# Advanced Complex Analysis - Part 1:Zeros of Analytic Functions,Analytic continuation, Monodromy, Hyperbolic Geometry and the Reimann Mapping Theorem

- Institution: IIT Madras
- Department: Mathematics
- Teacher: Dr. T.E. Venkata Balaji
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111106084)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111106084)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

## Lectures

1. [Fundamental Theorems Connected with Zeros of Analytic Functions](https://invisiblenetwork.ai/learn/nptel/111106084/44efcb74e559) (Video)
2. [The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem of Algebra](https://invisiblenetwork.ai/learn/nptel/111106084/a1dbdae7ba3e) (Video)
3. [Morera's Theorem and Normal Limits of Analytic Functions](https://invisiblenetwork.ai/learn/nptel/111106084/f2ac42f0a078) (Video)
4. [Hurwitz's Theorem and Normal Limits of Univalent Functions](https://invisiblenetwork.ai/learn/nptel/111106084/0fb475f8cf5b) (Video)
5. [Local Constancy of Multiplicities of Assumed Values](https://invisiblenetwork.ai/learn/nptel/111106084/f2b5eeb214a9) (Video)
6. [The Open Mapping Theorem](https://invisiblenetwork.ai/learn/nptel/111106084/d995ffa68ce4) (Video)
7. [Introduction to the Inverse Function Theorem](https://invisiblenetwork.ai/learn/nptel/111106084/ed7682d7db9e) (Video)
8. [Completion of the Proof of the Inverse Function Theorem: The Integral Inversion Formula for the Inverse Function](https://invisiblenetwork.ai/learn/nptel/111106084/b6edc0794f8f) (Video)
9. [Univalent Analytic Functions have never-zero Derivatives and are Analytic Isomorphisms](https://invisiblenetwork.ai/learn/nptel/111106084/fc5241b4f01b) (Video)
10. [Introduction to the Implicit Function Theorem](https://invisiblenetwork.ai/learn/nptel/111106084/d3b300bb5dcf) (Video)
11. [Proof of the Implicit Function Theorem: Topological Preliminaries](https://invisiblenetwork.ai/learn/nptel/111106084/fc31cc6d2c00) (Video)
12. [Proof of the Implicit Function Theorem: The Integral Formula for & Analyticity of the Explicit Function](https://invisiblenetwork.ai/learn/nptel/111106084/7f192d4a78fe) (Video)
13. [Doing Complex Analysis on a Real Surface: The Idea of a Riemann Surface](https://invisiblenetwork.ai/learn/nptel/111106084/c32c39c0ceec) (Video)
14. [F(z,w)=0 is naturally a Riemann Surface](https://invisiblenetwork.ai/learn/nptel/111106084/38be61405ea1) (Video)
15. [Constructing the Riemann Surface for the Complex Logarithm](https://invisiblenetwork.ai/learn/nptel/111106084/ae8acbbe19f6) (Video)
16. [Constructing the Riemann Surface for the m-th root function](https://invisiblenetwork.ai/learn/nptel/111106084/2bdcf7bf1f45) (Video)
17. [The Riemann Surface for the functional inverse of an analytic mapping at a critical point](https://invisiblenetwork.ai/learn/nptel/111106084/9d211a704568) (Video)
18. [The Algebraic nature of the functional inverses of an analytic mapping at a critical point](https://invisiblenetwork.ai/learn/nptel/111106084/7c40ce815a61) (Video)
19. [The Idea of a Direct Analytic Continuation or an Analytic Extension](https://invisiblenetwork.ai/learn/nptel/111106084/1a5fc9a2457b) (Video)
20. [General or Indirect Analytic Continuation and the Lipschitz Nature of the Radius of Convergence](https://invisiblenetwork.ai/learn/nptel/111106084/17ecbc17a163) (Video)
21. [Analytic Continuation Along Paths via Power Series Part A](https://invisiblenetwork.ai/learn/nptel/111106084/9e8081e7965e) (Video)
