# Complex Analysis

- Institution: IIT Kanpur
- Department: Mathematics
- Teacher: Prof. Kaushik Bal
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111104001)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111104001)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

Complex analysis studies holomorphic functions—complex differentiable everywhere—and their remarkable properties on the Riemann sphere. The course begins with Cauchy's theorem and integral formula, enabling powerful contour integration techniques that simplify real integrals via residues. Key results include Liouville's theorem (bounded entire functions are constant), the maximum modulus principle, and Laurent/Riemann mapping theorems revealing conformal invariance. Applications span physics (fluid dynamics, electrostatics) and pure math (argument principle, Rouche's theorem for zero counting).

Intended audience: Engineering Students (Electrical,Mechanical, Aerospace)

Prerequisites: 10+2

## Lectures

1. [Lecture 1: Complex Numbers and Basic Operations](https://invisiblenetwork.ai/learn/nptel/111104001/f156fb203af5) (YouTube)
2. [Lecture 2: Complex Numbers and Basic Operations II](https://invisiblenetwork.ai/learn/nptel/111104001/40729888fc00) (YouTube)
3. [Lecture 3: Complex Numbers and Basic Operations III](https://invisiblenetwork.ai/learn/nptel/111104001/adc20699f27c) (YouTube)
4. [Lecture 4: Topology of Complex Plane](https://invisiblenetwork.ai/learn/nptel/111104001/de9e206321a4) (YouTube)
5. [Lecture 5: Limit and Continuity](https://invisiblenetwork.ai/learn/nptel/111104001/dfd27854a521) (YouTube)
6. [Lecture 6: Differentiability: Definition and CR Equation](https://invisiblenetwork.ai/learn/nptel/111104001/9ffb2ff35fe3) (YouTube)
7. [Lecture 7: Differentiability: Harmonic Conjugate and Polar Form](https://invisiblenetwork.ai/learn/nptel/111104001/db9be9ef53ad) (YouTube)
8. [Lecture 8: Elementary Functions: Exponential and Trigonometric](https://invisiblenetwork.ai/learn/nptel/111104001/1b3232c3f882) (YouTube)
9. [Lecture 9: Elementary Functions: Complex Logarithm](https://invisiblenetwork.ai/learn/nptel/111104001/e6ef0c64c50b) (YouTube)
10. [Lecture 10: Elementary Functions: Inverse Trigonometric Functions](https://invisiblenetwork.ai/learn/nptel/111104001/066fd4444071) (YouTube)
11. [Lecture 11: Complex Line Integral 1](https://invisiblenetwork.ai/learn/nptel/111104001/a0ea008ec9ef) (YouTube)
12. [Lecture 12: Complex Line Integral 2](https://invisiblenetwork.ai/learn/nptel/111104001/ce43d7e042b8) (YouTube)
13. [Lecture 13: Worked Out Examples](https://invisiblenetwork.ai/learn/nptel/111104001/f1cd0acf8c04) (YouTube)
14. [Lecture 14: Cauchy Integral Theorem](https://invisiblenetwork.ai/learn/nptel/111104001/0b67983c67f6) (YouTube)
15. [Lecture 15: Cauchy-Gourset Theorem for Multiply Connected Domain](https://invisiblenetwork.ai/learn/nptel/111104001/47c9a24213fd) (YouTube)
16. [Lecture 16: Cauchy Integral Formula](https://invisiblenetwork.ai/learn/nptel/111104001/b33358da979c) (YouTube)
17. [Lecture 17: CIF: Examples and Applications](https://invisiblenetwork.ai/learn/nptel/111104001/363f2763dc76) (YouTube)
18. [Lecture 18: Complex Series: Convergence](https://invisiblenetwork.ai/learn/nptel/111104001/40537c34b61d) (YouTube)
19. [Lecture 19: Convergence Test](https://invisiblenetwork.ai/learn/nptel/111104001/df81d1d4082e) (YouTube)
20. [Lecture 20: Power Series](https://invisiblenetwork.ai/learn/nptel/111104001/409d6302ebe9) (YouTube)
21. [Lecture 21: Analytic Function](https://invisiblenetwork.ai/learn/nptel/111104001/67baae225e3b) (YouTube)
22. [Lecture 22: Taylor Series Representation](https://invisiblenetwork.ai/learn/nptel/111104001/c4c0863ddfca) (YouTube)
23. [Lecture 23: Taylor Series Continued](https://invisiblenetwork.ai/learn/nptel/111104001/266a862670c9) (YouTube)
24. [Lecture 24: Laurent Series](https://invisiblenetwork.ai/learn/nptel/111104001/5fe95f43e2ae) (YouTube)
25. [Lecture 25: Laurent Series Continued](https://invisiblenetwork.ai/learn/nptel/111104001/3db0b0516377) (YouTube)
26. [Lecture 26: Residue](https://invisiblenetwork.ai/learn/nptel/111104001/9c7f0eb418df) (YouTube)
27. [Lecture 27: Residue at Infinity](https://invisiblenetwork.ai/learn/nptel/111104001/73a7c4fa3336) (YouTube)
28. [Lecture 28: Types of Singularity](https://invisiblenetwork.ai/learn/nptel/111104001/17568b5f555e) (YouTube)
29. [Lecture 29: Residue Formula](https://invisiblenetwork.ai/learn/nptel/111104001/f279eeabf60d) (YouTube)
30. [Lecture 30: Application of Residue to Real Integral](https://invisiblenetwork.ai/learn/nptel/111104001/a3edcb4ee048) (YouTube)
31. [Lecture 31: Indentation Theorem](https://invisiblenetwork.ai/learn/nptel/111104001/27e1f6c9f888) (YouTube)
32. [Lecture 32: Meromorphic Function and Argument Principle](https://invisiblenetwork.ai/learn/nptel/111104001/91aeab6cc26a) (YouTube)
33. [Lecture 33: Rouche's Theorem](https://invisiblenetwork.ai/learn/nptel/111104001/7ac4c36ad7a0) (YouTube)
34. [Lecture 34: Open Mapping and Maximum Modulus Principle](https://invisiblenetwork.ai/learn/nptel/111104001/d74f3605cba9) (YouTube)
35. [Lecture 35: Mobius Transformation](https://invisiblenetwork.ai/learn/nptel/111104001/865b5cce8981) (YouTube)
36. [Lecture 36: Conformal Map](https://invisiblenetwork.ai/learn/nptel/111104001/46e1b9f1592e) (YouTube)
37. [Lecture 37: Transformation of Harmonic Functions](https://invisiblenetwork.ai/learn/nptel/111104001/7a0a27a227b8) (YouTube)