1. Fundamental Theorems Connected with Zeros of Analytic Functions
2. The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem
3. Morera's Theorem and Normal Limits of Analytic Functions
4. Hurwitz's Theorem and Normal Limits of Univalent Functions
5. Local Constancy of Multiplicities of Assumed Values
6. The Open Mapping Theorem
7. Introduction to the Inverse Function Theorem
8. Completion of the Proof of the Inverse Function Theorem: The Integral Inversion
9. Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms
10. Introduction to the Implicit Function Theorem
11. Proof of the Implicit Function Theorem: Topological Preliminaries
12. Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity
13. Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface
14. F(z,w)=0 is naturally a Riemann Surface
15. Constructing the Riemann Surface for the Complex Logarithm
16. Constructing the Riemann Surface for the m-th root function
17. The Riemann Surface for the functional inverse of an analytic
18. The Algebraic nature of the functional inverses of an analytic
19. The Idea of a Direct Analytic Continuation or an Analytic Extension
20. General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius
21. Analytic Continuation Along Paths via Power Series Part A
22. Analytic Continuation Along Paths via Power Series Part B
23. Continuity of Coefficients occurring in Families of Power Series defining Analytic
24. Analytic Continuability along Paths: Dependence on the Initial Function
25. Maximal Domains of Direct and Indirect Analytic Continuation: Second
26. Deducing the Second (Simply Connected) Version of the Monodromy Theorem
27. Existence and Uniqueness of Analytic Continuations on Nearby Paths
28. Proof of the First (Homotopy) Version of the Monodromy Theorem
29. Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse
30. The Mean-Value Property, Harmonic Functions and the Maximum Principle
31. Proofs of Maximum Principles and Introduction to Schwarz's Lemma
32. Proof of Schwarz's Lemma and Uniqueness of Riemann Mappings
33. Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains
34. Part A: Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic
35. Part B: Differential or Infinitesimal Schwarz's Lemma, Pick's Lemma, Hyperbolic
36. Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc
37. Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc
38. Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform
39. Completion of the Proof of the Arzela-Ascoli Theorem and Introduction
40. The Proof of Montel's Theorem
41. The Candidate for a Riemann Mapping
42. Completion of Proof of The Riemann Mapping Theorem - Part A
43. Completion of Proof of The Riemann Mapping Theorem - Part B

This is the first part of a series of lectures on advanced topics in Complex Analysis. By advanced, we mean topics that are not (or just barely) touched upon in a first course on Complex Analysis. The theme of the course is to study zeros of analytic (or holomorphic) functions and related theorems. These include the theorems of Hurwitz and Rouche, the Open Mapping theorem, the Inverse and Implicit Function theorems, applications of those theorems, behaviour at a critical point, analytic branches, constructing Riemann surfaces for functional inverses, Analytic continuation and Monodromy, Hyperbolic geometry and the Riemann Mapping theorem.

Check the official website for a detailed course outline: http://nptel.ac.in/syllabus/111106084/