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Functions of a Complex Variable

Hemant Kumar Pathak, Ravi Agarwal, and Yeol Je Cho
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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The book under review is intended as a text for an advanced undergraduate or early graduate course in the theory of functions of a complex variable. There are, of course, any number of other books, old and new, on this subject (see, for example, our reviews of the books by Muir, Bak and Newman, Sasane and Sasane, Howie, and Ahlfors) and, in broad outline, this book covers much the same topics that others do: the basic definitions and topology of the field of complex numbers, analytic and harmonic functions, the Cauchy-Riemann equations, power series, conformal mappings, complex integration, singularities and residues, uniform convergence, analytic continuation, and a peek at Riemann surfaces. In the manner of presentation, however, this book has some features that distinguish it from competing texts. Whether these features are improvements or not, however, will largely depend on the tastes and goals of individual instructors.

For one thing, this book (like many other books published by CRC Press, it seems) is a big one, weighing in at 700+ pages. This makes the book more than twice as long as all of the ones listed above. The length of this text is largely based on the fact that the authors explore matters in much greater detail than one typically sees in a text, or, for that matter, in a classroom. For example:

  • The geometric significance of complex numbers is discussed in much more detail than is customary, including the introduction of topics (such as the orthocenter and circumcenter of a triangle) that an average undergraduate encounters, if at all, in a course in advanced Euclidean geometry
  • While many undergraduate complex analysis texts discuss how the residue calculus can be used to evaluate real integrals, and give several examples, the authors here spend about 90 pages on the topic
  • Möbius transformations are discussed in considerable detail (about 60 pages), including topics that are rarely mentioned in beginning courses, including classification theorems
  • An entire chapter, also about 60 pages long, is devoted to the study of other particular transformations and their mapping properties
  • A fair amount of space is spent discussing specific techniques of finding an analytic function whose real part is a given harmonic function, including methods, such as those of L. M. Milne-Thomson, that do not appear to be common currency in most undergraduate textbooks

Features like this certainly enhance the value of this book as a reference for faculty members looking for interesting examples or other information for lectures. On the other hand, they also slow the pace of exposition: at best, they require the instructor to make hard choices about what to cover and what to omit; at worst, they make it difficult to ever get to certain material. And of course students studying this book on their own, without faculty assistance, may get hopelessly bogged down and never reach the really important stuff.

In this book, for example, we don’t even get to the simplest form of Cauchy’s integral theorem until page 341; Liouville’s theorem is proved on page 383; the Fundamental Theorem of Algebra, typically proved right after Liouville’s theorem, is further delayed until page 476. Contrast this with Bak and Newman’s book, where Liouville’s theorem is proved on page 62. (As a side note, I should also record here one mild criticism connected with the Fundamental Theorem: the authors state in a footnote that all proofs are “purely analytic”, a statement that I believe can be misinterpreted, especially since proofs exist that, although using some analysis, also use Sylow and Galois theory from algebra. See, for example, Dummit and Foote’s Abstract Algebra.)

Apart from its length, there is a lot to recommend about this book. In connection with topic coverage, I liked the fact that the authors do the usual topological material about the complex plane in the more general context of metric spaces. (I don’t think this is any more difficult, and the added generality is useful.) I also liked the fact that a number of different versions of Cauchy’s theorem (including the homotopic version) are given.

Other pedagogical benefits of the text include a writing style that is generally quite clear, numerous worked-out examples (especially concerning the mapping properties of functions), and a number of end-of-chapter historical discussions. And, although I did encounter several typos (e.g., the definition of a Mobius transformation is garbled, Milne-Thomson’s name is sometimes not hyphenated, and a lower case “x” in the definition of a metric should be upper case) these were non-substantive in nature; I didn’t notice any genuine mathematical mistakes.

In addition, there are a lot of exercises (some calling for computation, others for proofs) covering a reasonable span of difficulty that appear throughout the text, though not at the end of every section; each chapter, though, has at least one, and frequently more, sections of exercises. For some reason, the exercise sections are not listed in the Table of Contents, so an instructor looking to assign some will have to flip through the book until the next exercise set appears. There is also a very short appendix (less than five pages long) giving solutions to a small subset of the exercises.

Speaking of solutions: the publisher’s website states that a “solutions manual and figure slides are available upon qualifying course adoption.” This is, at least as of a few days ago, incorrect: when I emailed a publisher’s representative inquiring about a solutions manual, I was advised that one was not yet available. This is not the first time that this has happened, and CRC Press should perhaps be reminded that the words “forthcoming” and “available” are not synonyms; use of the latter term to mean the former may prove misleading to an instructor making adoption decisions.

Despite its length, there are some topics that the authors have chosen not to cover. This is not, for example, an applications-oriented text, and thus many applications, either to other areas of mathematics or other disciplines, are not discussed at all. Once again, contrast the book by Bak and Newman, which covers a number of applications to mathematics (including a proof of the Prime Number theorem) and also mentions physics-related applications (such as the heat equation). Although Möbius transformations are discussed and it is mentioned that they form a non-abelian group, the connection with matrices is not discussed.

