Math617 - 600: Theory of Functions of a Complex Variable I, Fall 2019

Math-617 constitutes the first half of the Complex Analysis qualifier sequence (Math-617, Math-618). More information on qualifiers, including old exams.

• time/location: MWF 12:40 - 1:30, BLOC 148
  

• instructor: Emil Straube, BLOC 601M, straube@math.tamu.edu
  

• office hours: MWF 1:45 - 2:45, but feel free to come by any time.
  

• text: John B. Conway, Functions of One Complex Variable, Springer GTM 11, 1978. You can download a free copy here. This site also has a link to purchase a softcover copy for USD 24.99 (a considerable reduction form the list price).
  You should also regularly consult Reinhold Remmert, Theory of Complex Functions, Springer GTM 122, 1991 (free download), especially for historical aspects of the subject.

• Prerequisites: This is a first year graduate course. Accordingly, a solid background in undergraduate analysis is expected. The officially listed prerequisite is Math-410.

• Course Content: We will cover chapters I through VI in Conway's book. A brief list of topics is as follows: complex numbers, metric spaces and the topology of the complex plane, elementary properties of analytic functions, Möbius transformations, complex integration, Cauchy's theorem, Cauchy estimates, Liouville's theorem, power series representation, the argument principle, the open mapping theorem, Goursat's theorem, singularities, residues, the maximum modulus principle, Schwarz's lemma, Hadamard's three circles theorem.

• Grading Policy: There will be three exams worth 100 points during the semester, and a comprehensive final exam worth 100 points also. All exams will be takehome.

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Semester log

• Monday, Aug.26: Chapter I, § 1-3; The field of complex numbers, the complex plane. HW is pages 2-3, problems 1-5, page 4, problems 1,2; read pages 1-7 in Theory of Complex Functions

• Wednesday, Aug. 28: Chapter I, § 4; Polar representation and roots. HW is page 6, problems 2, 5, 7; have a look at Chapter 3 of Numbers.

• Friday, Aug. 30: Chapter I, § 5, 6; Lines and half planes, stereographic projection and Riemann sphere. HW is page 7, problem 1, page 10, problems 2, 3, 4.

• Monday, Sept. 2: Chapter II, § 1; Definition and examples of metric spaces. HW is pages 13-14, problems 1, 2, 7, 8, 11.

• Wednesday, Sept. 4: Chapter II, § 2; Connectedness. HW is page 17, problems 3, 4, 5. See email with some remarks about connectedness.

• Friday, Sept. 6: Chapter II, § 3; Sequences and completeness. HW is page 20, problems 3, 4, 6, 8.

• Monday, Sept.9: Chapter II, § 4; Compactness. HW is page 24, problems 4, 5, 6.

• Wednesday, Sept. 11: Chapter II, § 4; Finish compactness. HW as above.

• Friday, Sept. 13: Chapter II, § 5: Continuity. HW is page 28, problems 2, 3, 6, 8, 9. Read § 6.

• Monday, Sept. 16: Chapter III, § 1: Power series. HW is page 33, problems 1, 2, 6, 7; read Chapter 4, § 1, section 5 in Theory of Complex Functions.

• Wednesday, Sept. 18: Chapter III, § 2: Analytic functions. HW is pages 43--44, problems 1, 3, 5.

• Friday, Sept. 20: Chapter III, continue § 2: Analytic functions. HW is pages 43--44, problems 9, 12 -- 15, 17.

• Monday, Sept. 23: Chapter III, finish § 2: Analytic functions. HW is page 44, problems 16, 17, 20, 21. See page 49--50 of Theory of Complex Functions for historical remarks on the Cauchy-Riemnn equations.

• Wednesday, Sept. 25: Chapter III, § 3: Analytic functions as mappings, Möbius transformations. HW is page 54, problems 1-5. A very nice geometric treatment of Möbius transformations in Ahlfors' book Complex Analysis; see in particular section 3.5 of Chapter 3.

• Friday, Sept.27: Chapter III, § 3: continue Analytic functions as mappings, Möbius transformations. HW is pages 54-57, problems 8, 10, 11, 14, 16, 18, 30.

• Monday, Sept. 30: Chapter III, § 3: finish Analytic functions as mappings, Möbius transformations. HW from Friady now due Wednesday, Oct. 2. , also 21, 23, 24.

