# TMA 4175 - Complex Analysis - Spring 2016

### Course Information

Instructor:

Winston Heap - winston [dot] heap [at] math [dot] ntnu [dot] no, Room 942 Sentralbygg II

- Lectures: Tuesday 12:15-14:00 R20, Friday 08:15-10:00 R41
- Exercise sessions: Thursday 14:15-15:00, 734, Sentralbygg II
- Office hours: Thursday, 13:00-14:00 or by appointment

Course material:

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

Tentative syllabus:

Topology of C, complex functions, complex integration, series and products, entire functions, the Riemann zeta function and analytic number theory.

Course plan:

- Week 1: We covered most of chapter 3, section 1.2 of Ahlfors: metric spaces, open and closed sets, closure and interior. We did subspaces and made a start on connectedness.
- Week 2: We finished connectedness and covered compactness (chapter 3, section 1.4)
- Week 3: We covered continuous functions and uniform convergence (chapter 3, section 1.5; chapter 2, section 2.3)
- Week 4: Power series, analytic functions, the complex exponential and trig. functions, the complex logarithm, the Cauchy-Riemann equations and harmonic functions (all in Chapter 2 apart from differentiability of the log: chapter 3, section 2.2 )
- Week 5: Conformal mappings and linear fractional transformations (chapter 3; sections 2.1, 2.3, 3.1, 3.2). Complex integration: definition in terms of parametrised curves, basic properties (Chapter 4, section 1.1)
- Week 6: Cauchy's Theorem (Ch. 4, sections 1.4, 4.4) and Cauchy's integral formula with applications (Ch. 4, section 2).
- Week 7: More applications of Cauchy's integral formula: Taylor series & analytic continuation, maximum modulus principle. Singularities and Laurent's Theorem.
- Week 8: The residue theorem and evaluation of definite integrals (Ch 4, sec 5.3), the argument principle and Rouche's Theorem (sec 5.2).
- Week 9: Functions defined by series, integrals and infinite products; the Gamma function (Chapter 5, sections 1.1, 2.2).
- Week 10: Entire functions; product representation (Chapter 5, section 2.3, 3.2)
- Week 11: More properties of the Gamma function: reflection formula, Weierstrass product, duplication formula (Chapter 5, section 2.4)
- Week 12: The Riemann zeta function: Euler product representation (continued), extension to the whole plane, functional equation, number of zeros formula (all in Ch. 5 Section 4) and the Hadamard product (not in Ahlfors, but a straightforward application of the product formula for entire functions of order 1). For some extra reading material on this stuff see Davenport's "Multiplicative number theory".
- Week 13: The distribution of primes. Explicit formula for prime counting functions.
- Week 14: Recap lectures
- Week 15: Final recap lecture on Tuesday

Detailed Syllabus:

- Topology of C: Chapter 3, Sections 1.2-1.5
- Analytic functions: Chapter 2, Sections 1.1-1.2, 2.1-2.4, 3.1-3.4
- Conformality and Linear Transformations: Chapter 3, Sections 2.1-2.3, 3.1-3.2
- Integration: Chapter 4, Sections 1.1, 1.4, 2.1-2.3, 3.1-3.2, 3.4, 4.2, 4.4, 5.1-5.3
- Series and Products: Chapter 5, Sections 1.1-1.3, 2.2-2.5, 3.1-3.2
- The Riemann zeta function and the Prime Number Theorem (non-examinable): Chapter 5, Section 4.1-4.4 + Chapters 9, 12, 15, 17, 18 of Davenport's "Multiplicative Number Theory"
- Extra stuff not in Ahlfors: Analytic continuation via Taylor series. Proving that certain functions defined by integrals are analytic (which requires Fubini's Theorem)

### Exercises

Problems: (These do not count towards your final grade)

- problems_1_ca.pdf (Questions 4-7 non-examinable)
- problems_2_ca.pdf (Questions 1-2 non-examinable)
- problems_5_ca.pdf (Question 7 non-examinable)

Solutions:

Here are some exercises to help those who are not familiar with big `Oh' notation: big_oh_fest.pdf. If you are familiar, then attempt the later exercises.

### Exam

The 2016 exam with solutions: tma_4175_complex_analysis_-_exam_solutions_2016.pdf

The time, date and room for the exam can be found on the main page here: https://www.ntnu.edu/studies/courses/TMA4175#tab=omEksamen

Any material from the course besides that covered in weeks 12-13 may appear on the exam. This includes the exercises (with the above stated exceptions). The examination code is B, meaning you can bring any written material and a basic calculator to the exam (consequently you will probably not be asked to prove theorems from the course). A good idea is to bring the book along with a few sides of A4 containing some key info/results e.g. how to map regions of the plane to one another using combinations of exponential, power and linear fractional maps (upper half plane to unit disk would be a useful map to have written down for example), how to prove a given integral is analytic (fubini+Morera) etc.

Here are some past exams: past_exams_ca.pdf. The questions marked with a cross require material we haven't covered so you needn't attempt these. Solutions may be found on the course pages for previous years (apart from the 2011 exam).