In addition, the gamma function and the Riemann zeta function are mentioned only briefly, with no hint of their applications. (The Riemann hypothesis is also not mentioned.) Picard’s theorem is stated but not proved; this is usually the case for undergraduate textbooks, but an instructor thinking about using this as a graduate text might object to this, or to the exclusion of other topics (elliptic functions, infinite products, functional-analytic approach to complex variables) that are often associated with graduate-level courses. Instructors of such courses might find books like Function Theory of One Complex Variable by Greene and Krantz, and Conway’s Functions of One Complex Variable I more suitable to a graduate-level course.

To summarize and conclude: if I were to teach a first-semester complex analysis course, I would be inclined to select a text that proceeds more directly to the essentials of the subject. But I would, at the same time, keep this book handy as a valuable reference source.

Mark Hunacek ( teaches mathematics at Iowa State University.

Complex Numbers and Their Geometrical Representation
Complex Numbers
Modulus and Argument of Complex Numbers
Geometrical Representations of Complex Numbers
Modulus and Argument of Complex Numbers
Properties of Moduli
Properties of Arguments
Equations of Straight Lines
Equations of Circles
Inverse Points
Relations between Inverse Points with Respect to Circles
Riemann Spheres and Point at Infinity
Cauchy–Schwarz’s Inequality and Lagrange’s Identity
Historical Remarks


Analytic Functions
Metric Spaces and Topology of C
Functions of Complex Variables
Uniform Continuity
Analytic and Regular Functions
Cauchy–Riemann Equations
Methods of Constructing Analytic Functions
Historical Remarks


Power Series and Elementary Functions
Power Series
Certain Theorems on Power Series
Elementary Functions of a Complex Variable
Many-Valued Functions: Branches
Logarithms and Power Functions
The Riemann Surfaces for Log z
Historical Remarks


Conformal Representation
Mappings or Transformations
Jacobian of Transformations
Conformal Mappings
Sufficient Condition for w = f(z) to Represent Conformal Mappings
Necessary Conditions for w = f(z) to Represent Conformal Mappings
Superficial Magnification
Some Elementary Transformations
Linear Transformations
Bilinear or Möbius Transformations
Product or Resultant of Two Bilinear Transformations
Every Bilinear Transformation Is the Resultant of Elementary Transformations
Bilinear Transformation as the Resultant of an Even Number of Inversions
The Linear Groups
Preservation of Cross-Ratio under Bilinear Transformations
Preservation of the Family of Circles and Straight Lines under Bilinear Transformations
Two Important Families of Circles
Fixed Point of Bilinear Transformations
Normal Form of a Bilinear Transformation
Elliptic, Hyperbolic, and Parabolic Transformations
Special Bilinear Transformations
Historical Remarks


Special Transformations
The Transformation w = za Where a Is a Complex Number
The Inverse Transformation z = √w
The Exponential Transformation w = ez
The Logarithmic Transformation w = log z
The Trigonometrical Transformation z = c sin w
The Transformation w = tan z
The Transformation w = tan2 (π/4a√z)
The Transformation w = 1/2 (z + 1/z)
The Transformation z = 1/2 (w + 1/w)
Historical Remarks


Complex Integrations
Complex Integrations
Complex Integrals
Cauchy's Theorem
Indefinite Integrals of Primitives
Cauchy's Integral Formula
Derivatives of Analytic Functions
Higher-Order Derivatives
Morera’s Theorem
Poisson's Integral Formula for Circles
Cauchy's Inequality
Liouville’s Theorem
Cauchy's Theorem and Integral Formulas
The Homotopic Version of Cauchy's Theorem and Simple Connectivity
Expansion of Analytic Functions as Power Series
Historical Remarks


Zeros of Analytic Functions
Singular Points
The "Point at Infinity"
Characterization of Polynomials
Characterization of Rational Functions


Residue Theory and Principle of Argument
The Residues at Singularities
Calculation of Residues in Some Special Cases
Residues at Infinity
Some Residue Theorems
Argument Principle and Rouché’s Theorem
Schwarz’s Lemma
The Inverse Functions
Formulas of Poisson, Hilbert, and Bromwich


Calculus of Residues
Evaluations of Definite Integrals by Contour Integrations
Integrations around the Unit Circle
Evaluations of Integrals of Type ∫-∞f(x) dx
Jordan's Inequality .
Jordan's Lemma
Evaluations of Integrals of Forms
Cases of Poles on the Real Axis
Cases of Poles on the Real Axis (Indenting Method)
Integrals of Many-Valued Functions
Quadrants or Sectors of Circles as Contours
Rectangular Contours


Uniform Convergence
Uniform Convergence of Sequences
Uniform Convergence of Series
Hardy's Tests for Uniform Convergence
Continuity of Sum Functions of Series
Term-by-Term Integrations
Analyticity of Sum Functions of Series (Term-by-Term Differentiations)
Uniform Convergence of Power Series
Formulas of Parseval, Schwarz, and Poisson
Functions Defined by Integrals


Harmonic Functions
Harmonic Functions
Inverse Mappings and Univalent Functions
Global Mapping Theorem
Riemann's Mapping Theorem
Historical Remarks


Analytic Continuation
Analytic Continuation
Power Series Methods of Analytic Continuation


Answers to Selected Questions