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An Introduction to Complex Analysis

Ravi P. Agarwal, Kanishka Perera and Sandra Pinelas
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
P. N. Ruane
, on

The structure and presentation of this introduction to complex analysis is most unusual. It consists of 50 ‘class-tested lectures’ in which the subject matter has been organized in the form of theorems, proofs and examples. Most of the lectures are covered in about four or five pages, and each one is followed by graded exercises that go from the routine to the richly informative. Solutions and hints are provided for nearly all of these, which means that the book is highly suited for self-tuition purposes.

The first 25 lectures (I nearly said ‘chapters’) cover the familiar early aspects of complex analysis. These include the algebra of complex numbers, basic topological ideas, the Cauchy-Riemann equations, analytic functions, mappings and contour integrals. Then there is discussion of Cauchy’s integral formulae, analytic functions in terms of power series and Laurent series. The maximum modulus principle is dealt with in the 20th lecture, and Cauchy’s residue theorem is explained in five of the eight pages devoted to lecture 31.

Lectures32 to 50 cover a range of topics not usually associated with an introductory course: harmonic functions, Weierstrass factorisation, the Mittag-Leffler theorem, Schwarz-Christoffel transformations, and the Riemann zeta function, for instance. The course concludes with lectures on Bieberbach’s conjecture, Riemann surfaces and topological aspects of the complex plane associated with dynamical systems. All this is neatly rounded off with a brief historical overview of complex numbers.

Throughout the course, the material is developed with a level of clarity and accuracy that is entirely consistent with the claim it has evolved from 40 years of combined teaching experience. Indeed, I worked through the text, and all the exercises in the first half of the book, and could find very few typos or errors. This level of accuracy, together with the amenable self-tuition structure of the book, means that it is also suited to the needs of non-specialists, such as those concerned with the applied sciences. There are, however, very few examples or exercises to illustrate how ideas of complex analysis may be applied.

Peter Ruane’s career was centred upon primary and secondary mathematics education.

-Preface. -Complex Numbers I. - Complex Numbers II. - Complex Numbers III. - Set Theory in the Complex Plane. - Complex Functions. -Analytic Functions I. - Analytic Functions II. - Elementary Functions I. - Elementary Functions II. - Mappings by Functions I. - Mappings by Functions II. - Curves, Contours, and Simply Connected Domains. - Complex Integration. -Independence of Path. - Cauchy-Goursat Theorem. - Deformation Theorem. - Cauchy’s Integral Formula. - Cauchy’s Integral Formula for Derivatives. - The Fundamental Theorem of Algebra. - Maximum Modulus Principle. - Sequences and Series of Numbers. - Sequences and Series of Functions. - Power Series. -Taylor’s Series. -Laurent’s Series. - Zeros of Analytic Functions. -Analytic Continuation. -Symmetry and Reflection. -Singularities and Poles I. -Singularities and Poles II. - Cauchy’s Residue Theorem. - Evaluation of Real Integrals by Contour Integration I. - Evaluation of Real Integrals by Contour Integration II. -Indented Contour Integrals. -Contour Integrals Involving Multi-valued Functions. -Summation of Series. -Argument Principle and Rouch´e and Hurwitz Theorems. -Behavior of Analytic Mappings. - Conformal Mappings. -Harmonic Functions. -The Schwarz-Christoffel Transformation. -Infinite Products. - Weierstrass’s Factorization Theorem. - Mittag-Leffler Theorem. -Periodic Functions. -The Riemann Zeta Function. -Bieberbach’s Conjecture. -Riemann Surfaces. -Julia and Mandelbrot Sets. -History of Complex Numbers. -References for Further Reading. -Index