This book is a comprehensive resource for students of undergraduate postgraduate courses in mathematics, physics and engineering. It makes use of numerous worked-out examples to show how the study of complex numbers and their derivatives and properties helps in solving many physical problems. Beginning with the algebraic and analytic properties of complex numbers, the reader is introduced to topological notions of sets in the complex plane, sequence and series representation of complex numbers, limit, continuity and differentiability of complex functions, and branch cut and branch points in multi-valued functions. Important theorems such as Ascoli–Arzela theorem, Montel’s theorem, Riemann mapping theorem, and the concept of Schawarz–Cristoffel transformations widely used in various fields are established. The notion of entire functions and their properties and direct and indirect analytic continuation of an analytic function, too, are covered.

The book contains an interesting range of chapter-end review exercises that will be of help to students and teachers alike. The inclusion of multiple-choice questions, in particular, will be of interest to those preparing for competitive examinations such as the NET, SET and GATE.*Salient points:*

- Many solved examples
- Chapter-end exercises of varied kinds, including multiple-choice questions
- Large number of interesting properties of complex differentiable functions
- Reveals many nice results of complex analysis
- Highlights practical applications of complex analysis in solving physical problems

*Prasun Kumar Nayak,*** **an assistant professor in the Department of Mathematics, Midnapore College, West Bengal, has more than 15 years of teaching experience. He has an MSc in Applied Mathematics from Calcutta University and a PhD from Vidyasagar University, West Bengal. He is the author of several publications, including 9 books and 42 research papers.

*Mijanur Rahaman Seikh*** **is** **an** a**ssistant professor in the Department of Mathematics, Kazi Nazrul University, Asansol, West Bengal. He has an MSc in Applied Mathematics from Calcutta University and a PhD from Vidyasagar University, West Bengal. He has published several research papers in national and international journals of repute.

