TMA 4175 - Complex Analysis - Spring 2016
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
Lars V. Ahlfors. Complex Analysis, 3rd edition. The book can be found on amazon.co.uk
Topology of C, complex functions, complex integration, series and products, entire functions, the Riemann zeta function and analytic number theory.
- 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
- 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)
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)
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.
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).