Math618 - 600: Theory of Functions of a Complex Variable II, Spring 2020

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

• time/location: TR 9:35 - 10:50, BLOC 202
  

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

• office hours: TR 2:00 - 3:30 , 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: Math-617.

• Course Content: We will cover chapters VII through X, and parts of XII, in Conway's book, with some additions. A brief list of topics is as follows: spaces of analytic and meromorhic functions, the Riemann mapping theorm, the Weierstrass factorization theorem, the inhomogeneous ∂ equation, Runge's theorem, simple connectedness, Mittag - Leffler's theorem, the Schwarz reflection principle, analytic continuation along a path, the monodromy theorem, the modular function and lifting of analytic functions, the Picard theorems, harmonic, subharmonic, and superharmonic functions, the Dirichlet problem.

• Grading Policy: There will be three exams worth 100 points during the semester; these exams will be takehome.

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Semester log

• Tuesday, Jan. 14: no class, will be made up on Friday, Jan.24, 9:00-10:15 in BLOC202.

• Thursday, Jan. 16: Chapter VII § 1; The space of continuous functions C(G,Ω), covered through Proposition 1.6, gave proof of Lemma 1.5. HW is problems 4-6 on page 150.

• Tuesday, Jan. 21: finish § 1; HW is problems 7,8, pp. 150-151.

• Thursday, Jan. 23: Chapter VII § 2; Spaces of analytic functions, HW is pp. 154-155, problems 1, 4, 6, 10, 13.

• Friday, Jan. 24: (make up for Jan. 14), Chapter VII, begin §4; The Riemann mapping theorem.

• Tuesday, Jan. 28: Chapter VII, finish §4; The Riemann mapping theorem, HW is page 163, problems 2, 4, 5, 9. Prove Poincaré's theorem on biholomorphic inequivalence of the bidisc and the ball in C².
  Some classical texts on conformal mapping: G.M.Goluzin, Geometric Theory of Functions of a Complex Variable, Z.Nehari, Conformal Mapping, P.Henrici, Applied and Computational Complex Analysis, Vol. 1. See also part B of Remmert's book mentioned above under text.

• Thursday, Jan. 30: finish proof of the biholomorphic inequivalence of the bi-disc and the unit ball in C². Chapter VII, begin § 5; The Weierstrass factorization theorem. HW is page 173, problems 1, 2, 7.

• Tuesday, Feb. 4: continue § 5; The Weierstrass factorization theorem. HW is problem 4, page 173.

• Thursday, Feb.6: continue § 5; The Weierstrass factorization theorem. HW is pages 173-174, problems 6, 8, 10, 12.

• Tuesday, Feb. 11: finish § 5; The Weierstrass factorization theorem. HW is pages 173-174, problems 3, 9. Read Chapter VII, § 6; Factorization of the sine function, do problem 4 (hint: evaluate 6.2 at z=1/2).

• Thursday, Feb. 13: give out Exam 1, due Tuesday, Feb. 18.
  Chapter VIII
  Note: the organization of the material for this chapter differs from that in the book, we emphasize the role of the (inhomogeneous) ∂-equation. Two references for this approach are Hö rmander: Complex analysis in several variables, Chapter I, and Berenstein-Gay: Complex variables, Chapter 3. As a result, homework problems are typically not from the book and are assigned in class.
  § 0; The Cauchy-Pompeiu formula

• Tuesday, Feb. 18: Chapter VIII, finish § 0; The Cauchy-Pompeiu formula

• Thursday, Feb. 20: discuss Exam 1, start Chapter VIII, § 1; Runge's theorem

• Tuesday, Feb. 25: continue Chapter VIII, § 1; Runge's theorem, continue proof of Runge's theorem.

• Thursday, Feb. 27: no class (to be made up later).

• Tuesday, Feb. March 3: continue Chapter VIII, § 1; Runge's theorem, finish proof of Runge's theorem, start corollaries (including Theorem 1.7, page 198, the book's version of Runge's theorem).

• Thursday, March 5: continue Chapter VIII, § 1; Runge's theorem, continue corollaries to Runge's theorem, then discuss H(Ω)-hulls.
  Lectures on complex approximation by Dieter Gaier is a reference (available for free download) that contains a lot of additional classical information on complex approximation; see in particular part II:

Spring Break, March 9 - March 13.

• Tuesday, March 17: no class due to covid-19 precautions.

• Thursday, March 19: no class due to covid-19 precautions.

Rest of the semester will be on line meetings only.

• Tuesday, March24: finish H(Ω)-hulls; solvability of the inhomogeneous Cauchy-Riemann equations on domains in the plane in the smooth category. HW: read § 2, chapter VIII, do problem 1, page 204.

• Thursday, March 26: Chapter VIII, §3; Mittag-Leffler's theorem, HW is pages 206-209, problems 1, 5, 6. Start Chapter IX Analytic Continuation;
  § 1; The Schwarz reflection principle, HW is page 213, problem 2, plus additional problems assigned in class.

• Tuesday, March 31: Chapter VIII, §2; Analytic continuation along a path, HW is p. 216--217: problems 1, 4--7. Begin § 3; The monodromy theorem. More information on the monodromy theorem is in Conway, §:7. See alo chapter 2 of N.Narasimhan Complex analysis in one variable. This is also an overall very nice book.

• Thursday, April 2: finish § 3; § 4; The elliptic modular function, HW is assigned in class. In the construction of the modular function, we use (a special case of) Caratheodory's theorem on the continuous extension to the boundary of the Riemann map of a region bounded by a Jordan curve. A classcal reference for this theorem is Goluzin's book mentioned above; for the elliptic modular function, a classical reference is Nehari's book. For modern references, see this Math Wiki article.

• Tuesday, April 7: Exam 2 due; continue § 4, The elliptic modular function, HW assigned in class.

• Thursday, April 9: discuss Exam 2 problems.

• Tuesday, April 14: finish § 4, The elliptic modular function, start Chapter 10 The Montel-Carathéodory and Picard theorems, HW assigned in class notes.

• Thursday, April 16: finish Chapter 10 The Montel-Carathéodory and Picard theorems, HW is page 301, problems 1-3, plus a problem form the class notes.

• Tuesday, April 21: return to Chapter VIII, § 4 Further approximation theorems, discuss the Mergelyan theorems for polynomial and rational approximation, prove a special case for polynomial approximation. A reference for the Mergelyan theorems is Lectures on complex approximation by Dieter Gaier (available for free download). For the theorem on "just" polynomial approximation, see also Topics in complex analysis by Mats Andersson (available for free download).