Complex Analysis
TIFR-CAM, odd semester, 2021.

Classes will be held from 9:30 AM to 10:55 AM on Wednesdays and Fridays
Recitation, Discussions and Office Hours will be held from 9:30 AM to 10:55 AM on Mondays (Room number 112). If the students have questions, they are encouraged to ask their queries by Friday lunch so that they can be taken up on Monday.
The syllabus that we will follow can be found by clicking the following link. In addition we may (depending on the time available) cover some additional topics.
Grading Scheme: 40 (Final term)+30 (midterm)+ 30 (assignments and recitation); solutions from bonus problems can be used to improve the net midterm+assignment score.
Here is the conversion chart from marks to letter grades. More about the grading scheme can be found by following this link.
Letter Grade Marks range
O 95-100
A+ 90-94
A 80-89
B+ 75-79
B 60-74
C 50-59
F Below 50
Finally, we will be following the book "Complex Analysis" by Stein and Shakarchi.
The midterm will be held on Monday, 18th October from 9:30-11:00.The syllabus will be whatever we cover until Monday, 11th October
Tentatively classes will run till 26th November, and the final term will be held in the third week of December.
Lecture notes (videos will be available only for 2-3 days. Please download them before I delete them.)

Lecture 1, 11th August: Notes A brief introduction and motivation.

Lecture 2, 13th August: Notes What is a holomorphic function and convergence of power series.

Lecture 3, 18th August: Notes Cauchy-Reimann equations and integration along curves

Lecture 4, 25th August: Notes Goursat's theorem

Lecture 5, 27th August: Notes Homotopy theory and Cauchy's theorem

Lecture 6, 1st of September: Notes An example of contour integration and started off Cauchy's integral formula

Lecture 7, 3rd of September: Notes Infinite differentiability and analyticity of holomorphic functions

Lecture 8, 8th of September: Notes Many consequences of differentiability, Morera's theorem, Liouville's theorem

Lecture 9, 15th of September: Notes Zeros of holomorphic functions, fundamental theorem of calculus

Lecture 10, 17th of September: Notes Schwarz Reflection principle

Lecture 11, 22nd of September: Notes Runge's theorem

Lecture 12, 24th of September: Notes Removable singularities

Lecture 13, 29th of September: Notes Poles

Lecture 14, 1st of October: Notes Residue theorem, Essential singularities, meromorphic functions

Lecture 15, 6th of October: Notes Argument principle, Rouche's theorem, a little intuition behind the winding number.

Lecture 16, 8th of October: Notes Maximum modulus principle, open mapping theorem, branches of logarithm

Lecture 17, 13th of October: Notes Maximum modulus principle, open mapping theorem, branches of logarithm

Lecture 18, 20th of October: Notes Jensen's formula

Lecture 19, 22nd of October: Notes Product formula for sine.

Lecture 20, 27th of October: Notes Weierstrass's product.

Lecture 21, 29th of October: Notes Hadamard's factorisation theorem.

Lecture 22, 3rd of November: Notes Introduction to Riemann mapping theorem.

Lecture 23, 5th of November: Notes Introduction to Möbius maps.

Lecture 24, 10th of November: Notes Möbius transformations map circles to circles.

Lecture 25, 12th of November: Notes Automorphisms of the disc and the upper half plane.

Lecture 26, 15th of November: Notes Video Riemann mapping theorem.

Lecture 27, 17th of November: Notes Riemann mapping theorem.

Lecture 28, 22th of November: Notes Video Mapping to Polygons part 1.

Lecture 28, 24th of November: Notes Video 1 Video 2 Mapping to Polygons part 2.

Assignment

Please feel free to discuss with your fellow students but the solution must be understood and written on your own. At any point the student can be asked to explain a given solution during the office hours. Please do not search for the solutions on the internet. It helps no one (including you). The assignment needs to submitted by email to my Tifr address before the class (scan/tex either will do).

The book often uses open sets when it means regions. By cautious!

Assignment 1, due on 25th of August before class: Link

Assignment 2, due on 1st of September before class: Link

Assignment 3, due on Thursday, 9th of September before class: Link

Assignment 4, due on the end of Sunday, 19th of September: Link

Assignment 5, due on the end of Thursday, 30th of September: Link

Assignment 6, due on the end of 11th of October: Link

Assignment 7, due on the end of 30th of October: Link

Assignment 8, due on the end of 15th of November: Link

Some Resources

We will perhaps not be able to cover the material necessary to solve all the questions. So be a bit careful before struggling too hard. I will try to mark out the ones we can solve when I get time.

Prof. K. Rama Murthy's problem collection in complex analysis (this is where I learnt my complex analysis from). Link

Some Problems in complex analysis meant to help students prepare for their comprehensive exams at Princeton university. This is where I picked up Fermat's last theorem from. Link