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# Functions of a Complex Variable

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## Book Description

**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.

## Table of Contents

**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 = z*Where a Is a Complex Number

^{a}The Inverse Transformation z = √w

The Exponential Transformation

*w = e*

^{z}The Logarithmic Transformation

*w =*log

*z*

The Trigonometrical Transformation

*z = c*sin

*w*

The Transformation

*w =*tan

*z*

The Transformation

*w*= tan

^{2}(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

## Author(s)

### 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.