# TMA 4175 - Complex Analysis - Spring 2014

Announcements

Instructor:

Berit Stensønes - beritste@math.ntnu.no, Room 950 Sentralbygg II Lectures and exercises

• Lectures: Monday 12:15-14:00 R4 Friday 10:15-12:00 G21
• Exercise sessions:
• Office hours: Tuesdays, 14:00-15:00

Course material Exam syllabus

Theodore W. Gamelin, Complex Analysis All homework tasks are included in the syllabus. One of the homework exercises will be literally on the exam.

Chapter 1, Section 1 - 8
Chapter 2, Section 1 - 7
Chapter 3, Section 1 - 5
Chapter 4, Section 1 - 5
Chapter 5, Section 1 - 5, 7
Chapter 6, Section 1 - 2
Chapter 7, Section 1 - 4, 7,8
Chapter 8, Section 1 - 3
Chapter 9, Section 1
Chapter 10, Section 1,3
Chapter 11, Section 1 - 2, 6

Lecture plan 06/01 Week 2. This week we refresh our knowledge of complex analysis from Matte 4.

Week 3. In week 3 and parts of week 4 we have and will discuss material from Chapter 2
in the text boo. We have expanded section 7 a bit, problems adressing the expanded part
will be in the first project. In addition to lectures we have also worked at Problems
from the textbook: Section 4 Problem 7, Section 5 Prolem 5, Section 6 Problems 4           and 6 problem 4 from section 7

Week4 In week 4 and part of week 5 we have covered naterial from Chapter 3 section 6 and 7 have not and will not be covered.

Two Projects will be graded by the lecturer. Handed out: 1. Monday February 3, due Monday February 10 2. Monday March 24, due Monday March 31.

There will be exercise solving and discussions during the lecture hours.

Examination CODE A: ALLWRITTEN MATERIAL IS ALLOWED. Written, 4 hours, 16.05, 9:00-13:00 Project 1, Complex Analysis due Monday February 10, 2014 ￼a)Showthatf(z)=e mapsDontoA={z∈C;Re(z)andIm(z)>0}. b)Showthatz→z2 mapsAontoH+ ={z;Im(z)>0}. c) Use a) and b) to find a map from D onto the unit disc ∆ = {z ∈ C; |z| < 1}. Problem 2: Let u(x,y) = xex cosy − yex siny a) Show that u is harmonic in the entire plane. b) Find a function v so that f = u + iv is analytic. ∫ 2π c) Let h(θ) = cos θecos θ cos(sin θ)−sin θecos θ sin(sin θ) and show that 0 h(θ) = 0. (Hint: Observe that u(cos θ, sin θ) = h(θ).) Problem 3: Let f be analytic in ∆ = {z;|z| < 1} and show that if f(z) ∈ R for every z ∈ ∆, then f is constant. Problem 4: Find ∮ sin z dz where C is the circle with center 0 and radius C z3(z−1) 2 travelled counterclockwise. Problem 5: a) Prove that if z ≠ 1, then 1+z+···+zn = 1−zn+1. 1−z ￼￼b) Use this to show that 1+2z+3z +···+nz =(1−z)2 −1−z

Project 1, Complex Analysis due Monday February 10, 2014

Project 1: ￼project1.pdf

Project 1, Solutions: project1solutions.pdf

Project 2, problems:problem_2.pdf

Project 2, Solutions: tma4175_losningsforslag.pdf

Previous Exams:

Previous exams for the course can be found here: eksamen_tma4175.pdf EXTRA OFFICE HOUR MONDAY MAY 12 AT 11