Modules / Lectures
Module NameDownloadDescriptionDownload Size
Unit 8: The Inversion-invariant Spherical Derivative for Meromorphic FunctionsMid-Course ExamMid-Course Exam88 kb
Unit 14: Zalcman\'s Lemma, Montel\'s Normality Criterion and Theorems of Picard, Royden and SchottkyEnd-Course ExamEnd-Course Exam87 kb

Sl.No Chapter Name English
1Properties of the Image of an Analytic Function: Introduction to the Picard TheoremsPDF unavailable
2Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable, Pole and Essential SingularitiesPDF unavailable
3Recalling Riemann's Theorem on Removable SingularitiesPDF unavailable
4Casorati-Weierstrass Theorem; Dealing with the Point at Infinity -- Riemann Sphere and Riemann Stereographic Projection PDF unavailable
5Neighborhood of Infinity, Limit at Infinity and Infinity as an Isolated SingularityPDF unavailable
6Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits and Limits at InfinityPDF unavailable
7When is a function analytic at infinity ?PDF unavailable
8Laurent Expansion at Infinity and Riemann\'s Removable Singularities Theorem for the Point at InfinityPDF unavailable
9The Generalized Liouville Theorem: Little Brother of Little Picard and Analogue of Casorati-Weierstrass; Failure of Cauchy\'s Theorem at InfinityPDF unavailable
10Morera\'s Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity of Rational and Meromorphic FunctionsPDF unavailable
11Residue at Infinity and Introduction to the Residue Theorem for the Extended Complex Plane: Residue Theorem for the Point at InfinityPDF unavailable
12Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane and Applications of the Residue at InfinityPDF unavailable
13Infinity as an Essential Singularity and Transcendental Entire FunctionsPDF unavailable
14Meromorphic Functions on the Extended Complex Plane are Precisely Quotients of PolynomialsPDF unavailable
15The Ubiquity of Meromorphic Functions: The Nerves of the Geometric Network Bridging Algebra, Analysis and TopologyPDF unavailable
16Continuity of Meromorphic Functions at Poles and Topologies of Spaces of FunctionsPDF unavailable
17Why Normal Convergence, but Not Globally Uniform Convergence, is the Inevitable in Complex AnalysisPDF unavailable
18Measuring Distances to Infinity, the Function Infinity and Normal Convergence of Holomorphic Functions in the Spherical MetricPDF unavailable
19The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane PDF unavailable
20Introduction to Hurwitz\'s Theorem for Normal Convergence of Holomorphic Functions in the Spherical MetricPDF unavailable
21Completion of Proof of Hurwitz\'s Theorem for Normal Limits of Analytic Functions in the Spherical MetricPDF unavailable
22Hurwitz\'s Theorem for Normal Limits of Meromorphic Functions in the Spherical MetricPDF unavailable
23What could the Derivative of a Meromorphic Function Relative to the Spherical Metric Possibly Be ?PDF unavailable
24Defining the Spherical Derivative of a Meromorphic FunctionPDF unavailable
25Well-definedness of the Spherical Derivative of a Meromorphic Function at a Pole and Inversion-invariance of the Spherical DerivativePDF unavailable
26Topological Preliminaries: Translating Compactness into BoundednessPDF unavailable
27Introduction to the Arzela-Ascoli Theorem: Passing from abstract Compactness to verifiable EquicontinuityPDF unavailable
28Proof of the Arzela-Ascoli Theorem for Functions: Abstract Compactness Implies EquicontinuityPDF unavailable
29Proof of the Arzela-Ascoli Theorem for Functions: Equicontinuity Implies CompactnessPDF unavailable
30Introduction to the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli Theorem & Why you get Equicontinuity for FreePDF unavailable
31Completion of Proof of the Montel Theorem - the Holomorphic Avatar of the Arzela-Ascoli TheoremPDF unavailable
32Introduction to Marty\'s Theorem - the Meromorphic Avatar of the Montel & Arzela-Ascoli Theorems PDF unavailable
33Proof 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
34Proof 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 DerivativesPDF unavailable
35Normal Convergence at Infinity and Hurwitz\'s Theorems for Normal Limits of Analytic and Meromorphic Functions at InfinityPDF unavailable
36Normal Sequential Compactness, Normal Uniform Boundedness and Montel\'s & Marty\'s Theorems at InfinityPDF unavailable
37Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s LemmaPDF unavailable
38Characterizing Normality at a Point by the Zooming Process and the Motivation for Zalcman\'s LemmaPDF unavailable
39Local Analysis of Normality and the Zooming Process - Motivation for Zalcman\'s LemmaPDF unavailable
40Montel\'s Deep Theorem: The Fundamental Criterion for Normality or Fundamental Normality Test based on Omission of ValuesPDF unavailable
41Proofs of the Great and Little Picard TheoremsPDF unavailable
42Royden\'s Theorem on Normality Based On Growth Of DerivativesPDF unavailable
43Schottky\'s Theorem: Uniform Boundedness from a Point to a Neighbourhood & Problem Solving SessionPDF unavailable


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