1st Edition

# Functions of a Complex Variable

742 Pages 192 B/W Illustrations
by Chapman & Hall

742 Pages
by Chapman & Hall

Also available as eBook on:

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
Introduction
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
Differentiability
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
Cross-Ratios
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
Introduction
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

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

Residue Theory and Principle of Argument
Introduction
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
Introduction
Analytic Continuation
Power Series Methods of Analytic Continuation

Answers to Selected Questions

Bibliography

Index

### Biography

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.