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September 23, 2023

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Description

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)

Course Curriculum

    • Lecture 01 – Introduction Unlimited
    • Lecture 02 – Introduction to Complex Numbers Unlimited
    • Lecture 03 – de Moivre’s Formula and Stereographic Projection Unlimited
    • Lecture 04 – Topology of the Complex Plane Part I Unlimited
    • Lecture 05 – Topology of the Complex Plane Part II Unlimited
    • Lecture 06 – Topology of the Complex Plane Part III Unlimited
    • Lecture 07 – Introduction to Complex Functions Unlimited
    • Lecture 08 – Limits and Continuity Unlimited
    • Lecture 09 – Differentiation Unlimited
    • Lecture 10 – Cauchy-Riemann Equations and Differentiability Unlimited
    • Lecture 11 – Analytic Functions; the Exponential Function Unlimited
    • Lecture 12 – Sine, Cosine and Harmonic Functions Unlimited
    • Lecture 13 – Branches of Multifunctions; Hyperbolic Functions Unlimited
    • Lecture 14 – Problem Solving Session I Unlimited
    • Lecture 15 – Integration and Contours Unlimited
    • Lecture 16 – Contour Integration Unlimited
    • Lecture 17 – Introduction to Cauchy’s Theorem Unlimited
    • Lecture 18 – Cauchy’s Theorem for a Rectangle Unlimited
    • Lecture 19 – Cauchy’s Theorem (cont.) Unlimited
    • Lecture 20 – Cauchy’s Theorem (cont.) Unlimited
    • Lecture 21 – Cauchy’s Integral Formula and its Consequences Unlimited
    • Lecture 22 – The First and Second Derivatives of Analytic Functions Unlimited
    • Lecture 23 – Morera’s Theorem and Higher Order Derivatives of Analytic Functions Unlimited
    • Lecture 24 – Problem Solving Session II Unlimited
    • Lecture 25 – Introduction to Complex Power Series Unlimited
    • Lecture 26 – Analyticity of Power Series Unlimited
    • Lecture 27 – Taylor’s Theorem Unlimited
    • Lecture 28 – Zeroes of Analytic Functions Unlimited
    • Lecture 29 – Counting the Zeroes of Analytic Functions Unlimited
    • Lecture 30 – Open Mapping Theorem Part I Unlimited
    • Lecture 31 – Open Mapping Theorem Part II Unlimited
    • Lecture 32 – Properties of Mobius Transformations Part I Unlimited
    • Lecture 33 – Properties of Mobius Transformations Part II Unlimited
    • Lecture 34 – Problem Solving Session III Unlimited
    • Lecture 35 – Removable Singularities Unlimited
    • Lecture 36 – Poles Classification of Isolated Singularities Unlimited
    • Lecture 37 – Essential Singularity and Introduction to Laurent Series Unlimited
    • Lecture 38 – Laurent’s Theorem Unlimited
    • Lecture 39 – Residue Theorem and Applications Unlimited
    • Lecture 40 – Problem Solving Session IV Unlimited

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