. A 1-point poly-quadrature domain of order 1 not biholomorphic to a complete circular domain. Analysis and Mathematical Physics, 2018.


. A note on the smoothness of the Minkowski function. arXiv preprint arXiv:1805.11023, 2018.


. Proper holomorphic mappings onto symmetric products of a Riemann surface. Doc. Math., 2018.


. Schwarz lemmas via the pluricomplex Green's function. arXiv preprint arXiv:1812.10337; J. Geom. Anal., to appear, 2018.

. Finiteness theorems for holomorphic mapping from products of hyperbolic Riemann surfaces. Internat. J. Math., 2017.


. Proper holomorphic mappings of balanced domains in $ℂ^n$. Math. Z., 2015.


. Proper holomorphic mappings between hyperbolic product manifolds. Internat. J. Math., 2014.


. Proper holomorphic maps between bounded symmetric domains revisited. Pacific J. Math., 2014.



Hyperbolicity in Complex Analysis
Aug 23, 2018 3:00 PM
The Alexander phenomenon
May 24, 2017 3:00 PM
Proper Holomorphic Mappings of Balanced Domains
Nov 16, 2016 2:00 PM
Continuous Computation and Applications
Mar 4, 2016 11:00 AM


“The function of the expert is not to be more right than other people, but to be wrong for more sophisticated reasons.” — …

Note: This is a brief summary of my understanding. I am neither a statistician nor a differential psychologist. As my understanding …


Jan – May 2019

Transformation Techniques MA5460

Announcements and notifications



Jan – May 2018

Complex Analysis MA5360

Announcements and notifications

  • First class test from Assignment 1 on Tuesday, 06-02-2018.
  • You can come and discuss any questions/clarifications on Assignment 1 on Thursday 01-02-2018 between 2 and 3:30 PM.
  • We will have the first non-optional problem-solving session on Thursday 25-01-2018.
  • Course Information -- Read this first.



Jan -- May 2017

Series and Matrics MA1102

Jul -- Nov 2016

Multivariable Calculus MA5371

Announcements and notifications




Jan – May 2016

Complex Analysis MA5360

Announcements and notifications 

  • The final class of the course will be on 22-04-2016 at 11:00 AM.  There will be an optional doublt-clearing session on 26-04-2016 (Tuesday) at 11:00 AM.
  • Please submit problems 1,3, and 6 of Assignment 4 by 26-04-2016 (Tuesday).
  • Please submit problem 1,3 and 4 of Assignment 3 on 29-03-2016 (Tuesday).
  • Solutions to Quiz 2.
  • Reminder: Quiz 2 on Saturday 19-03-2016.
  • The class on 18-03-2016 (Friday) will be at 10:00 AM.
  • For Problem 5 in Assignment 2, you may assume, without proof, that the function f has a power series expansion about 0 with radius of convergence infinity.
  • Please submit Problems 1, 2 and 5 of Assignment 2 on 09-03-2016 (Wednesday).
  • There will be no lecture on March 4th (Friday). Instead there will be an optional lecture on the proof of Green’s theorem on March 10th (Thursday) 9:00-10:00 AM.
  • The second quiz will be held on 19th March (Saturday) from 9:30 - 11:00 AM. Venue will be announced later. The syllabus for the quiz will Chapters 2 and 3 with the exception of the proof of Green’s theorem.
  • I will be done correcting Quiz 1 this weekend. Please come to my office between 10:00 AM to 4:00 PM on Monday to have a look at your corrected answer books.
  • There will be an extra class on 27-02-2016 (Saturday) at 10:00 AM. I will lecture on vector fields.
  • There is no lecture on 19-02-2016 (Friday). I will announce when there will be a makeup lecture at a later time.
  • The special optional class on Riemann Integration scheduled on 18-02-2016 (Thursday) has been postponed to 25-02-2016 (Thurday) after the regular class. Please have a look at some of the material (especially the slides) posted in the miscellanous section before you come for this class.
  • Solutions to Quiz 1.
  • A short proof that the winding number is a continuous function.
  • The first quiz will be on 18-02-2016 (Thursday). The topics for the quiz will be Chapter 1 and the part of Chapter 2 dealing with holomorphic functions.
  • On 11-02-2016 (Thursday), we will have an informal and optional discussion session after class from 9 AM onwards.
  • There is an error in Problem 1 of Assignment 1. I have posted a corrected copy.
  • Please submit Problems 1, 2 and 6 of Assignment 1 on 09-02-2016 (Tuesday) 16-02-2016 (Tuesday).
  • Outline of solution of Problem 6 of Assignment 1. The due date for Assignment 1 is 09-02-2016 (Tuesday). On Monday, I will announce which problem you have to submit. Problem 6 is part of this.
  • There is an error in question 9 of Assignment 1. I have uploaded a fixed copy of the assignment.
  • We will have an optional and informal discussion session on 04-02-2016 (Thursday) after class from 9 AM onwards. Please work out the questions in the assignments and let me know the questions in which you are having difficulties.
  • There will be no lecture on 21-01-2016 (Thursday). Instead the lecture on 22-01-2016 (Friday) will be 90 minutes long starting at 2:00 PM.
  • Course Information — Read this first.

