Complex Analysis – Joseph Bak, Donald J. Newman – 3rd Edition


This unusual and lively textbook offers a clear and intuitive approach to the classical and beautiful theory of complex variables. With very little dependence on advanced concepts from several-variable calculus and topology, the text focuses on the authentic complex-variable ideas and techniques. Accessible to students at their early stages of mathematical study, this full first year course in complex analysis offers new and interesting motivations for classical results and introduces related topics stressing motivation and technique. Numerous illustrations, examples, and now 300 exercises, enrich the text.

Students who master this textbook will emerge with an excellent grounding in complex analysis, and a solid understanding of its wide applicability. Beginning with the first edition of Complex Analysis, we have attempted to present the classical and beautiful theory of complex variables in the clearest and most intuitive form possible. The changes in this edition, which include additions to ten of the nineteen chapters, are intended to provide the additional insights that can be obtained by seeing a little more of the “big picture”. This includes additional related results and occasional generalizations that place the results in a slightly broader context.

The Fundamental Theorem of Algebra is enhanced by three related results. Section 1.3 offers a detailed look at the solution of the cubic equation and its role in the acceptance of complex numbers. While there is no formula for determining the roots of a general polynomial, we added a section on Newton’s Method, a numerical technique for approximating the zeroes of any polynomial. And the Gauss-Lucas Theorem provides an insight into the location of the zeroes of a polynomial and those of its derivative.

A series of new results relate to the mapping properties of analytic functions. A revised proof of Theorem 6.15 leads naturally to a discussion of the connection between critical points and saddle points in the complex plane. The proof of the Schwarz Reflection Principle has been expanded to include reflection across analytic arcs, which plays a key role in a new section (14.3) on the mapping properties of analytic functions on closed domains. And our treatment of special mappings has been enhanced by the inclusion of Schwarz-Christoffel transformations.

A single interesting application to number theory in the earlier editions has been expanded into a new section (19.4) which includes four examples from additive number theory, all united in their use of generating functions.

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  • 1. The Complex Numbers
    1.1 The Field of Complex Numbers
    1.2 The Complex Plane
    1.3 The Solution of the Cubic Equation
    1.4 Topological Aspects of the Complex Plane
    1.5 Stereographic Projection; The Point at Infinity

    2. Functions of the Complex Variable z
    2.1 Analytic Polynomials
    2.2 Power Series
    2.3 Differentiability and Uniqueness of Power Series

    3. Analytic Functions
    3.1 Analyticity and the Cauchy-Riemann Equations
    3.2 The Functions ez, sin z, cos z

    4. Line Integrals and Entire Functions
    4.1 Properties of the Line Integral
    4.2 The Closed Curve Theorem for Entire Functions

    5. Properties of Entire Functions
    5.1 The Cauchy Integral Formula and Taylor Expansion for Entire Functions
    5.2 Liouville Theorems and the Fundamental Theorem of Algebra; The Gauss-Lucas Theorem
    5.3 Newton's Method and Its Application to Polynomial Equations

    6. Properties of Analytic Functions
    6.1 The Power Series Representation for Functions Analytic in a Disc
    6.2 Analytic in an Arbitrary Open Set
    6.3 The Uniqueness, Mean-Value, and Maximum-Modulus Theorems; Critical Points and Saddle Points

    7. Further Properties of Analytic Functions
    7.1 The Open Mapping Theorem; Schwarz' Lemma
    7.2 The Converse of Cauchy's Theorem: Morera's Theorem; The Schwarz Reflection Principle and Analytic Arcs

    8. Simply Connected Domains
    8.1 The General Cauchy Closed Curve Theorem
    8.2 The Analytic Function log z

    9. Isolated Singularities of an Analytic Function
    9.1 Classification of Isolated Singularities; Riemann's Principle and the Casorati-Weierstrass Theorem
    9.2 Laurent Expansions

    10. The Residue Theorem
    10.1 Winding Numbers and the Cauchy Residue Theorem
    10.2 Applications of the Residue Theorem

    11. Applications of the Residue Theorem to the Evaluation of Integrals and Sums
    11.1 Evaluation of Definite Integrals by Contour Integral Techniques
    11.2 Application of Contour Integral Methods to Evaluation and Estimation of Sums

    12. Further Contour Integral Techniques
    12.1 Shifting the Contour of Integration
    12.2 An Entire Function Bounded in Every Direction

    13. Introduction to Conformal Mapping
    13.1 Conformal Equivalence
    13.2 Special Mappings
    13.3 Schwarz-Christoffel Transformations

    14. The Riemann Mapping Theorem
    14.1 Conformal Mapping and Hydrodynamics
    14.2 The Riemann Mapping Theorem
    14.3 Mapping Properties of Analytic Functions on Closed Domains

    15. Maximum-Modulus Theorems for Unbounded Domains
    15.1 A General Maximum-Modulus Theorem
    15.2 The Phragmén-Lindelöf Theorem

    16. Harmonic Functions
    16.1 Poisson Formulae and the Dirichlet Problem
    16.2 Liouville Theorems for Re f; Zeroes of Entire Functions of Finite Order

    17 Different Forms of Analytic Functions
    17.1 Infinite Products
    17.2 Analytic Functions Defined by Definite Integrals
    17.3 Analytic Functions Defined by Dirichlet Series

    18. Analytic Continuation; The Gamma and Zeta Functions
    18.1 Power Series
    18.2 Analytic Continuation of Dirichlet Series
    18.3 The Gamma and Zeta Functions

    19. Applications to Other Areas of Mathematics
    19.1 A Variation Problem
    19.2 The Fourier Uniqueness Theorem
    19.3 An Infinite System of Equations
    19.4 Applications to Number Theory
    19.5 An Analytic Proof of The Prime Number Theorem
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