Math 542 Lecture Log and Homework Assignments

Math 542: Lecture Log and Homework Assignments

All references are to the course textbook (Palka), by chapter, section and subsection, unless otherwise specified.

References to book by Conway are denoted [C]; references to the book by Stein-Shakarchi are denoted [SS]; references to the book by Ahlfors are denoted [A]; references to the book by Ransford are denoted [R].

Date Topics covered Reference Remarks

8/22 Introduction and overview, the complex number system, matrix presentation, Argand plane, polar coordinates, complex roots I.1 Welcome to Math 542!

8/24 Elementary functions of a complex variable: polynomials, rational functions, exponential, logarithm, Mobius transformations. Functions as mappings I.2, I.3

8/27 Basic point-set topology in the complex plane: open and closed sets, relatively open and closed sets, the extended complex number system, the Riemann sphere, stereographic projection II.1, II.2, VIII.4.1, VIII.4.2, [C] I.6

8/29 Continuity, connectedness, path connectedness, Jordan Curve Theorem, Jordan-Schoenflies theorem, analytic functions, Cauchy-Riemann equations II.2, II.3, IV.1.1, III.1, III.2

8/31 Cauchy-Riemann equations, z and z bar derivatives, polarization of analytic identities, polynomials, Lucas' theorem III.5.2, VI.1, [C] III.2, [A] II.1.3 Homework #1 due

9/5 rational functions, linear fractional transformations, cross ratio [A] II.1.4, IX.2, [C] III.3, [A] III.3

9/7 sequences, series, modes of convergence, Weierstrass M-test, Taylor series, radius of convergence VII.1, VII.2, VII.3.2, VII.3.3 Homework #2 due

9/10 piecewise smooth paths, contour integrals, arc length integrals IV.1, IV.2

9/12 primitives of continuous functions, Cauchy's Theorem, applications V.1

9/14 Cauchy's theorem (continued), winding numbers V.2

9/17 Cauchy integral formula, analytic functions are infinitely differentiable, Morera's theorem, Liouville's theorem, Fundamental Theorem of Algebra V.2.3, V.3.1, V.3.2 Homework #3 due

9/19 applications of the Cauchy integral formula, power series, exponential and logarithm via power series VII.3.1, VII.3.3, III.3, III.4

9/21 inverse branches of analytic functions III.4

9/24 singularities (removable, poles, essential), residues, discreteness of analytic functions, applications, Residue Theorem VIII.1, VIII.2, VIII.3.1 Homework #4 due

9/26 using residues to compute integrals I VIII.3.2 guest lecturer: Prof. J. P. D'Angelo

9/28 using residues to compute integrals II VIII.3.2 guest lecturer: Prof. J. P. D'Angelo

10/1 Argument Principle, Rouche's theorem VIII.3.3 Homework #5 due

10/3 analytic mappings as branched coverings, Open Mapping Theorem, Maximum Principle, Schwarz Lemma VIII.3.3, V.3.3

10/5 homology, global Cauchy theorem, connectivity of domains V.5.2, V.5.3, V.6.1

10/8 homotopy, Maximum Modulus theorem, Hadamard Three Lines and Three Circles Theorem V.7, V.3.3

10/10 Hadamard Three Lines and Three Circles Theorem (continued), Laurent series V.3.3, VII.3.4 Homework #6 due

10/12 Laurent series (continued) VII.3.4

10/15 Normal families, Arzela-Ascoli theorem, Montel's theorem VII.4

10/17 Normal families, examples, Weierstrass and Hurwitz theorems VII.4

10/19 conformal maps, examples, statement of the Riemann mapping theorem IX.1 Homework #7 due

10/22 proof of the Riemann mapping theorem IX.3

10/24 moduli of Riemann maps, Schwarz reflection principle, Schwarz-Christoffel mappings IX.5.2, IX.5.3

10/26 Schwarz-Christoffel mappings (examples) IX.5.3 guest lecturer: Prof. R. Laugesen

10/29 Mittag-Leffler theorem, examples X.1 Homework #8 due

10/31 elliptic functions, Weierstrass P-function X.2

11/2 MIDTERM EXAM

11/5 Weierstrass P-function: differential equation, infinite products X.2.1, [A] VII.3.3

11/7 Infinite products of analytic functions, Weierstrass Factorization Theorem, Gamma function X.2.3, X.2.4, [A] V.2.4

11/9 Gamma function, Riemann zeta function [A] V.2.4, [A] V.4

11/12 Runge approximation theorem [C] VIII.1

11/14 Runge approximation theorem (continued), Pade approximation [C] VIII.1

11/16 analytic continuation, introduction to Riemann surfaces X.3 Homework #9 due

11/26 harmonic functions, Mean Value Property VI.1, VI.2, [R] I.1

11/28 subharmonic functions, Dirichlet problem [R] II

11/30 Dirichlet problem (continued), Harnack's inequalities, Liouville theorem for harmonic functions VI.3, [R] I.2, I.3

12/3 Perron's method, barriers [R] IV.1, IV.2

12/5 Green's function, potential-theoretic proof of Picard's theorem [R] VI.4, I.3

