Topic outline
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Information
Teacher: Salvador Rodriguez-Lopez
You can reach me by using the message system in the course-page, or by writing a message in the course-forum.
Teaching assistant: Tim Schmatzler, timotheus.schmatzler@math.su.se
Time and place
Please see Timedit in case there are changes to the schedule.
Lecture: Tuesdays and Fridays 10:00 – 12:00 (starting September 1 until September 25). After that date, 9:00-11:00
Exercise sessions: Tuesdays and Fridays 9:00 – 10:00 (starting September 6 until September 25). After that date, 11:00-12:00.
Place: Albano Hus 2, Lärosal 8
Course book: E. B. Saff and A. D. Snider, Fundamentals of complex analysis: with Applications to Engineering and Science, 3rd edition, 2013, Pearson, ISBN 9781292023755.
Further reading:
L. V. Ahlfors - Complex analysis;
Jiří Lebl, Guide to Cultivating Complex Analysis
L. Hörmander - An introduction to complex analysis in several variables
Jiří Lebl, Tasty Bits of Several Complex Variables
Examination
Examiner: Salvador Rodriguez-Lopez
Examination Form: Written exam, October 25 (room to be announced, re-examination on December 16)
DO NOT FORGET TO REGISTER IN THE EXAM, at least 15 days before the exam date!
Old Exams: can be found here, and below under Gamla tentor (old exams)
Grading criteria: the grade is based on a final exam. Grades A, B, C, D, E respectively are guaranteed after obtaining at least 90%, 80%, 70%, 60%, and 50% ofthe points on the exam,
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If you are interested in the history of how complex analysis developed, then you might like this book "Hidden Harmony - Geometric Fantasies: The Rise of Complex Function Theory" by Umberto Bottazzini and Jeremy Gray.
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News forum
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Material:
- complex numbers and how to multiply them
- polar coordinates
- review of power series
- exponential function
- domains (connected open sets)
- point at infinity, Riemann sphere, a.k.a extended complex numbers
Suggested reading and exercises:
Please complete the quiz! It closes on September 10.
Read sections 1.1 - 1.7 in the course book. The following exercises from the book are suggested:
- 1.1: 10, 25.
- 1.2: 7, 16.
- 1.3: 7(b, d, f, g, h), 13, 17.
- 1.4: 12.
- 1.5: 5, 7(b), 10.
- 1.6: 2, 3, 4, 5, 6, 7, 8, 11, 16, 17.
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Material:
- domains (connected sets, open sets)
- (complex) differentiability and analyticity
- Cauchy-Riemann equations
- Relation between analytic functions and harmonic functions (harmonic conjugate)
After this course day:
Read sections 2.1 - 2.5 in the course book and work on the following exercises from the book:
- 2.2: 18 (16 in the second edition).
- 2.3: 11(a, c, f, g), 13(a, c, e, g).
- 2.4: 1, 2, 3, 7, 8, 10, 11.
- 2.5: 2, 3(a, b, c, e), 6.
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This is a video about the Cauchy-Riemann equations, made by Martin Tamm for another version of the course.
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MM5022_Lecture_02 File
The password to open the file will be given during the next lecture.
- domains (connected sets, open sets)
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Material:
- more on the Cauchy-Riemann equations
- harmonic functions
- logarithm(s), exponential function, powers, trigonometric functions
- point at infinity, Riemann sphere, a.k.a extended complex numbers
Suggested reading and exercises:
Chapters 1, 2, 3 in the course book (you can skip sections 3.4 and 3.6, which discuss applications we will not explore). Work on the following exercises from the book:
- 3.2: 5(a, c, e), 6, 11, 13(a), 17(a, c)
- 3.3: 1, 3, 4, 5(b), 8
- 3.5: 1(a,c,d), 3, 8.
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This video from 3Blue1Brown contains some visuals that might help you understand complex numbers and stereographic projection, especially from 4:15 to 17:30. The rest of the video is just for fun if you want to learn about quaternions.
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This is part of a series of videos created by Martin Tamm when the course was taught previously. It discusses the Riemann sphere and stereographic projection.
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This is a video about elementary functions (exponentials, logarithms, square roots, etc) by Martin Tamm from a previous version of the course.
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MM5022_Lecture_03 File
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Material:
- simply connected domains
- review of Green's theorem
- integral over a path in the complex plane
- Cauchy's Integral Theorem
- homotopy
Suggested reading and exercises:
Read sections 4.1 - 4.4 in the course book, and have a look at the following exercises:
- 4.1: 1, 4, 8
- 4.2: 3(b, c), 6(a), 7, 9, 13, 14(b, c)
- 4.3: 5, 6
- 4.4: 3, 5, 9, 10(b, d)
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Martin Tamm's video from a previous version of the course details some aspects of the proof of Cauchy's theorem (subdivisions of triangles, homotopy).
