Mathematics - Advanced Complex Analysis - Part 1 (English)

   Monday, May 12, 2025   

Starting a new Lecture Notes Series on Mathematics - Advanced Complex Analysis - Part 1

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Lecture 1: Mod-01 Lec-01 Fundamental Theorems Connected with Zeros of Analytic Functions

Lecture 2: Mod-01 Lec-02 The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem

Lecture 3: Mod-01 Lec-03 Morera's Theorem and Normal Limits of Analytic Functions

Lecture 4: Mod-01 Lec-04 Hurwitz's Theorem and Normal Limits of Univalent Functions

Lecture 5: Mod-02 Lec-05 Local Constancy of Multiplicities of Assumed Values

Lecture 6: Mod-02 Lec-06 The Open Mapping Theorem

Lecture 7: Mod-03 Lec-07 Introduction to the Inverse Function Theorem

Lecture 8: Mod-03 Lec-08 Completion of the Proof of the Inverse Function Theorem: The Integral Inversion

Lecture 9: Mod-03 Lec-09 Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms

Lecture 10: Mod-04 Lec-10 Introduction to the Implicit Function Theorem

Lecture 11: Mod-04 Lec-11 Proof of the Implicit Function Theorem: Topological Preliminaries

Lecture 12: Mod-04 Lec-12 Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity

Lecture 13: Mod-05 Lec-13 Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface

Lecture 14: Mod-05 Lec-14 F(z,w)=0 is naturally a Riemann Surface

Lecture 15: Mod-05 Lec-15 Constructing the Riemann Surface for the Complex Logarithm

Lecture 16: Mod-05 Lec-16 Constructing the Riemann Surface for the m-th root function

Lecture 17: Mod-05 Lec-17 The Riemann Surface for the functional inverse of an analytic

Lecture 18: Mod-05 Lec-18 The Algebraic nature of the functional inverses of an analytic

Lecture 19: Mod-06 Lec-19 The Idea of a Direct Analytic Continuation or an Analytic Extension

Lecture 20: Mod-06 Lec-20 General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius

Lecture 21: Mod-06 Lec-21A Analytic Continuation Along Paths via Power Series Part A

Lecture 22: Mod-06 Lec-21B Analytic Continuation Along Paths via Power Series Part B

Lecture 23: Mod-06 Lec-22 Continuity of Coefficients occurring in Families of Power Series defining Analytic

Lecture 24: Mod-07 Lec-23 Analytic Continuability along Paths: Dependence on the Initial Function

Lecture 25: Mod-07 Lec-24 Maximal Domains of Direct and Indirect Analytic Continuation: Second

Lecture 26: Mod-07 Lec-25 Deducing the Second (Simply Connected) Version of the Monodromy Theorem

Lecture 27: Mod-07 Lec-27 Existence and Uniqueness of Analytic Continuations on Nearby Paths

Lecture 28: Mod-07 Lec-28 Proof of the First (Homotopy) Version of the Monodromy Theorem

Lecture 29: Mod-07 Lec-30 Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse

Lecture 30: Mod-08 Lec-31 The Mean-Value Property, Harmonic Functions and the Maximum Principle

Lecture 31: Mod-08 Lec-32 Proofs of Maximum Principles and Introduction to Schwarz's Lemma

Lecture 32: Mod-08 Lec-33 Proof of Schwarz's Lemma and Uniqueness of Riemann Mappings

Lecture 33: Mod-08 Lec-34 Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains

Lecture 34: Mod-09 Lec 35 Part A Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic

Lecture 35: Mod-09 Lec-35 Part B Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic

Lecture 36: Mod-09 Lec-36 Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc

Lecture 37: Mod-09 Lec-37 Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc

Lecture 38: Mod-10 Lec-38 Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform

Lecture 39: Mod-10 Lec-39 Completion of the Proof of the Arzela-Ascoli Theorem and Introduction

Lecture 40: Mod-10 Lec-40 The Proof of Montel's Theorem

Lecture 41: Mod-11 Lec-41 The Candidate for a Riemann Mapping

Lecture 42: Mod-11 Lec-42A Completion of Proof of The Riemann Mapping Theorem

Lecture 43: Mod-11 Lec-42B Completion of Proof of The Riemann Mapping Theorem