IIT Madras Course , Prof. T.E. Venkata Balaji

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IIT Madras Course , Prof. T.E. Venkata Balaji

Theorems of Rouche and Hurwitz:Fundamental Theorems Connected with Zeros of Analytic Functions - The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem of Algebra - Morera's Theorem and Normal Limits of Analytic Functions - Hurwitz's Theorem and Normal Limits of Univalent Functions;Open Mapping Theorem:Local Constancy of Multiplicities of Assumed Values - The Open Mapping Theorem;Inverse Function Theorem:Introduction to the Inverse Function Theorem - Completion of the Proof of the Inverse Function Theorem: The Integral Inversion Formula for the Inverse Function - Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms;Implicit Function Theorem:Introduction to the Implicit Function Theorem - Proof of the Implicit Function Theorem: Topological Preliminaries - Proof of the

Implicit Function Theorem: The Integral Formula for & Analyticity of the Explicit Function

Riemann Surfaces for Multi-Valued Functions:Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface - F(z,w)=0 is naturally a Riemann Surface - Constructing the Riemann Surface for the Complex Logarithm - Constructing the Riemann Surface for the m-th root function - The Riemann Surface for the functional inverse of an analytic mapping at a critical point - The Algebraic nature of the functional inverses of an analytic mapping at a critical point;Analytic Continuation:The Idea of a Direct Analytic Continuation or an Analytic Extension - General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius of Convergence - Analytic Continuation Along Paths via Power Series - Continuity of Coefficients occurring in Families of Power Series defining Analytic Continuations along Paths

Implicit Function Theorem: The Integral Formula for & Analyticity of the Explicit Function

Riemann Surfaces for Multi-Valued Functions:Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface - F(z,w)=0 is naturally a Riemann Surface - Constructing the Riemann Surface for the Complex Logarithm - Constructing the Riemann Surface for the m-th root function - The Riemann Surface for the functional inverse of an analytic mapping at a critical point - The Algebraic nature of the functional inverses of an analytic mapping at a critical point;Analytic Continuation:The Idea of a Direct Analytic Continuation or an Analytic Extension - General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius of Convergence - Analytic Continuation Along Paths via Power Series - Continuity of Coefficients occurring in Families of Power Series defining Analytic Continuations along Paths

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Advanced Complex Analysis - Part 1 by Dr. T.E. Venkata Balaji,Department of Mathematics,IIT Madras.For more details on NPTEL visit httpnptel.ac.in

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- 1.Fundamental Theorems Connected with Zeros of Analytic Functions
- 2.The Argument (Counting) Principle, Rouches Theorem and The Fundamental Theorem
- 3.Moreras Theorem and Normal Limits of Analytic Functions
- 4.Hurwitzs Theorem and Normal Limits of Univalent Functions
- 5.Local Constancy of Multiplicities of Assumed Values
- 6.The Open Mapping Theorem
- 7.Introduction to the Inverse Function Theorem
- 8.Completion of the Proof of the Inverse Function Theorem The Integral Inversion
- 9.Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms
- 10.Introduction to the Implicit Function Theorem
- 11.Proof of the Implicit Function Theorem Topological Preliminaries
- 12.Proof of the Implicit Function Theorem The Integral Formula for & Analyticity
- 13.Doing Complex Analysis on a Real Surface The Idea of a Riemann Surface
- 14.F(z,w)=0 is naturally a Riemann Surface
- 15.Constructing the Riemann Surface for the Complex Logarithm
- 16.Constructing the Riemann Surface for the m-th root function
- 17.The Riemann Surface for the functional inverse of an analytic
- 18.The Algebraic nature of the functional inverses of an analytic
- 19.The Idea of a Direct Analytic Continuation or an Analytic Extension
- 20.General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius
- 21.Analytic Continuation Along Paths via Power Series Part A
- 22.Analytic Continuation Along Paths via Power Series Part B
- 23.Continuity of Coefficients occurring in Families of Power Series defining Analytic

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