976), but this book reaches out to a wide audience. Where most books on this topic are on just a single focus, Walker develops applications in physics, geometry and number theory. Thus, one finds the Jacobian functions and elliptic integrals familiar to physicists side-by-side with the theta functions and modular functions familiar to number theorists. Needham's book should attract upper-division undergraduate students, but one should expect other authors to adopt his approach and improve on the particulars. Walker's book, though very nicely done, has many fine competitors and will find readers ranging from upper-division undergraduates through faculty. D. V. Feldman; University of New Hampshire

Author notes provided by Syndetics

Tristan Needham is Associate Professor of Mathematics at the University of San Francisco. For part of the work in this book, he was presented with the Carl B. Allendoerfer Award by the Mathematical Association of America.