Math618 - 600: Theory of Functions of a Complex Variable II, Spring 2021

This is the semesterlog for the course. It will contain information such as material covered, homework assignments, exams, etc. Course information and syllabus are here.

Semester log

Wednesday, January 20: start Chapter VII, Section 1: The space of continuous functions C(G,Ω), HW is page 150, problems 1,2,4,5.
Friday, January 22: a metric on C(G,Ω), completeness of C(G,Ω), normal families; HW is page 150, problems 6,8.
Monday, January 25: more on normal families, equicontinuity, Arzela-Ascoli theorem; HW is problem 7 on page 150, read § 2 through the middle of page 152.
Wednesday, January 27: §2;Spaces of analytic functions, H(G), completeness, Hurwitz' theorem, locally bounded families, Montel's theorem; HW is page2 154-155: 4,6,10,13.
Friday, January 29: § 3, the Riemann mapping theorem; HW is pages 163-164: 2,5,9.
Monday, February 1: finish Riemann mapping theorem; biholomorphic inequivalence of bidisc and unit ball; HW is exercise on page 5 of the notes.
Wednesday, February 3: finish biholomorphic inequivalence of bidisc and unit ball; HW is to read pages B,C,D of notes.
Friday, February 5: begin Weierstrass factorization theorem, read page 166 (up to Lemma 5.6 plus its proof); HW is page 173, problems 1,2.
Monday, February 8: continue Weierstrass factorization theorem; HW is page 173, problems 5,6,7.
Class moved to BLOC116 from now on.
Wednesday, February 10: class cancelled due to technical problems in BLOC161. Make up is on March 31.
Friday, February 12: finish Weierstrass factorization theorem for the case of entire functions; HW is page 173, problems 1,4.
Monday, February 15 through Friday, February 19: classes cancelled due to inclement weather; make up times to be announced.
Monday, February 22: factorization on a general region; HW is pages 173-174, problems 8,10,12.
Wednesday, February 24: the full Cauchy formula (Pompeiu formula), the ∂-equation.
Friday, February 26: finish the ∂-equation; begin Runge's theorem.
Monday, March 1: proof of Runge's theorem, first part.
Wednesday, March 3: proof of Runge's theorem, second part, begin corollaries of Runge's theorem; HW is problem 2 on page 201, plus the two exercises from notes 13.
Friday, March 5: Go over problems on Exam 2.
Monday, March 8: begin corollaries to Runge's theorem.
Wednesday, March 10: further corollaries to Runge's theorem, H(Ω)-hulls; HW is to read Theorem 2.2, page 202, characterizations of simple connectedness.
Friday, March 12: finish H(Ω)-hulls; HW is again to read Theorem 2.2, page 202, characterizations of simple connectedness.
Monday, March 15: solvability of the inhomogeneous ∂-equation in C^∞(Ω) (another application of Runge's theorem); begin Mittag-Leffler theorem.
Wednesday, March 17: finish Mittag-Leffler theorem; Schwarz reflection principle; HW is problems 5,6, page 209.
Thursday, March 18: redefined day, students attend Friday classes; reflection principle for circles; begin analytic continuation along a path; HW is problem 2, page 213 plus the two problems on page 4 of the notes.
Monday, March 22: continue analytic continuation along a path; begin monodromy theorem; HW is problems 1, 5-7 on pages 216-217.
Wednesday, March 24: continue monodromy theorem; give out Exam 2, due Monday.
For more on monodromy (in the abstract), see Conway, §7,Chapter IX, or R. Narasimhan, Complex Analysis in One Variable (Birkhäuser), downloadable version is here.
Friday, March 26: finish monodromy theorem; begin construction of the elliptic modular function,
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. Classcal reference for this theorem are Goluzin's book Geometric Theory of Functions of a Complex Variable and Nehari's book Conformal Mapping; the latter also for the elliptic modular function. For modern references on Caratheodory's theorem, see this Math Wiki article.
Monday, March 29: the elliptic modular function, covering maps. Exam 2 due today.
Wednesday, March 31: two lectures (one make up for February 10); lifting of analytic functions through a covering map; begin Chapter X, (part of) the Montel-Caratheodory and Picard theorems; little Picard theorem, Montel-Caratheodory theorem; HW as assigned in class notes.
Friday, April 2: reading day, no classes.
Monday, April 5: big Picard theorem, finish the Montel-Caratheodory theorem, HW is page 301, problems 1-3, problem 4 from class notes.
Wednesday, April 7: harmonic functions, mean value property, maximum principles; HW is pages 255-256, problems 4, 6-8.
Friday, April 9: maximum principles, harmonic functions in a disc, Poisson kernel; HW reassigned form Wednesday.
Monday, April 12: Poisson kernel as an approximation to the identity, solution of the Dirichlet problem on a disc; HW is to read Corollaries 2.9 and 2.10.
Wednesday, April 14: the space of harmonic functions on a domain; Harnack's inequality, Harnack's theorem; HW is pages 262-263, problems 2, 4, 5.
Friday, April 16: start Chapter XI, Entire Functions, Jensen and Poisson-Jensen formulas, genus of an entire function; HW is pages 281-282, problems 1-3.
Monday, April 19: order of an entire function, finite genus implie sfinite order.
Wednesday, April 21: finish finite genus, begin Hadamard factorization theorem; HW is problem 2, page 282.
Friday, April 23: continue Hadamard factorization theorem; HW is page 286, problems 1,2,4,7.
Monday, April 26: finish Hadamard factorization theorem. Discuss HW.
Wednesday, April 28: entire functions with non-integer finite order; discuss HW; this is the last class this spring; Exam 3 sent out after class, due Monday, May 3, 1:00pm.