Lecture notes

  1. Arithmetic, Geometry and Topology the Complex Plane – Draft updated and proof checked on 29-01-2016. Please report any errors that remain. This is an incomplete draft version probably filled with errors. I shall upload a complete proof-checked version in the next few days.

  2. Functions of a Complex Variable – Draft updated on 11-03-2016. Still requires some minor changes. Incomplete draft. The first section has been proof-checked and is what you need for Quiz 1.

  3. Complex Integration – Almost complete draft. Requires proof checking.

  4. Local Properties – Almost complete draft. Requires proof checking.

  5. Please wait!

  6. Harmonic Functions – Complete draft.


  1. Assignment 1 – Due date 09-02-2016.

  2. Assignment 2 – Due date 01-03-2016.

  3. Assignment 3 – Due date 29-03-2016.

  4. Assignment 4 – Due date 26-04-2016.


A proof of Liouville’s theorem by Edward Nelson. Edward Nelosn’s short proof of Liouville’s theorem for Harmonic functions.

Some Simplifications in Basic Complex Analysis by Oswaldo Rio Branco de Oliveira A systematic and simple development of the local theory of holomorphic functions.

The 3 stooges of vector calculus and their impersonators: A viewer’s guide to the classic episodes by J Buskin, P Prosapio, SA Taylor. An article where Green’s theorem is proved in a different way.

Constructing Möbius Transformations with Spheres by Rob Siliciano. A nice paper that covers material about rigid motions and linear fractional transformations.

Möbius Transformations Revealed:- An excellent video to understand the geoemtry of linear fractional transformations. Please read the companion article also.

Stereographic Projection by Bill Casselman. A nice treatment of Stereographic projection from the viewpoint of classical geometry.

Slides on the Riemann–Stieltjes integral by Dr. Aditya Kaushik

  1. Lecture 1 — Definitions and basic properties.
  2. Lecture 2 — More properties.
  3. Lecture 3 — Fundamental theorem of Calculus.

Here are some resources for learning Real Analysis rigoroursly:

  1. A companion to Rudin’s Principles of Mathematical Analysis by Professor Evelyn Silvia . Excellent set of notes that will teach you how to think and write mathematics clearly. Highly recommended!
  2. Real Analysis Lectures by Professor Francis Su. Excellent video lectures that follows Rudin’s textbook. Learnstream Rudinium is also very useful.

The Wikipedia article on the winding number has nice animated gif explaning the winding number. I am reproducing it here.

Winding Number

Many students wanted a book with a lot of worked out problems. Andreas Kleefeld has a written a partial solution manual to Conway’s book on Complex Analysis. There are more than a hundred problems worked out in this solution manual. You can also have a look at A Complex Analysis Problem Book by Daniel Alpay for even more worked out problems.

When is a Function that Satisfies the Cauchy-Riemann Equations Analytic? by J. D. Gray and S. A. Morris. A very nice paper that gives a brief survey of the many results obtained on the question. 

Avinash Sathaye has given many courses on the history of mathematics (especially contributions by Indian mathematicians). Here are handouts that deal with the history of the solution of equations from a course he gave at the Universty of Kentucky. 

  1.  al-Khwārizmī, Quadratic Equations, and the Birth of Algebra

  2.  Cardano’s Ars Magna

  3.  Cubics, Trigonometric Methods, and Angle Trisection

  4. The Cubic and Quartic from Bombelli to Euler

  5. The Theory of Equations from Cardano to Galois

Aug – Oct 2015

I gave a series of lectures on Manifolds. Here are lecture notes.