Advanced Complex Analysis Part 1. Instructor: Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras.
43
This is the first part of a series of lectures on advanced topics in Complex Analysis. By advanced, we mean topics that are not (or just barely) touched upon in a first course on Complex Analysis. The theme of the course is to study zeros of analytic (or holomorphic) functions and related theorems. These include the theorems of Hurwitz and Rouche, the open mapping theorem, the inverse and implicit function theorems, applications of those theorems, behaviour at a critical point, analytic branches, constructing Riemann surfaces for functional inverses, analytic continuation and monodromy, hyperbolic geometry and the Riemann mapping theorem. (from nptel.ac.in)
Course Currilcum

 Lecture 01 – Fundamental Theorems Connected with Zeroes of Analytic Functions Unlimited
 Lecture 02 – The Argument (Counting) Principle, Rouche’s Theorem and … Unlimited
 Lecture 03 – Morera’s Theorem and Normal Limits of Analytic Functions Unlimited
 Lecture 04 – Hurwitz’s Theorem and Normal Limits of Univalent Functions Unlimited

 Lecture 05 – Local Constancy of Multiplicities of Assumed Values Unlimited
 Lecture 06 – The Opening Mapping Theorem Unlimited

 Lecture 07 – Introduction to the Inverse Function Theorem Unlimited
 Lecture 08 – Completion of the Proof of the Inverse Function Theorem Unlimited
 Lecture 09 – Univalent Analytic Functions have NeverZero Derivatives and are Analytic Isomorphisms Unlimited

 Lecture 10 – Introduction to the Implicit Function Theorem Unlimited
 Lecture 11 – Proof of the Implicit Function Theorem: Topological Preliminaries Unlimited
 Lecture 12 – Proof of the Implicit Function Theorem: The Integral Formula for Analyticity … Unlimited

 Lecture 13 – Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface Unlimited
 Lecture 14 – F(z,w)=0 is Naturally a Riemann Surface Unlimited
 Lecture 15 – Constructing the Riemann Surface for the Complex Logarithm Unlimited
 Lecture 16 – Constructing the Riemann Surface for the mth Root Function Unlimited
 Lecture 17 – The Riemann Surface for the Functional Inverse of an Analytic Mapping … Unlimited
 Lecture 18 – The Algebraic Nature of the Functional Inverse of an Analytic Mapping … Unlimited

 Lecture 19 – The Idea of a Direct Analytic Continuation or an Analytic Extension Unlimited
 Lecture 20 – General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius … Unlimited
 Lecture 21a – Analytic Continuation along Paths via Power Series Part A Unlimited
 Lecture 21b – Analytic Continuation along Paths via Power Series Part B Unlimited
 Lecture 22 – Continuity of Coefficients Occurring in Families of Power Series defining Analytic … Unlimited

 Lecture 23 – Analytic Continuability along Paths: Dependence on the Initial Function and … Unlimited
 Lecture 24 – Maximal Domains of Direct and Indirect Analytic Continuation Unlimited
 Lecture 25 – Deducing the Second Version of the Monodromy Theorem from … Unlimited
 Lecture 27 – Existence and Uniqueness of Analytic Continuations on Nearby Paths Unlimited
 Lecture 28 – Proof of the First (Homotopy) Version of the Monodromy Theorem Unlimited
 Lecture 30 – Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse of … Unlimited

 Lecture 31 – The Mean Value Property, Harmonic Functions and the Maximum Principle Unlimited
 Lecture 32 – Proofs of Maximum Principles and Introduction to Schwarz Lemma Unlimited
 Lecture 33 – Proof of Schwarz Lemma and Uniqueness of Riemann Mappings Unlimited
 Lecture 34 – Reducing Existence of Riemann Mappings to Hyperbolic Geometry of … Unlimited

 Lecture 35a – Differential and Infinitesimal Schwarz’s Lemma, Pick’s Lemma, … Unlimited
 Lecture 35b – Differential and Infinitesimal Schwarz’s Lemma, Pick’s Lemma, … (cont.) Unlimited
 Lecture 36 – Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc Unlimited
 Lecture 37 – SchwarzPick Lemma for the Hyperbolic Metric on the Unit Disc Unlimited

 Lecture 38 – ArzelaAscoli Theorem: Under Uniform Boundedness, Equicontinuity and … Unlimited
 Lecture 39 – Completion of the Proof of the ArzelaAscoli Theorem and … Unlimited
 Lecture 40 – The Proof of Montel’s Theorem Unlimited

 Lecture 41 – The Candidate for a Riemann Mapping Unlimited
 Lecture 42a – Completion of Proof of the Riemann Mapping Theorem Unlimited
 Lecture 42b – Completion of Proof of the Riemann Mapping Theorem (cont.) Unlimited