Complex Analysis

By Prof. Pranav Haridas | Kerala School of Mathematics

Learners enrolled: 3808

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

INDUSTRIES SUPPORT : Almost all engineering-based companies

Summary

Course Status :	Completed
Course Type :	Core
Duration :	12 weeks
Category :	Mathematics
Credit Points :	3
Level :	Undergraduate/Postgraduate
Start Date :	14 Sep 2020
End Date :	04 Dec 2020
Enrollment Ends :	25 Sep 2020
Exam Date :	20 Dec 2020 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:Construction and algebra of the complex numbers

Week 2:Geometry of the complex numbers

Week 3: Complex differentiation and power series, Convergence of power series

Week 4:Differentiability and the Cauchy-Riemann equations, Maximum principle

Week 5:Integration along a contour, Integration in rectifiable curves

Week 6:The fundamental theorem of calculus, Integration by parts

Week 7: Homotopy, Cauchy’s theorem

Week 8:Cauchy integral formula, Analytic continuation,

Week 9:Cauchy’s inequalities, Uniform limit of holomorphic functions

Week 10:Winding number, General Cauchy integral formula

Week 11: Singularities of a holomorphic function, Laurent series

Week 12:The residue theorem, Argument principle, Rouche’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:20 December 2020, 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.

CRITERIA TO GET A CERTIFICATE:
• 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.

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