| Unit 1: Theorems of Picard, Casorati-Weierstrass and Riemann on Removable Singularities |
| Lecture 01 - Properties of the Image of an Analytic Function: Introduction to the Picard Theorems |
| Lecture 02 - Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable, Pole and Essential Singularities |
| Lecture 03 - Recalling Riemann's Theorem on Removable Singularities |
| Lecture 04 - Casorati-Weierstrass Theorem; Dealing with the Point at Infinity - Riemann Sphere and Riemann Stereographic Projection |
| Unit 2: Neighborhoods of Infinity, Limits at Infinity and Infinite Limits |
| Lecture 05 - Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity |
| Lecture 06 - Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits and Limits at Infinity |
| Unit 3: Infinity as a Point of Analyticity |
| Lecture 07 - When is a Function Analytic at Infinity? |
| Lecture 08 - Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem for the Point at Infinity |
| Lecture 09 - The Generalized Liouville Theorem: Little Brother of Little Picard and Analogue of Casorati-Weierstrass; Failure of Cauchy's Theorem at Infinity |
| Lecture 10 - Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity of Rational and Meromorphic Functions |
| Unit 4: Residue at Infinity and Residue Theorem for the Extended Complex Plane |
| Lecture 11 - Residue at Infinity and Introduction to the Residue Theorem for the Extended Complex Plane: Residue Theorem for the Point at Infinity |
| Lecture 12 - Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane and Applications of the Residue at Infinity |
| Unit 5: The Behavior of Transcendental and Meromorphic Functions at Infinity |
| Lecture 13 - Infinity as an Essential Singularity and Transcendental Entire Functions |
| Lecture 14 - Meromorphic Functions on the Extended Complex Plane are Precisely Quotients of Polynomials |
| Lecture 15 - The Ubiquity of Meromorphic Functions: The Nerves of the Geometric Network Bridging Algebra, Analysis and Topology |
| Lecture 16 - Continuity of Meromorphic Functions at Poles and Topologies of Spaces of Functions |
| Unit 6: Normal Convergence in the Inversion-Invariant Spherical Metric on the Extended Plane |
| Lecture 17 - Why Normal Convergence, but Not Globally Uniform Convergence, is the Inevitable in Complex Analysis |
| Lecture 18 - Measuring Distances to Infinity, the Function Infinity and Normal Convergence of Holomorphic Functions in the Spherical Metric |
| Lecture 19 - The Invariance under Inversion of the Spherical Metric on the Extended Complex Plane |
| Unit 7: Hurwitz Theorems on Normal Limits of Holomorphic and Meromorphic Functions under the Spherical Metric |
| Lecture 20 - Introduction to Hurwitz's Theorem for Normal Convergence of Holomorphic Functions in the Spherical Metric |
| Lecture 21 - Completion of Proof of Hurwitz's Theorem for Normal Limits of Analytic Functions in the Spherical Metric |
| Lecture 22 - Hurwitz's Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric |
| Unit 8: The Inversion-Invariant Spherical Derivative for Meromorphic Functions |
| Lecture 23 - What could the Derivative of a Meromorphic Function Relative to the Spherical Metric Possibly Be? |
| Lecture 24 - Defining the Spherical Derivative of a Meromorphic Function |
| Lecture 25 - Well-definedness of the Spherical Derivative of a Meromorphic Function at a Pole and Inversion-invariance of the Spherical Derivative |
| Unit 9: From Compactness to Boundedness via Equicontinuity |
| Lecture 26 - Topological Preliminaries: Translating Compactness into Boundedness |
| Lecture 27 - Introduction to the Arzela-Ascoli Theorem: Passing from Abstract Compactness to Verifiable Equicontinuity |
| Lecture 28 - Proof of the Arzela-Ascoli Theorem for Functions: Abstract Compactness Implies Equicontinuity |
| Lecture 29 - Proof of the Arzela-Ascoli Theorem for Functions: Equicontinuity Implies Compactness |
| Unit 10: The Montel Theorem - The Holomorphic Avatar of the Arzela-Ascoli Theorem |
| Lecture 30 - Introduction to the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem and Why you get Equicontinuity for Free |
| Lecture 31 - Completion of Proof of the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem |
| Unit 11: The Marty Theorem - The Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems |
| Lecture 32 - Introduction to Marty's Theorem - the Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems |
| Lecture 33 - Proof of One Direction of Marty's Theorem - the Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems - Normal Uniform Boundedness of Spherical Derivatives Implies Normal Sequential Compactness |
| Lecture 34 - Proof of the Other Direction of Marty's Theorem - the Meromorphic Avatar of the Montel and Arzela-Ascoli Theorems - Normal Sequential Compactness Implies Normal Uniform Boundedness of Spherical Derivatives |
| Unit 12: The Hurwitz, Montel and Marty Theorems at Infinity |
| Lecture 35 - Normal Convergence at Infinity and Hurwitz's Theorems for Normal Limits of Analytic and Meromorphic Functions at Infinity |
| Lecture 36 - Normal Sequential Compactness, Normal Uniform Boundedness and Montel's and Marty's Theorems at Infinity |
| Unit 13: Local Analysis of Normality by the Zooming Process and Zalcman's Lemma |
| Lecture 37 - Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma |
| Lecture 38 - Characterizing Normality at a Point by the Zooming Process and the Motivation for Zalcman's Lemma |
| Unit 14: Zalcman's Lemma, Montel's Normality Criterion and Theorems of Picard, Royden and Schottky |
| Lecture 39 - Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma |
| Lecture 40 - Montel's Deep Theorem: The Fundamental Criterion for Normality or Fundamental Normality Test based on Omission of Values |
| Lecture 41 - Proofs of the Great and Little Picard Theorems |
| Lecture 42 - Royden's Theorem on Normality based on Growth of Derivatives |
| Lecture 43 - Schottky's Theorem: Uniform Boundedness from a Point to a Neighbourhood and Problem Solving Session |