# TMA 4175 - Kompleks analyse - Vår 2012

### Beskjeder

 30/05 Exam and solutions can be found here
 23/05 You will be allowed to use one A5 yellow sheet stampled by the institute in which you may write what you think you need. On the both sides of course (-:. You can get empty sheets in the institute office.
 22/05 Test exam 3 is on the web. Do not panic: I made it more complicated than usual exam. Yet some problems of such level of complexity you can meet in your exam
 22/05 Preexam meetings: Wednesday, 23.05, 14:15-16:00,room 922 SB2;Saturday, 26.05, 10:15-12:00, room 734, SB2;and by appointment
 22/05 Pensum correction ! See addition in red. Sorry for misprint.
 08/02 No øvingstime on 29.02 ! Those who have questions are welcomed to my office after Tuesday's lecture on 28.02
 08/02 IMPORTANT MESSAGE ! One of the homework exercises will be literarly included into the final exam (:
 30/01 FLYTTEMELDING ! Starting from tomorrow 31.01 our lectures on Tuesdays will be in F3, not in F2. Do not ask me why. Time is still the same. Yu.L.
 26/01 MISPRINT ! I made a mistake yesterday when wrote numbers of homework exercises for week 4. It is corrected now. Sorry. Yu.L.
 23/01 Exercise meetings are rescheduled to Wednesdays, 14:15-15:00, room 922, SB2
 10/01 First exercise meeting will be on week 3 i.e. 19.01

### Kursbeskrivelse:

"Emnet gir en innføring i grunnleggende teori for kompleks integrasjon, konforme avbildninger og harmoniske funksjoner. Utvalgte videregående emner som f.eks. analytisk fortsettelse, analytisk tallteori, harmoniske funksjoner, interpolasjon og approksimasjon, og anvendelser innen fluid-dynamikk." (Fra studiehandboken)

### Foreleser:

Yurii Lyubarskii - yura [at] math [dot] ntnu [dot] no, Rom 954 Sentralbygg II.

    Forelesninger: Mandag  12:15 - 14:00 i F4,  Tirsdag 8:15 - 10:00 i  F3
Øving: Onsdag 14:15 - 15:00 room 922, SB2
Treffetid:   Tirsdag 10:15 - 11:00 og etter avtale

### EXAM PENSUM

Theodore W. Gamelin: “Complex Analysis”:
all homework is a part of pensum
one of the homework exercises will be literally taken to 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
Chapter 13, Section 3 - 4

### Forelesningsplan

10/01 Week 2. This week we refresh our knowledge of comlpex analysis from Matte 4. See "what to know list" from the course materials section.

We have discussed items 1-3, 5-7, and 8a-8c in the "what to know" list. This corresponds to the following pieces from the Gamelin book:
Chapter 1, 1-8; Chapter 2, 1-3; Chapter 4, 1-4.

17/01 Week 3. This week we keep moving along "what to know" list and also study some new material, related to mean value theorem, maximum modulus principle, entire and harmonic functions, see Notes for week 3 for more details. We also proved the fundamantal theorem of algebra.

The material corresponds to the following pieces of the Gamelin book:
Chapter 3, 1-5; Chapter 4, 1-5; Chapter 5, 1-4, 7. See also notes about fundamenta theorem of algbera.

24/01 Week 4. This week we learned about harmonic functions (see my notes in the course materials) harmonicity of real and imaginary parts of analytic functions, harmonic conjugate, simply/multiconnected domains, mean value theorems and max/min principle for harmonic functions, power series exapnsions,Dirichlet problem for the disk (not included into notes). We also repeat Laurent expansions, classification of isolated singularities and began to study behavior of analytic function near isolated singular point.

This material corresponds to the following pieces from Gamelin book:
Ch. II.5; Ch. VI, 1,2

31/01 Week 5. This week we completed our study of singular points and proved the Casoratti-Weierstrass theorem. We defined residues, proved the Cauchy residues theorem, and established formulas for residues in case when the singular point is a pole of any order. We saw how one can use residues in order to find the sums of series, and obtained the Lagrange interpolation formula. Afterward we discussed notion of bandlimited functions and proved in a simple case the Whittaker–Kotel'nikov-Shannon interpolation formula.

This material corresponds to the following pieces from Gamelin book:

08/02 Week 6. This week we learned how one can eveluate integrals by using comlpex analys techniques. In particular we proved Jordan lemma. We also discussed a little techniques of generating functions.

This material corresponds to the following pieces from Gamelin book: Ch. VII.2 - VII.4.

14/02 Week 7. This week we discussed the Argument principle, Rouchet and Hurwitz theorems. We also discussed some topological notions related to compacteness. Then we discussed the compactwise convergence and boundedness of families of analytic function and proved that each compactwise bounded sequence of analytic functions contains a convergent subsequence. The Arzela-Ascoli theorem has also been proved.

