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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)

Introduction


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

References
Complex Analysis
Instructor: Prof. P. A. S. Sree Krishna, Department of Mathematics, IIT Guwahati. This course provides an introduction to complex analysis.