Advanced Complex Analysis


Lecture 1 - Fundamental Theorems Connected with Zeros of Analytic Functions


Lecture 2 - The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem of Algebra


Lecture 3 - Morera's Theorem and Normal Limits of Analytic Functions


Lecture 4 - Hurwitz's Theorem and Normal Limits of Univalent Functions


Lecture 5 - Local Constancy of Multiplicities of Assumed Values


Lecture 6 - The Open Mapping Theorem


Lecture 7 - Introduction to the Inverse Function Theorem


Lecture 8 - Completion of the Proof of the Inverse Function Theorem: The Integral Inversion Formula for the Inverse Function


Lecture 9 - Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms


Lecture 10 - Introduction to the Implicit Function Theorem


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


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


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


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


Lecture 15 - Constructing the Riemann Surface for the Complex Logarithm


Lecture 16 - Constructing the Riemann Surface for the m-th root function


Lecture 17 - The Riemann Surface for the functional inverse of an analytic mapping at a critical point


Lecture 18 - The Algebraic nature of the functional inverses of an analytic mapping at a critical point


Lecture 19 - The Idea of a Direct Analytic Continuation or an Analytic Extension


Lecture 20 - General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius of Convergence


Lecture 21 - Analytic Continuation Along Paths via Power Series Part A


Lecture 22 - Analytic Continuation Along Paths via Power Series Part B


Lecture 23 - Continuity of Coefficients occurring in Families of Power Series defining Analytic Continuations along Paths


Lecture 24 - Analytic Continuability along Paths: Dependence on the Initial Function and on the Path - First Version of the Monodromy Theorem


Lecture 25 - Maximal Domains of Direct and Indirect Analytic Continuation: Second Version of the Monodromy Theorem


Lecture 26 - Deducing the Second (Simply Connected) Version of the Monodromy Theorem from the First (Homotopy) Version


Lecture 27 - Existence and Uniqueness of Analytic Continuations on Nearby Paths


Lecture 28 - Proof of the First (Homotopy) Version of the Monodromy Theorem


Lecture 29 - Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse of an Analytic Function at a Critical Point


Lecture 30 - The Mean-Value Property, Harmonic Functions and the Maximum Principle


Lecture 31 - Proofs of Maximum Principles and Introduction to Schwarz Lemma


Lecture 32 - Proof of Schwarz Lemma and Uniqueness of Riemann Mappings


Lecture 33 - Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains of the Unit Disc


Lecture 34 - Differential or Infinitesimal Schwarzs Lemma, Picks Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc


Lecture 35 - Differential or Infinitesimal Schwarzs Lemma, Picks Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc


Lecture 36 - Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc


Lecture 37 - Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc


Lecture 38 - Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform Sequential Compactness are Equivalent


Lecture 39 - Completion of the Proof of the Arzela-Ascoli Theorem and Introduction to Montels Theorem


Lecture 40 - The Proof of Montels Theorem


Lecture 41 - The Candidate for a Riemann Mapping


Lecture 42 - Completion of Proof of The Riemann Mapping Theorem


Lecture 43 - Completion of Proof of The Riemann Mapping Theorem