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MM5022 Lecture 04 File
- simply connected domains
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Material:
- Cauchy's theorem
- Cauchy's integral formula
- Cauchy estimates for derivatives
- Mean value property (for holomorphic functions)
- Liouville's theorem
Suggested reading and exercises:
Read sections 4.5 - 4.6 in the course book and work on the following exercises:
- 4.5: 3(a, c, e), 4, 6, 15.
- 4.6: 3, 4, 11.
(optional) Just for fun, if you want to see what can go wrong with Green's theorem, see Fesq's article giving a Counterexample to Green's formula.
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Cauchy's integral formula explained in a video by Martin Tamm from a previous version of the course.
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A proof of the Cauchy-Goursat theorem can be found, for instance in the fourth chapter of the book:
Since it is not easy to access the book from our library, I include my notes here for those interested.Churchill, Ruel V.; Brown, James W.Complex variables and applications. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg
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Material:
- More on integration, Cauchy's theorem and formula
- Morera's theorem
- Liouville's theorem
- Power series
- Laurent series
Suggested reading and exercises:
Read sections 5.1 - 5.3 in the course book and work on the following exercises:
- 5.1: 3, 4, 7(a, c, e)
- 5.2: 4, 11, 13
- 5.3: 3(a, b, d, f), 4, 5, 14, 16.
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Video about power series by Martin Tamm from a previous version of the course.
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Material:
- Expansions in power series and Laurent series
- Annular domains
Suggested reading and exercises:
Read sections 5.4 - 5.5 in the course book. The following exercises are suggested:
- 5.4: 1(a, c), 3(a, b, c, d), 5(a).
- 5.5: 1, 4, 5, 7(a, b), 9.
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Video on Laurent series by Martin Tamm from when the course was taught previously.
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Material:
- zero of a function
- removable singularity
- pole
- essential singularity
- order of zero or pole
- residue at a pole
Suggested reading and exercises:
Read sections 5.6 - 5.8 in the course book. The following exercises are suggested:
- 5.6: 1(a, d, f, g), 2, 3(a, c), 6
- 5.8: 1, 4
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Video on zeros and isolated singularities by Martin Tamm, from when the course was taught previously.
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Material:
- Residue at a pole
- Residue Theorem
- Evaluating integrals using the residue theorem
Suggested reading and exercises:
Read sections 6.1 - 6.3 in the course book. Work on the following exercises from the book:
- 6.1: 1, 3(a, c, f, g), 5, 6
- 6.2: 1, 5
- 6.3: 1, 3, 10, 11, 14
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Video about residue calculus by Martin Tamm, from when the course was taught previously.
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Material:
- more on residue theory
- "generalized" integrals (indented contours; branch cuts)
- principal value (of an integral that does not converge)
- Jordan's Lemma
- argument principle
- Rouché's theorem
After this course day:
Read sections 6.4 - 6.7 in the course book. Work on the following exercises from the book:
- 6.4: 3, 6
- 6.5: 1(b, c, d), 4
- 6.6: 1, 2, 6
- 6.7: 1, 2, 4, 6, 8, 10
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Video about the argument principle, by Martin Tamm from when the course was taught previously.
- more on residue theory
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Material:
- more on residues, argument principle
- Rouché's theorem
- maximum principle
- open mapping theorem
- preservation of angle
- conformal mappings
After this course day:
Read sections 7.2 - 7.3 in the course book carefully and work on the following exercises from the book:
- 7.2: 3, 6, 8, 9.
- 7.3: 1, 2, 3, 4, 7, 10.
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Video on conformal maps, by Martin Tamm from when the course was taught before.
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Material:
- open mapping theorem
- preservation of angle
- conformal mappings, automorphism
- Möbius transformation, cross ratio
- harmonic functions, Poisson formula, Dirichlet problem.
Suggested reading and exercises:
Read sections 3.4, 4.7, 7.1, 7.2, 7.3, and 7.4 in the course book and work on the following exercises:
- 7.1: 7
- 7.4: 1, 2, 5, 6, 7, 9
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Video on harmonic functions, by Martin Tamm from when the course was taught before.
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Material:
- Poisson integral formula
- maximum principle
- Dirichlet problem
- physical applications of harmonic functions
- conformal mappings
- Riemann mapping theorem (statement)
After this course day:
Start/continue reviewing for the exam!
Read sections 7.6--7.7 in the course book. The following exercises are suggested:
- 7.6: 1, 6, 9
- 7.7: 6
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"Applications of conformal mappings" video by Martin Tamm from when the course was taught before.
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Material:
- harmonic functions, conformal mappings
- Riemann mapping theorem
- holomorphic functions of several variables
- ball and polydisk
- outlook: where to go from this course
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"Several variables" video by Martin Tamm from when the course was taught before.