This material corresponds to the following pieces from Gamelin book: Ch. VIII.1 -3; XI.5

23/02 Week 8. This week we constructed the extended comlpex plane (Riemanian sphere) and discussed analyticity, singularity at infinity. The stereographic projection was also reminded.Then we discussed the residii at infinity: definition, expression via Laurent expansion at infinity, the theorem about the sum of residii. The next topic was the priciple of preservation of domain, and inverse function theorem. We discussed the behavior of analytic function near the regular point (one to one correspondence, angle preservation) and also behavior near the critical point. We proved the principle of correspondence of boundaries. Then we introduced notion of conformal mapping, formulated the Riemann mappipng theorem and started with fractional-linear mappings.

This material corresponds to the following pieces from Gamelin book: Ch. VIII.1 -3; XI.5

28/02 Week 9. This week we studied farctional linear mappings and also developed a "conformal mapping toolbox" as well as some examples. We proved Morera theorem and theorem about removing of singularities. Afterward we proved the Schwartz reflection principle.

This material corresponds to the following pieces from Gamelin book: Ch. II.6-7; Ch. IV.9; Ch.X.3; Ch XI. 1

06/03 Week 10. This week we proved the Riemann mapping theorem. We discussed (once again) notion of simply connected domain, proved the Schwartz lemma, formulated the corresponding extremal problem, and proved that it has a solution which realizes the corresponding mapping.

This material corresponds to the following pieces from Gamelin book: Ch.IX.1; Ch XI. 2,6

13/03 Week 11. This week we completed discussion around the Riemann mapping theorem. We also consider conformal mapping of annulus and prove they are not conformly equivalent unless have the same radii ratio. We started to discuss harmonic functions, reminded various versions of the Green formula and solved the Dirichlet problem in simply connected domains by using conformal mapping. Also we introduced notion of the Green function. We proved Poisson representation formula for disk and half-plane. We returned to the theorem on boundary correspondence (which I missed on the previous week) and prepared to prove it.

This material corresponds to the following pieces from Gamelin book: Ch.XV. See also my notes (to come later)

21/03 Week 12. This week we completed the proof of the theorem on correspondence of the boundaries under conformal mapping. Then we started discussion of entire functions. We introduced infinite products and then proved the Weierstass theorem.

See my notes on boundary correspondence. See also Chapter XIII 3,4 from Gamelin book.

### Homeworks

 12/01 Homework for week 2 (to be discussed on week 3): From Gamelin book: I.1 - nn 1d),e),i), 3, 6; I.2 - 2 c),d), 6; I.4 - 1d),f); II.3 - 2,5

Those who do not have Gamelin book see "Exercise set for week 2" in the course materials section.

 19/01 Homework for week 3 (to be discussed on week 4): From Gamelin book: II.2 -5; III.3 - 1b) III.5 - 2,4; IV.2 - 1; IV.4 4 b), e); IV.5 - 1,2

Those who do not have Gamelin book see "Exercise set for week 3" in the course materials section. ¨

 25/01 Homework for week 4 (to be discussed on week 5): From Gamelin book: II.3 - 4,5; II.4 - 7; II.5 - 5,6,7; VI.1 1,2; VI.2 - 1 a), c) g), 5

Those who do not have Gamelin book see "Exercise set for week 4" in the course materials section.

 02/02 Homework for week 5 (to be discussed on week 6): From Gamelin book: VI.2 - 1 a),c),e) VII.1 - 1 e),g),i); 2 a),b); 3 f)

Those who do not have Gamelin book see "Exercise set for week 5" in the course materials section.

 08/02 Homework for week 6 (to be discussed on week 7): From Gamelin book: VII.2 - 4,5; VII.3 - 1,5, VII.4 - 2,3,5

Those who do not have Gamelin book see "Exercise set for week 6" in the course materials section.

 17/02 Homework for week 7 (to be discussed on week 8): From Gamelin book: VIII.1 - 1,6,8; VIII.2 - 2,5, XI.5 - 1,3

Those who do not have Gamelin book see "Exercise set for week 7" in the course materials section.

 23/02 Homework for week 8 (to be discussed on week 9): From Gamelin book: VII.8 1 e), d), 2,9; VIII.4 1,8; VIII.5 1,5

Those who do not have Gamelin book see "Exercise set for week 8" in the course materials section.

 02/03 Homework for week 9 (to be discussed on week 10): From Gamelin book: II.6 3 a),b), 4, 6, 8; II.7 1 a), 2,4,5

Those who do not have Gamelin book see "Exercise set for week 9" in the course materials section.

 08/03 Homework for week 10 (to be discussed on week 11): From Gamelin book: VII.7 - 2,4,5; VII.8 - 4; X.3 - 6; XI.1 - 5,11;

Those who do not have Gamelin book see "Exercise set for week 10" in the course materials section.

 15/03 Homework for week 11 (to be discussed on week 12): From Gamelin book: XV.1: 2,5,7; VX.3 - 2; III.5 - 3

Those who do not have Gamelin book see "Exercise set for week 12" in the course materials section.

### Eksamen

Skriftlig, 4 timer. 29.05, 9:00-13:00

### Lærebok:

Theodore W. Gamelin: "Complex Analysis".