Complex Analysis

By Prof. Pranav Haridas   |   Kerala School of Mathematics
Learners enrolled: 1862
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.

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

PREREQUISITES Real Analysis, Linear Algebra

INDUSTRIES  SUPPORT     : Almost all engineering-based companies
Course Status : Completed
Course Type : Core
Duration : 12 weeks
Category :
  • Mathematics
  • Foundations of Mathematics
Credit Points : 3
Level : Undergraduate/Postgraduate
Start Date : 25 Jul 2022
End Date : 14 Oct 2022
Enrollment Ends : 08 Aug 2022
Exam Date : 30 Oct 2022 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1: Algebra and Topology of the complex plane
Week 2:Geometry of the complex plane, Complex differentiation
Week 3: Power series and its convergence, Cauchy-Riemann equations
Week 4: Harmonic functions, Möbius transformations
Week 5:Integration along a contour, The fundamental theorem of calculus
Week 6: Homotopy, Cauchy’s theorem
Week 7: Cauchy integral formula, Cauchy’s inequalities and other consequences
Week 8: Winding number, Open mapping theorem, Schwarz reflection Principle
Week 9: Singularities of a holomorphic function, Laurent series
Week 10: The residue theorem, Argument principle, Rouche’s theorem
Week 11: Branch of the Complex logarithm, Automorphisms of the Unit disk
Week 12: Covering spaces, Picard's theorem

Books and references

  1. Comlex Analysis by Elias M. Stein and Rami Shakarchi
  2. Functions of one complex variable - I by John B Conway
  3. Complex Analysis by Lars Ahlfors
  4. Complex Analysis by Serge Lang

Instructor bio

Prof. Pranav Haridas

Kerala School of Mathematics
The instructor is an Assistant Professor at the Kerala School of Mathematics. His research interests broadly lie in Complex Analysis and more specifically quadrature domains in several complex variables. He is also interested in the study of quasiconformal mappings and Teichmller spaces. He completed his doctoral studies from the Indian Institute of Sciences, Bangalore

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 30 October 2022 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.


Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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