# Advanced Complex Analysis Part 1

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3 STUDENTS

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

### Course Curriculum

 Fundamental Theorems Connected with Zeros of Analytic Functions Details 58:10 The Argument (Counting) Principle, Rouche’s Theorem and The Fundamental Theorem Details 52:29 Morera’s Theorem and Normal Limits of Analytic Functions Details 52:9 Hurwitz’s Theorem and Normal Limits of Univalent Functions Details 56:54 Local Constancy of Multiplicities of Assumed Values Details 0:52 The Open Mapping Theorem Details 1:8:37 Introduction to the Inverse Function Theorem Details 45:21 Completion of the Proof of the Inverse Function Theorem: The Integral Inversion Details 44:37 Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms Details 47:53 Introduction to the Implicit Function Theorem Details 40:31 Proof of the Implicit Function Theorem: Topological Preliminaries Details 47:52 Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity Details 53:50 Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface Details 55:45 F(z,w)=0 is naturally a Riemann Surface Details 47:11 Constructing the Riemann Surface for the Complex Logarithm Details 1:3:23 Constructing the Riemann Surface for the m-th root function Details 1:27 The Riemann Surface for the functional inverse of an analytic Details 48:6 The Algebraic nature of the functional inverses of an analytic Details 48:25 The Idea of a Direct Analytic Continuation or an Analytic Extension Details 51:31 General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius Details 36:57 Analytic Continuation Along Paths via Power Series Part A Details 30:33 Analytic Continuation Along Paths via Power Series Part B Details 35:38 Continuity of Coefficients occurring in Families of Power Series defining Analytic Details 52:15

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