*Preface*

**1 Complex Numbers**

1.1 Complex Numbers

1.1.1 Fundamental Operations

1.1.2 Conjugation

1.1.3 Modulus

1.2 Point Representation of Complex Numbers

1.2.1 Polar Form of Complex Number

1.2.2 nth Root of Complex Numbers

1.2.3 Square Roots

1.2.4 Inequalities

1.3 Stereographic Projection

1.4 Topological Aspects in C

1.4.1 Diameter

1.4.2 Open and Closed Discs

1.4.3 Neighbourhood of a Point

1.4.4 Interior, Exterior and Boundary Points

1.4.5 Open Set

1.4.6 Limit Point

1.4.7 Isolated Point

1.4.8 Closed Set

1.5 Compactness

1.6 Connected Sets

1.7 Domain and Region

**2 Sequence and Series**

2.1 Complex Sequences

2.1.1 Convergent Sequences

2.1.2 Bounded sequence

2.1.3 Complex Cauchy’s Sequence

2.2 Completeness

2.3 Compact Sets

2.4 Infinite Series

2.4.1 Test of Convergence

2.5 Complex Power Series

2.5.1 Radius of Convergence

2.5.2 Behaviour of Power Series on the Circle of Convergence

2.6 Sequence and Series of Functions

2.6.1 Pointwise Convergence

2.6.2 Uniform Convergence

**3 Complex Differentiation**

3.1 Functions of Complex Variables

3.1.1 Univalent and Inverse Functions

3.1.2 Conjugation and Composition of Functions

3.2 Limits

3.2.1 Limit Involving Point of Infinity

3.2.2 Limit of Polynomial Functions

3.3 Continuity

3.3.1 Uniform Continuity

3.4 Derivability

3.5 Analytic Functions

3.5.1 Inverse Function

3.5.2 Orthogonal System

3.6 Harmonic Function

3.7 Singular Points

3.7.1 Isolated Singularity

3.7.2 Pole

3.7.3 Removable Singularity

3.7.4 Essential Singularity

3.8 Meromorphic Functions

3.9 Entire Function

3.10 Multi valued Functions

3.10.1 Branch

3.10.2 Riemann Surfaces

**4 Elementary Transcendental Functions**

4.1 Complex Exponential Function

4.2 Complex Trigonometric Functions

4.3 Complex Logarithmic Function

4.3.1 The general Power Function z

4.4 Complex Hyperbolic Functions

4.4.1 Inverse Trigonometric Functions

4.4.2 Inverse Hyperbolic Functions

**5 Complex Integration**

5.1 Integrals of a Function

5.2 Contours

5.2.1 Curve

5.2.2 Path (Arc)

5.2.3 Contour (Piecewise Smooth Curve)

5.2.4 Rectifiable Curves

5.2.5 Homotopy of Curves

5.3 Contour Integrals

5.3.1 Estimation of Contour Integrals

5.4 Winding Number

5.5 Cauchy’s Theorem

5.5.1 Cauchy’s Theorem for Triangles

5.5.2 Cauchy’s Theorem for Rectangles

5.5.3 Cauchy’s Theorem for Discs

5.5.4 Cauchy’s Theorem for a Closed Polygon

5.5.5 Cauchy’s Theorem for a Simple Closed Curve

5.5.6 Cauchy’s Theorem for Multiply connected Regions

5.5.7 Deformation Theorem

5.5.8 Homotopy Version of Cauchy’s Theorem

5.6 Cauchy’s Integral Formula

5.7 Modulus Theorems

5.8 Series Expansions

5.8.1 Taylor’s Series

5.9 Singularities of a Function

5.9.1 Zeros of Analytic Functions

5.9.2 Isolated Singular Points

5.9.3 Removable Singularity

5.9.4 Poles

5.9.5 Essential Singularity

5.9.6 Singularities at Infinity

5.10 Schwarz’s Lemma and Its Consequences

5.10.1 Some Consequences of Schwarz’s Lemma

**6 Linear Fractional Transformations**

6.1 Transformation

6.2 Conformal Mapping

6.3 Elementary Transformations

6.3.1 Translation

6.3.2 Rotation–Dilation

6.3.3 Magnification or Contraction

6.3.4 Inversion

6.4 Linear Fractional Transformations

6.4.1 Properties of Linear Fractional Transformation

6.4.2 Fixed Points

6.4.3 Normal/Canonical Form

6.5 Cross Ratio

6.6 Mappings by Elementary Functions

6.6.1 The Mapping *w* = *z*^{2}

6.6.2 The Transformation *w* = ½( *z* + 1/*z*)

6.6.3 The Transformation *w* = e^{z}

6.6.4 The Transformation *w* = log *z*

6.6.5 The Transformation *w* = sin *z*

6.6.6 The Transformation *w* = cosh *z*

6.6.7 The Transformation of *w* = tan *z*

6.7 The Schawarz–Cristoffel Transformations

6.7.1 Transformation of the Real Axis Onto a Polygon

6.7.2 Transformation of Schawarz–Cristoffel

6.7.3 Schwarz–Christoffel Transformation for Triangles

6.7.4 Schwarz–Christoffel Transformation for Rectangles

**7 Calculus of Residues **

7.1 Residues

7.1.1 Cauchy’s Residue Theorem

7.1.2 The Residue at Infinity

7.2 The Argument Principle

7.2.1 Rouche’s Theorem

7.2.2 Local Mapping Theorem

7.2.3 Inverse Function Theorem

7.2.4 Hurwitz’s Theorem

7.2.5 Open Mapping Theorem

7.3 Evaluation of Definite Integrals

7.3.1 Definite Integral of the Type

7.3.2 Definite Integrals of the Type

7.3.3 Integrals of the Form

7.3.4 Poles on the Real Axis

7.3.5 Integrals of Multi valued Functions

7.3.6 Other Types of Contours

7.4 Estimation of Infinite Sums

**8 Some Relevant Theorems**** **

8.1 Sequence of Functions in a Compact Set

8.2 Convergence in the Space of Analytic Functions

8.2.1 Arzela`–Ascoli Theorem

8.2.2 Montel’s Theorem

8.3 Riemann Mapping Theorem

8.4 Harmonic Functions

8.4.1 Poisson’s Integral Formula

8.4.2 Subharmonic and Superharmonic Functions

**9 Entire and Meromorphic Functions**

9.1 Infinite Products

9.2 Infinite Product of Complex Numbers

9.2.1 Absolute Convergence of Infinite Products

9.2.2 Semi-convergence

9.2.3 Infinite Product of Functions

9.3 Factorization of Entire Functions

9.3.1 Weierstrass’ Primary Factor

9.3.2 The Weierstrass Theorem

9.3.3 Weierstrass Factorization Theorem

9.3.4 Canonical Product

9.4 Counting Zeros of Analytic Functions

9.5 Order of an Entire Function

9.5.1 The Maximum Modulus of an Entire Function

9.5.2 Entire Function of Finite Order

9.5.3 Estimation of the Number of Zeros

9.5.4 Convergent Exponent

9.6 Hadamard’s Factorization Theorem

9.7 Meromorphic Functions

9.7.1 Partial Fraction Decomposition of Meromorphic Functions

9.7.2 Mittag–Leffler Theorem

**10 Analytic Continuation**

10.1 Analytic Continuation

10.1.1 Direct Analytic Continuation

10.1.2 Indirect Analytic Continuation

10.1.3 Indirect Analytic Continuation using Power Series

10.1.4 Indirect Analytic Continuation along a Curve

10.1.5 Regular and Singular Points

10.1.6 Natural Boundary

10.2 Monodromy Theorem

10.3 Reflection Principle

*Bibliography*

*Index*