1st Edition

Functions of a Complex Variable

    742 Pages 192 B/W Illustrations
    by Chapman & Hall

    Functions of a Complex Variable provides all the material for a course on the theory of functions of a complex variable at the senior undergraduate and beginning graduate level. Also suitable for self-study, the book covers every topic essential to training students in complex analysis. It also incorporates special topics to enhance students’ understanding of the subject, laying the foundation for future studies in analysis, linear algebra, numerical analysis, geometry, number theory, physics, thermodynamics, or electrical engineering.

    After introducing the basic concepts of complex numbers and their geometrical representation, the text describes analytic functions, power series and elementary functions, the conformal representation of an analytic function, special transformations, and complex integration. It next discusses zeros of an analytic function, classification of singularities, and singularity at the point of infinity; residue theory, principle of argument, Rouché’s theorem, and the location of zeros of complex polynomial equations; and calculus of residues, emphasizing the techniques of definite integrals by contour integration.

    The authors then explain uniform convergence of sequences and series involving Parseval, Schwarz, and Poisson formulas. They also present harmonic functions and mappings, inverse mappings, and univalent functions as well as analytic continuation.

    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




    Hemant Kumar Pathak is a professor and head of the School of Studies in Mathematics and the Director of the Center for Basic Sciences at Pt. Ravishankar Shukla University. Dr. Pathak is the author of nearly 50 textbooks for undergraduate and post-graduate students and the author or co-author of nearly 250 publications. Dr. Pathak is an editorial board member of the American Journal of Computational and Applied Mathematics and the Journal of Calculus of Variations as well as a reviewer for the American Mathematical Society. His research interests include nonlinear analysis, general topology, Banach frames, and integration theory.

    Ravi P. Agarwal is a professor and the chair of the Department of Mathematics at Texas A&M University–Kingsville. Dr. Agarwal is the author or co-author of more than 1,000 scientific papers. His research interests include nonlinear analysis, differential and difference equations, fixed point theory, and general inequalities.

    Yeol Je Cho is a professor in the Department of Mathematics Education at Gyengsang National University and a fellow of The Korean Academy of Science and Technology. Dr. Cho is the author or co-author of nearly 400 publications. His research interests include nonlinear analysis with applications, inequality theory, and the geometry of Banach spaces.