Course11110400137 lectures
Complex Analysis
Prof. Kaushik Bal · IIT Kanpur
About
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
Week 1
- Lecture 1: Complex Numbers and Basic Operations
- Lecture 2: Complex Numbers and Basic Operations II
- Lecture 3: Complex Numbers and Basic Operations III
- Lecture 4: Topology of Complex Plane
Week 2
- Lecture 5: Limit and Continuity
- Lecture 6: Differentiability: Definition and CR Equation
- Lecture 7: Differentiability: Harmonic Conjugate and Polar Form
Week 3
- Lecture 8: Elementary Functions: Exponential and Trigonometric
- Lecture 9: Elementary Functions: Complex Logarithm
- Lecture 10: Elementary Functions: Inverse Trigonometric Functions
Week 4
Week 5
- Lecture 13: Worked Out Examples
- Lecture 14: Cauchy Integral Theorem
- Lecture 15: Cauchy-Gourset Theorem for Multiply Connected Domain
Week 6
- Lecture 16: Cauchy Integral Formula
- Lecture 17: CIF: Examples and Applications
- Lecture 18: Complex Series: Convergence
Week 7
Week 8
- Lecture 22: Taylor Series Representation
- Lecture 23: Taylor Series Continued
- Lecture 24: Laurent Series
Week 9
- Lecture 25: Laurent Series Continued
- Lecture 26: Residue
- Lecture 27: Residue at Infinity
- Lecture 28: Types of Singularity
Week 10
- Lecture 29: Residue Formula
- Lecture 30: Application of Residue to Real Integral
- Lecture 31: Indentation Theorem
Week 11
- Lecture 32: Meromorphic Function and Argument Principle
- Lecture 33: Rouche's Theorem
- Lecture 34: Open Mapping and Maximum Modulus Principle
Week 12
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IIT Kanpur
- Department
- Mathematics
- Teacher
- Prof. Kaushik Bal
- Level
- Undergraduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed