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Complex Analysis

N.B. Singh

This audiobook is narrated by a digital voice. This audiobook, Complex Analysis, is a beginner-friendly guide that introduces you to the fascinating world of complex numbers and functions in a clear and simple way. Starting with the basics, it...

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Preface

1/19/2025
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Introduction to Complex Numbers

1/19/2025
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Historical Background

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Definition and Properties of Complex Numbers

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The Complex Plane and Argand Diagram

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Polar and Exponential Forms

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Operations with Complex Numbers

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Roots of Complex Numbers

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Complex Conjugates

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Complex Functions and Mappings

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Definition of Complex Functions

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Complex Mappings and Transformations

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Limits of Complex Functions

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Continuity of Complex Functions

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Elementary Complex Functions

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Complex Powers and Logarithms

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Special Mappings in Complex Analysis

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Limits and Continuity in Complex Analysis

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The Concept of Limits in the Complex Plane

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Properties of Complex Limits

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Uniform and Pointwise Convergence

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Epsilon-Delta Definition for Complex Functions

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Sequential Continuity

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Examples and Problems in Limits and Continuity

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Differentiation in the Complex Plane

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Definition of Complex Derivatives

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Rules of Differentiation for Complex Functions

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Geometric Interpretation of Differentiation

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Differentiability and Analyticity

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The Power Rule and Higher Order Derivatives

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Harmonic Conjugates

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Applications of Complex Differentiation

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Cauchy-Riemann Equations and Analyticity

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Deriving the Cauchy-Riemann Equations

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Implications of the Cauchy-Riemann Equations

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Analytic Functions and Their Properties

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Harmonic Functions and Their Relationship to Analyticity

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Applications of Cauchy-Riemann Equations

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Examples of Analytic and Non-Analytic Functions

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Singularities in Analytic Functions

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Complex Integration

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Introduction to Complex Integration

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Line Integrals in the Complex Plane

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Parametrization of Paths

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Properties of Complex Integrals

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Fundamental Theorem of Complex Integration

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Examples and Computation of Line Integrals

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Path Independence and Connected Domains

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Cauchy’s Theorem and Applications

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Statement and Proof of Cauchy’s Theorem

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Applications of Cauchy’s Theorem

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Deformation of Contours

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Simple and Multiply Connected Domains

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Cauchy-Goursat Theorem

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Morera’s Theorem

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Consequences of Cauchy’s Theorem

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Cauchy’s Integral Formula

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Statement of Cauchy’s Integral Formula

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Applications of the Integral Formula

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Extensions and Corollaries

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Higher Order Derivatives

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Integral Formula for Derivatives

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Proofs and Examples

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Implications of Cauchy’s Integral Formula

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Series Representations of Analytic Functions

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Power Series in Complex Analysis

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Convergence of Power Series

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Taylor Series for Complex Functions

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Laurent Series

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Examples of Series Expansions

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Analytic Continuation

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Applications of Series Representations

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Residues and Poles

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Isolated Singularities

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Classification of Singularities

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Residues and Their Computation

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Poles and Essential Singularities

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Residue at Infinity

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Examples and Applications of Residues

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Series Expansions and Residues

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The Residue Theorem and Applications

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Statement of the Residue Theorem

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Proof and Examples of the Residue Theorem

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Applications in Complex Integration

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Computing Real Integrals with Residues

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Residues and Improper Integrals

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Integral Representations of Special Functions

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Advanced Applications of the Residue Theorem

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Conformal Mappings

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Definition and Properties of Conformal Maps

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Examples of Conformal Mappings

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Riemann Mapping Theorem

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Applications in Physics and Engineering

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Mobius Transformations

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Bilinear Transformations

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Conformal Mapping Techniques

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Harmonic Functions and the Laplace Equation

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Definition of Harmonic Functions

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Laplace’s Equation in Complex Analysis

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Harmonic Conjugates and Analyticity

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The Mean Value Property

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Maximum Principle for Harmonic Functions

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Poisson’s Integral Formula

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Applications of Harmonic Functions

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Fluid Flow and Complex Potentials

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Electrostatics and Complex Analysis

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Heat Conduction Problems

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Applications in Mechanical Engineering

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Applications in Signal Processing

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Complex Analysis in Control Theory

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Miscellaneous Applications

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Advanced Topics in Complex Analysis

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Riemann Surfaces

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Analytic Continuation and Monodromy

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Elliptic Functions

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Hypergeometric Functions

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Complex Dynamics and Fractals

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Modular Forms and Applications

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Current Research and Open Problems

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