# InfoCoBuild

## 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 29 - Counting the Zeroes of Analytic Functions

Go to the Course Home or watch other lectures:

 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