22. [Analytic Continuation Along Paths via Power Series Part B](https://invisiblenetwork.ai/learn/nptel/111106084/b07267a5b44e) (Video)
23. [Continuity of Coefficients occurring in Families of Power Series defining Analytic Continuations along Paths](https://invisiblenetwork.ai/learn/nptel/111106084/0dd3c4ec6894) (Video)
24. [Analytic Continuability along Paths: Dependence on the Initial Function and on the Path - First Version of the Monodromy Theorem](https://invisiblenetwork.ai/learn/nptel/111106084/e433b262912d) (Video)
25. [Maximal Domains of Direct and Indirect Analytic Continuation: SecondVersion of the Monodromy Theorem](https://invisiblenetwork.ai/learn/nptel/111106084/05875a191163) (Video)
26. [Deducing the Second (Simply Connected) Version of the Monodromy Theorem from the First (Homotopy) Version](https://invisiblenetwork.ai/learn/nptel/111106084/2aa056ca4ab9) (Video)
27. [Existence and Uniqueness of Analytic Continuations on Nearby Paths](https://invisiblenetwork.ai/learn/nptel/111106084/43f7bf8b5e4c) (Video)
28. [Proof of the First (Homotopy) Version of the Monodromy Theorem](https://invisiblenetwork.ai/learn/nptel/111106084/8ae443d9d9b6) (Video)
29. [Proof of the Algebraic Nature of Analytic Branches of the Functional Inverse of an Analytic Function at a Critical Point](https://invisiblenetwork.ai/learn/nptel/111106084/503fb2345ea5) (Video)
30. [The Mean-Value Property, Harmonic Functions and the Maximum Principle](https://invisiblenetwork.ai/learn/nptel/111106084/96ba97889e1c) (Video)
31. [Proofs of Maximum Principles and Introduction to Schwarz Lemma](https://invisiblenetwork.ai/learn/nptel/111106084/25adf2a28430) (Video)
32. [Proof of Schwarz Lemma and Uniqueness of Riemann Mappings](https://invisiblenetwork.ai/learn/nptel/111106084/0362a5a7dff9) (Video)
33. [Reducing Existence of Riemann Mappings to Hyperbolic Geometry of Sub-domains of the Unit Disc](https://invisiblenetwork.ai/learn/nptel/111106084/f29738631b41) (Video)
34. [Differential or Infinitesimal Schwarzs Lemma, Picks Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc](https://invisiblenetwork.ai/learn/nptel/111106084/f093cd00b932) (Video)
35. [Differential or Infinitesimal Schwarzs Lemma, Picks Lemma, Hyperbolic Arclengths, Metric and Geodesics on the Unit Disc.](https://invisiblenetwork.ai/learn/nptel/111106084/4d32f98a6826) (Video)
36. [Hyperbolic Geodesics for the Hyperbolic Metric on the Unit Disc](https://invisiblenetwork.ai/learn/nptel/111106084/aeeffa626cfb) (Video)
37. [Schwarz-Pick Lemma for the Hyperbolic Metric on the Unit Disc](https://invisiblenetwork.ai/learn/nptel/111106084/97dabdae24cd) (Video)
38. [Arzela-Ascoli Theorem: Under Uniform Boundedness, Equicontinuity and Uniform Sequential Compactness are Equivalent](https://invisiblenetwork.ai/learn/nptel/111106084/7b583aff17ab) (Video)
39. [Completion of the Proof of the Arzela-Ascoli Theorem and Introduction to Montels Theorem](https://invisiblenetwork.ai/learn/nptel/111106084/beeb22401009) (Video)
40. [The Proof of Montels Theorem](https://invisiblenetwork.ai/learn/nptel/111106084/c342c4515b09) (Video)
41. [The Candidate for a Riemann Mapping](https://invisiblenetwork.ai/learn/nptel/111106084/99473cce5a8b) (Video)
42. [Completion of Proof of The Riemann Mapping Theorem](https://invisiblenetwork.ai/learn/nptel/111106084/4244f4c414e8) (Video)
43. [Completion of Proof of The Riemann Mapping Theorem.](https://invisiblenetwork.ai/learn/nptel/111106084/b7a851d95b3f) (Video)