• Wednesday, Oct. 2: Chapter III, § 3: discuss HW problems from § 3; give out Exam 1 (take home, due Monday, Oct. 7 at the beginning of class). See email message concerning Problem 9.

• Friday, Oct. 4: no class, work on take home exam.

• Monday, Oct. 7: exam 1 due; Chapter IV, § 1:Riemann-Stieltjes integrals (begin), HW: read 1.7 Proposition through 1.9 Theorem, page 67, problems 1-4. Also look at the historical remarks in Chapter 6 of Theory of Complex Functions, in particular Gauss' December 18, 1811, letter to Bessel quoted in the introduction, and section 4.

• Wednesday, Oct, 11: go over exam 1. Solutions sent via email.

• Friday, Oct. 11: Chapter IV, § 1; finish Riemann-Stieltjes integrals. HW is pages 67-68, problems 7, 9, 10, 12, 13, 16, 22.

• Monday, Oct. 14: Chapter IV, § 2: Power series representation of analytic functions. HW is pages 73-76, problems 2,3,6,7a,d, 9c,e.

• Wednesday, Oct. 16: Chapter IV, § 3: Zeros of an analytic function. HW is page 80, problems 6-9.

• Friday, Oct. 18: Chapter IV, § 4: The index of a closed curve. HW is page 83, problems 2, 3.

• Monday, Oct. 21: Chapter IV, § 5: Cauchy's theorem and integral formula. HW is page 87, problems 1, 3, 5, 6, 8. See email for solution to problem 1.

• Wednesday, Oct. 23: Chapter IV, § 6: The homotopy version of Cauchy's theorem and simple connectivity. HW is pages 95-96, problems 4, 5, 7, 9, 10, 11. Did not finish; this assignment is due only Monday, Oct.28. See email with solutions to problem 7, page 80, problem 3, page 83, problem 8, page 87.

• Friday, Oct. 25: Chapter IV, finish § 6; see above for HW. See email for solution to problem 1, page 87.

• Monday, Oct. 28: Chapter IV, § 7: Counting zeros; the open mapping theorem. HW is pages 99-100, problems 1-4; read §8 Goursat's theorem.

• Wednesday, Oct. 30: Chapter V, § 1: Classification of singularities. HW from this section not yet assigned. Give out Exam 2, due Monday, Nov.4 at the beginning of class. See email with solutions to problem 11, page 96, and a technical part in the proof of the argument principle.

• Friday, Nov. 1: Chapter V, finish § 1. HW is pages 110-111, problems 1befg, 4,5,10, 13, 17. This HW is due only Wed., Nov. 6 (because of Exam 2).

• Monday, Nov. 4: Exam 2 due, Chapter V, begin § 2: Residues,

• Wednesday, Nov. 6: discuss Exam 2, Chapter V, continue § Residues, HW is pages 121-122, problems 1bd, 2cg, 3, 4, 13.

• Friday, Nov. 8: Chapter V, § 3; The argument principle, HW is pages 126-127, problems 2, 6, 7, 9.

• Monday, Nov. 11: Chapter V, finish § 3; The argument principle

• Wednesday, Nov. 13: Chapter VI, § 1: The maximum principle, HW is pages 129-130, problems 1, 3, 4, 7.

• Friday, Nov. 15: Chapter VI, § 2: Schwarz's lemma , HW is page 133, problems 3, 5, 8.

• Monday, Nov. 18: Discuss Problem 3, page 130 in Conway's book (sublevel sets and level curves of |p(z)| for a polynomial p.

• Wednesday, Nov. 20: Chapter VI, § 3: The three lines theorem, Hadamard's three circles theorem, HW is pages 137-138, problems 3,5,6,7. Give out Exam 3 (due Monday, Nov. 25).

• Friday, Nov. 22: finish The three lines theorem, Hadamard's three circles theorem, HW is pages 137-138, problems 3,5,6,7.

• Monday, Nov. 25: Exam 3 due. Go over exam problems.

• Wednesday, Nov. 27: Reading day, no class.

• Friday. Nov. 29: Thanksgiving holiday, no class.

• Monday, Dec. 2: Go over exam problems.

• Wednesday, Dec. 4: Last day of classes. Discuss HW problems. Give out final exam (due Monday Dec. 9).