**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 |