# Complex Analysis - IITM

- Institution: Kerala School of Mathematics
- Department: Mathematics
- Teacher: Prof. Pranav Haridas
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111106141)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111106141)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

This is a first course in Complex Analysis focussing on holomorphic functions and its basic properties like Cauchy’s theorem and residue theorems, the classification of singularities, and the maximum principle. We shall study the singularities of holomorphic functions. If time permits, we shall also study Branches of the complex logarithm through covering spaces and attempt proving Picard’s theorem.

Intended audience: Third year Undergraduate or first year Master’s students in various universities.

Prerequisites: Real Analysis, Linear Algebra

## Lectures

1. [Field of Complex Numbers](https://invisiblenetwork.ai/learn/nptel/111106141/c1ce432a61ac) (YouTube)
2. [Conjugation and Absolute value](https://invisiblenetwork.ai/learn/nptel/111106141/ede437c94f35) (YouTube)
3. [Topology on Complex plane](https://invisiblenetwork.ai/learn/nptel/111106141/422bb5ac14cb) (YouTube)
4. [Topology on Complex Plane (Contd)](https://invisiblenetwork.ai/learn/nptel/111106141/0394c87076aa) (YouTube)
5. [Problem Session](https://invisiblenetwork.ai/learn/nptel/111106141/065d3083a088) (YouTube)
6. [Isometries on the Complex Plane](https://invisiblenetwork.ai/learn/nptel/111106141/05639d2747b8) (YouTube)
7. [Functions on the Complex Plane](https://invisiblenetwork.ai/learn/nptel/111106141/e7d63af2ccfa) (YouTube)
8. [Complex differentiability](https://invisiblenetwork.ai/learn/nptel/111106141/7ff850b44dd9) (YouTube)
9. [Power Series](https://invisiblenetwork.ai/learn/nptel/111106141/1ebf79f26d84) (YouTube)
10. [Differentiation of power series](https://invisiblenetwork.ai/learn/nptel/111106141/36f7bc0bacdd) (YouTube)
11. [Problem Session](https://invisiblenetwork.ai/learn/nptel/111106141/1a61f5200a1c) (YouTube)
12. [Cauchy-Riemann equations](https://invisiblenetwork.ai/learn/nptel/111106141/6d67a6d197ed) (YouTube)
13. [Harmonic functions](https://invisiblenetwork.ai/learn/nptel/111106141/42266e1bfe90) (YouTube)
14. [Möbius transformations](https://invisiblenetwork.ai/learn/nptel/111106141/d45b6d196cc8) (YouTube)
15. [Problem session](https://invisiblenetwork.ai/learn/nptel/111106141/e25a555dcb9a) (YouTube)
16. [Curves in the complex plane](https://invisiblenetwork.ai/learn/nptel/111106141/614a6f9be91f) (YouTube)
17. [Complex Integration over curves](https://invisiblenetwork.ai/learn/nptel/111106141/a4cebf2bbfc5) (YouTube)
18. [First Fundamental theorem of Calculus](https://invisiblenetwork.ai/learn/nptel/111106141/5dc7564f30d7) (YouTube)
19. [Second Fundamental theorem of Calculus](https://invisiblenetwork.ai/learn/nptel/111106141/18aa67e2d30c) (YouTube)
20. [Problem session](https://invisiblenetwork.ai/learn/nptel/111106141/335494853ba8) (YouTube)
21. [Homotopy of curves](https://invisiblenetwork.ai/learn/nptel/111106141/086f6046a9ef) (YouTube)
22. [Cauchy-Goursat theorem](https://invisiblenetwork.ai/learn/nptel/111106141/d9bd2d878a75) (YouTube)
23. [Cauchy's theorem](https://invisiblenetwork.ai/learn/nptel/111106141/189c7a1ca42c) (YouTube)
24. [Problem Session](https://invisiblenetwork.ai/learn/nptel/111106141/ac7a7236dfd5) (YouTube)
25. [Cauchy Integral Formula](https://invisiblenetwork.ai/learn/nptel/111106141/107cbe76076c) (YouTube)
26. [Principle of analytic continuation and Cauchy estimates](https://invisiblenetwork.ai/learn/nptel/111106141/1b0ad36ff2c1) (YouTube)
27. [Further consequences of Cauchy Integral Formula](https://invisiblenetwork.ai/learn/nptel/111106141/57279e36bd3c) (YouTube)
28. [Problem session](https://invisiblenetwork.ai/learn/nptel/111106141/d945297d28a0) (YouTube)
29. [Winding number](https://invisiblenetwork.ai/learn/nptel/111106141/f227d326b834) (YouTube)
30. [Open mapping theorem](https://invisiblenetwork.ai/learn/nptel/111106141/f90071f187f3) (YouTube)
31. [Schwarz reflection principle](https://invisiblenetwork.ai/learn/nptel/111106141/53f5d703cb71) (YouTube)
32. [Problem session](https://invisiblenetwork.ai/learn/nptel/111106141/de8f52070e3b) (YouTube)
33. [Singularities of a holomorphic function](https://invisiblenetwork.ai/learn/nptel/111106141/3a5b0fe6a21a) (YouTube)
34. [Pole of a function](https://invisiblenetwork.ai/learn/nptel/111106141/e18ae9193aa9) (YouTube)
35. [Laurent Series](https://invisiblenetwork.ai/learn/nptel/111106141/50c96b5e6dee) (YouTube)
36. [Casorati Weierstrass theorem](https://invisiblenetwork.ai/learn/nptel/111106141/2ad10e6a5edf) (YouTube)
37. [Problem Session](https://invisiblenetwork.ai/learn/nptel/111106141/e3efd8e8d4db) (YouTube)
38. [Residue theorem](https://invisiblenetwork.ai/learn/nptel/111106141/85ea8e6bb00b) (YouTube)
39. [Argument principle](https://invisiblenetwork.ai/learn/nptel/111106141/908e47fb7b91) (YouTube)
40. [Problem Session](https://invisiblenetwork.ai/learn/nptel/111106141/0cde1083893c) (YouTube)
41. [Branch of the Complex logarithm](https://invisiblenetwork.ai/learn/nptel/111106141/a2cca0efe6f3) (YouTube)
42. [Automorphisms of the Unit disk](https://invisiblenetwork.ai/learn/nptel/111106141/311bf422653e) (YouTube)
43. [Phragmen Lindelof method](https://invisiblenetwork.ai/learn/nptel/111106141/572aa40d6aaa) (YouTube)
44. [Problem Session](https://invisiblenetwork.ai/learn/nptel/111106141/22756870d7e0) (YouTube)
45. [Lifting of maps](https://invisiblenetwork.ai/learn/nptel/111106141/0ea9cbdd4011) (YouTube)
46. [Covering spaces](https://invisiblenetwork.ai/learn/nptel/111106141/1ae30daacbaf) (YouTube)
47. [Bloch's theorem](https://invisiblenetwork.ai/learn/nptel/111106141/af148a9cdbc8) (YouTube)
48. [Little Picard's theorem](https://invisiblenetwork.ai/learn/nptel/111106141/552c57f79bda) (YouTube)