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Complex Analysis

Complex Analysis. Instructor: Prof. P. A. S. Sree Krishna, Department of Mathematics, IIT Guwahati. This course provides an introduction to complex analysis, covering topics: Complex numbers, the topology of the complex plane, the extended complex plane and its representation using the sphere. Complex functions and their mapping properties, their limits, continuity and differentiability, analytic functions, analytic branches of a multiple-valued function. Complex integration, Cauchy's theorem, Cauchy's integral formulae. Power series, Taylor's series, zeroes of analytic functions, Rouche's theorem, open mapping theorem. Mobius transformations and their properties. Isolated singularities and their classification, Laurent's series, Cauchy's residue theorem, the argument principle. (from nptel.ac.in)

Lecture 24 - Problem Solving Session II

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The Arithmetic, Geometric and Topological Properties of the Complex Numbers

Lecture 01 - Introduction

Lecture 02 - Introduction to Complex Numbers

Lecture 03 - de Moivre's Formula and Stereographic Projection

Lecture 04 - Topology of the Complex Plane Part I

Lecture 05 - Topology of the Complex Plane Part II

Lecture 06 - Topology of the Complex Plane Part III

Complex Functions: Limits, Continuity and Differentiation

Lecture 07 - Introduction to Complex Functions

Lecture 08 - Limits and Continuity

Lecture 09 - Differentiation

Lecture 10 - Cauchy-Riemann Equations and Differentiability

Lecture 11 - Analytic Functions; the Exponential Function

Lecture 12 - Sine, Cosine and Harmonic Functions

Lecture 13 - Branches of Multifunctions; Hyperbolic Functions

Lecture 14 - Problem Solving Session I

Complex Integration Theory

Lecture 15 - Integration and Contours

Lecture 16 - Contour Integration

Lecture 17 - Introduction to Cauchy's Theorem

Lecture 18 - Cauchy's Theorem for a Rectangle

Lecture 19 - Cauchy's Theorem (cont.)

Lecture 20 - Cauchy's Theorem (cont.)

Lecture 21 - Cauchy's Integral Formula and its Consequences

Lecture 22 - The First and Second Derivatives of Analytic Functions

Lecture 23 - Morera's Theorem and Higher Order Derivatives of Analytic Functions

Lecture 24 - Problem Solving Session II

Further Properties of Analytic Functions

Lecture 25 - Introduction to Complex Power Series

Lecture 26 - Analyticity of Power Series

Lecture 27 - Taylor's Theorem

Lecture 28 - Zeroes of Analytic Functions

Lecture 29 - Counting the Zeroes of Analytic Functions

Lecture 30 - Open Mapping Theorem Part I

Lecture 31 - Open Mapping Theorem Part II

Mobius Transformations

Lecture 32 - Properties of Mobius Transformations Part I

Lecture 33 - Properties of Mobius Transformations Part II

Lecture 34 - Problem Solving Session III

Isolated Singularities and Residue Theorem

Lecture 35 - Removable Singularities

Lecture 36 - Poles Classification of Isolated Singularities

Lecture 37 - Essential Singularity and Introduction to Laurent Series

Lecture 38 - Laurent's Theorem

Lecture 39 - Residue Theorem and Applications

Lecture 40 - Problem Solving Session IV