Modules / Lectures

Module Name | Download | Description | Download Size |
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Unit 8: The Inversion-invariant Spherical Derivative for Meromorphic Functions | Mid-Course Exam | Mid-Course Exam | 88 kb |

Unit 14: Zalcman\'s Lemma, Montel\'s Normality Criterion and Theorems of Picard, Royden and Schottky | End-Course Exam | End-Course Exam | 87 kb |

Sl.No | Chapter Name | English |
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1 | Properties of the Image of an Analytic Function: Introduction to the Picard Theorems | PDF unavailable |

2 | Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable, Pole and Essential Singularities | PDF unavailable |

3 | Recalling Riemann's Theorem on Removable Singularities | PDF unavailable |

4 | Casorati-Weierstrass Theorem; Dealing with the Point at Infinity -- Riemann Sphere and Riemann Stereographic Projection | PDF unavailable |

5 | Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity | PDF unavailable |

6 | Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits and Limits at Infinity | PDF unavailable |

7 | When is a function analytic at infinity ? | PDF unavailable |

8 | Laurent Expansion at Infinity and Riemann\'s Removable Singularities Theorem for the Point at Infinity | PDF unavailable |

9 | The Generalized Liouville Theorem: Little Brother of Little Picard and Analogue of Casorati-Weierstrass; Failure of Cauchy\'s Theorem at Infinity | PDF unavailable |

10 | Morera\'s Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity of Rational and Meromorphic Functions | PDF unavailable |

11 | Residue at Infinity and Introduction to the Residue Theorem for the Extended Complex Plane: Residue Theorem for the Point at Infinity | PDF unavailable |

12 | Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane and Applications of the Residue at Infinity | PDF unavailable |

13 | Infinity as an Essential Singularity and Transcendental Entire Functions | PDF unavailable |

14 | Meromorphic Functions on the Extended Complex Plane are Precisely Quotients of Polynomials | PDF unavailable |

15 | The Ubiquity of Meromorphic Functions: The Nerves of the Geometric Network Bridging Algebra, Analysis and Topology | PDF unavailable |

16 | Continuity of Meromorphic Functions at Poles and Topologies of Spaces of Functions | PDF unavailable |

17 | Why Normal Convergence, but Not Globally Uniform Convergence, is the Inevitable in Complex Analysis | PDF unavailable |

18 | Measuring Distances to Infinity, the Function Infinity and Normal Convergence of Holomorphic Functions in the Spherical Metric | PDF unavailable |

19 | The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane | PDF unavailable |

20 | Introduction to Hurwitz\'s Theorem for Normal Convergence of Holomorphic Functions in the Spherical Metric | PDF unavailable |

21 | Completion of Proof of Hurwitz\'s Theorem for Normal Limits of Analytic Functions in the Spherical Metric | PDF unavailable |

22 | Hurwitz\'s Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric | PDF unavailable |

23 | What could the Derivative of a Meromorphic Function Relative to the Spherical Metric Possibly Be ? | PDF unavailable |

24 | Defining the Spherical Derivative of a Meromorphic Function | PDF unavailable |

25 | Well-definedness of the Spherical Derivative of a Meromorphic Function at a Pole and Inversion-invariance of the Spherical Derivative | PDF unavailable |

26 | Topological Preliminaries: Translating Compactness into Boundedness | PDF unavailable |

27 | Introduction to the Arzela-Ascoli Theorem: Passing from abstract Compactness to verifiable Equicontinuity | PDF unavailable |

28 | Proof of the Arzela-Ascoli Theorem for Functions: Abstract Compactness Implies Equicontinuity | PDF unavailable |

29 | Proof of the Arzela-Ascoli Theorem for Functions: Equicontinuity Implies Compactness | PDF unavailable |

30 | Introduction to the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem & Why you get Equicontinuity for Free | PDF unavailable |

31 | Completion of Proof of the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem | PDF unavailable |

32 | Introduction to Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems | PDF unavailable |

33 | Proof of one direction of Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems - Normal Uniform Boundedness of Spherical Derivatives Implies Normal Sequential Compactness | PDF unavailable |

34 | Proof of the other direction of Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems - Normal Sequential Compactness Implies Normal Uniform Boundedness of Spherical Derivatives | PDF unavailable |

35 | Normal Convergence at Infinity and Hurwitz\'s Theorems for Normal Limits of Analytic and Meromorphic Functions at Infinity | PDF unavailable |

36 | Normal Sequential Compactness, Normal Uniform Boundedness and Montel\'s & Marty\'s Theorems at Infinity | PDF unavailable |

37 | Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma | PDF unavailable |

38 | Characterizing Normality at a Point by the Zooming Process and the Motivation for Zalcman\'s Lemma | PDF unavailable |

39 | Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s Lemma | PDF unavailable |

40 | Montel\'s Deep Theorem: The Fundamental Criterion for Normality or Fundamental Normality Test based on Omission of Values | PDF unavailable |

41 | Proofs of the Great and Little Picard Theorems | PDF unavailable |

42 | Royden\'s Theorem on Normality Based On Growth Of Derivatives | PDF unavailable |

43 | Schottky\'s Theorem: Uniform Boundedness from a Point to a Neighbourhood & Problem Solving Session | PDF unavailable |

Sl.No | Language | Book link |
---|---|---|

1 | English | Not Available |

2 | Bengali | Not Available |

3 | Gujarati | Not Available |

4 | Hindi | Not Available |

5 | Kannada | Not Available |

6 | Malayalam | Not Available |

7 | Marathi | Not Available |

8 | Tamil | Not Available |

9 | Telugu | Not Available |