12/7 review for the final exam

Date	Topics covered	Reference	Remarks
8/22	Introduction and overview, the complex number system, matrix presentation, Argand plane, polar coordinates, complex roots	I.1	Welcome to Math 542!
8/24	Elementary functions of a complex variable: polynomials, rational functions, exponential, logarithm, Mobius transformations. Functions as mappings	I.2, I.3
8/27	Basic point-set topology in the complex plane: open and closed sets, relatively open and closed sets, the extended complex number system, the Riemann sphere, stereographic projection	II.1, II.2, VIII.4.1, VIII.4.2, [C] I.6
8/29	Continuity, connectedness, path connectedness, Jordan Curve Theorem, Jordan-Schoenflies theorem, analytic functions, Cauchy-Riemann equations	II.2, II.3, IV.1.1, III.1, III.2
8/31	Cauchy-Riemann equations, z and z bar derivatives, polarization of analytic identities, polynomials, Lucas' theorem	III.5.2, VI.1, [C] III.2, [A] II.1.3	Homework #1 due
9/5	rational functions, linear fractional transformations, cross ratio	[A] II.1.4, IX.2, [C] III.3, [A] III.3
9/7	sequences, series, modes of convergence, Weierstrass M-test, Taylor series, radius of convergence	VII.1, VII.2, VII.3.2, VII.3.3	Homework #2 due
9/10	piecewise smooth paths, contour integrals, arc length integrals	IV.1, IV.2
9/12	primitives of continuous functions, Cauchy's Theorem, applications	V.1
9/14	Cauchy's theorem (continued), winding numbers	V.2
9/17	Cauchy integral formula, analytic functions are infinitely differentiable, Morera's theorem, Liouville's theorem, Fundamental Theorem of Algebra	V.2.3, V.3.1, V.3.2	Homework #3 due
9/19	applications of the Cauchy integral formula, power series, exponential and logarithm via power series	VII.3.1, VII.3.3, III.3, III.4
9/21	inverse branches of analytic functions	III.4
9/24	singularities (removable, poles, essential), residues, discreteness of analytic functions, applications, Residue Theorem	VIII.1, VIII.2, VIII.3.1	Homework #4 due
9/26	using residues to compute integrals I	VIII.3.2	guest lecturer: Prof. J. P. D'Angelo
9/28	using residues to compute integrals II	VIII.3.2	guest lecturer: Prof. J. P. D'Angelo
10/1	Argument Principle, Rouche's theorem	VIII.3.3	Homework #5 due
10/3	analytic mappings as branched coverings, Open Mapping Theorem, Maximum Principle, Schwarz Lemma	VIII.3.3, V.3.3
10/5	homology, global Cauchy theorem, connectivity of domains	V.5.2, V.5.3, V.6.1
10/8	homotopy, Maximum Modulus theorem, Hadamard Three Lines and Three Circles Theorem	V.7, V.3.3
10/10	Hadamard Three Lines and Three Circles Theorem (continued), Laurent series	V.3.3, VII.3.4	Homework #6 due
10/12	Laurent series (continued)	VII.3.4
10/15	Normal families, Arzela-Ascoli theorem, Montel's theorem	VII.4
10/17	Normal families, examples, Weierstrass and Hurwitz theorems	VII.4
10/19	conformal maps, examples, statement of the Riemann mapping theorem	IX.1	Homework #7 due
10/22	proof of the Riemann mapping theorem	IX.3
10/24	moduli of Riemann maps, Schwarz reflection principle, Schwarz-Christoffel mappings	IX.5.2, IX.5.3
10/26	Schwarz-Christoffel mappings (examples)	IX.5.3	guest lecturer: Prof. R. Laugesen
10/29	Mittag-Leffler theorem, examples	X.1	Homework #8 due
10/31	elliptic functions, Weierstrass P-function	X.2
11/2			MIDTERM EXAM
11/5	Weierstrass P-function: differential equation, infinite products	X.2.1, [A] VII.3.3
11/7	Infinite products of analytic functions, Weierstrass Factorization Theorem, Gamma function	X.2.3, X.2.4, [A] V.2.4
11/9	Gamma function, Riemann zeta function	[A] V.2.4, [A] V.4
11/12	Runge approximation theorem	[C] VIII.1
11/14	Runge approximation theorem (continued), Pade approximation	[C] VIII.1
11/16	analytic continuation, introduction to Riemann surfaces	X.3	Homework #9 due
11/26	harmonic functions, Mean Value Property	VI.1, VI.2, [R] I.1
11/28	subharmonic functions, Dirichlet problem	[R] II
11/30	Dirichlet problem (continued), Harnack's inequalities, Liouville theorem for harmonic functions	VI.3, [R] I.2, I.3
12/3	Perron's method, barriers	[R] IV.1, IV.2
12/5	Green's function, potential-theoretic proof of Picard's theorem	[R] VI.4, I.3
12/7	review for the final exam

Homework assignments

Homework 1: due Friday, August 31
Homework 2: due Friday, September 7
Homework 3: due Monday, September 17
Homework 4: due Monday, September 24
Homework 5: due Monday, October 1
Homework 6: due Wednesday, October 10
Homework 7: due Friday, October 19
Homework 8: due Monday, October 29
Homework 9: due Friday, November 16
Homework 10: due Friday, December 7