# A Textbook of Complex Analysis

Authors: Prasun Kumar Nayak – Mijanur Rahaman Seikh

Publisher: Universities Press

Contents:

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
2.5.2 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 = z2
6.6.2 The Transformation w = ½( z + 1/z
6.6.3 The Transformation w = ez
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
9.7 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

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