### Lecture 1: Fundamental Theorems Connected with Zeros of Analytic Functions

##### Lecture Details :

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

##### Course Description :

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

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