# TMA4175 - Complex Analysis: Spring term 2021

## Messages

- Lectures do start on Monday January 11
- Project 1 is now available
- Project 2 is now available

## Instructor

Berit Stensønes

## Lectures and exercises

In the beginning of the semester the lectures will be on zoom. Sorry about todays lecture. One thing I had not leared is that the laptop needs to be charged all throug the lecture. I planned to finish the examples today. Then show that a linear fractional transformation is determined completely by the value of 3 points . Next I will discuss symmetry and reflection. From here we will go to chap 4. My plan is ti pick up from where we where stopped by lack of juice and go from there. We will become perfect at the technology ?????????? ENDA EN TIRSDAG KATASTROFE: Det ser ut til at jeg må snike meg inn på kontoret og forelese fra nå av. Vi holder god fart så det skal gå bra etter hvert

- Lectures: Monday 10:15-12:00 R 10 Tuesday 14:15-16:00 H1
- Zoom info: https://NTNU.zoom.us/j/95108980200?pwd=K25hSythRXhFODRqMHYrR3oyNTU2QT09. Meeting ID: 951 0898 0200. Passcode: 330351
- Office hours: Wendesday, 13:00-14:00
- REFERANSE GRUPPE: The following are on the referanse gruppe: Guro Rio Berge gurorb@stud.ntnu.no, Emil Gautesen emilga@stud.ntnu.no, Olav Hellebust Haaland olavhha@stud.ntnu.no

MARCH 8: THE TEACHER IS SICK TODAY

MARCH9 ;THE TEACHER IS STILL SICK

## Course material

### Exam syllabus

Lars V. Ahlfors. Complex Analysis, 3rd edition. The book can be found on amazon.co.uk

Material to be covered:

Chapter 1 and Chapter 2, section 1 is independent reading.

Chapter 3

Chapter 4

Chapter 2, Section 2

Chapter 5

Chapter 6, Section 1-3

Chapter 8, Section 1

Progress plans:
Week 10:

In the week 18/1 and 19/1 we have covered the following material. Section 3.2, Cross Ratio, Section 3.3 Reflection . This is in Chapter 3 From Chapter 4 we have talked about integration and covered the proof of Theorem 2 on page 109. Also I have talked about using Greens Theorem together With the Cauchy Riemann equation to prove Cauchy´s theorem in the case when f is analytic and ALSO have continuous partiell derivatives. In week 5 I expect to cover Singularities on Page 124 Zeroes and poles page 126-129. Holomorphic functions as mappings page 132. If time permits I will talk about the Maximum principle and Schwarz lemma page 137-148 In week 6 we will finish the the proof of the Maximum principle and the Schwarz lemma. Then we will jump to chapter 5 section 1 and talk about convergence of sequences of analytic functions and Laurant Laurant series. In week 7 we shall talk about Residues and theorems related to this this section starts on page 148 Further we will talk about the argument principle and the Rouches theorem (page 152) Finally we will use the residues to calculate real integrals. In the coming week, week 11, we will review some of the material we did on harmonic function, then we will prove Schwarz theorem the next topic will be on reflection In the last part of the semester we will talk about Normal families page 219-225. Then we will go through the proof of the Riemann mapping theorem page 229-234

Review:

Chapter 1:

Section 1:

1.3, #2

1.4, #4

1.5, #4

Section 2:

2.2, #3

Chapter 2:

Section 1:

1.2, #5,7

Examples:

### Problems

Page 83: 2 Page 108 : 3,6,8 Page 123 :1,2,3 Solutions

Problem set 2: Page 129:2,3,5 Page 133:1 Page 136:1,2,5 Solutions

Problem set 3: Page 154:Problems 1 and 2 Page 161 : Problems 1 and 3 a,d,e Page 178 :Problems 1 and 5

At this point we have covered ,the material in Chapter 3 up to and including reflection. In the last part of the lecture today (18/1) we introduced integration and worked our way towards Cauchy´s theorem. I suggest that for practice problems you look at the following problems in the book: Page 83 problem 2, Page 108 , problems 3,6,8 and page 123 problems 1,2,3 NEW PROBLEMS: Page 173 problems1,4,5

PLAN FOR THE REMAINDER OF THE SEMESTER: We shall talk about some aspects of conformal mappings. Next we shall go through a list of what have been covered during the semester. Finally we shall cover some examples. these can be found in the website for spring 2015 as project 1 and project 2

## Projects

There will be 2 projects during the term, both will count towards the final grade. The first will be posted on February 22 and will be due on March 1 The second will be posted on April 12 and will be due on April 19 Project 1 counts for 20% towards the final grade. Project 2 counts for 20% towards the final grade The oral Exam counts for 60 % towards the final grade. One must have a passing grade in each of the 3 in order to pass the course.

## Exam

The oral examination will be on May 18,19 and 20 from 9-16. Each candidate will be examined for about 30 minutes. The students will be notified about their time in an email E-mails with time and place for the oral exam have been sent . If you did not get one get in touch with me