MATH 230 - Complex Analysis II

Gauss

Riemann

When & Where: MWF 11-noon on Zoom

Email: roberto DOT deleo AT howard DOT edu

Textbook

Ahlfors, Complex Analysis, McGraw Hill, 1979

Suggested Books/Lecture Notes

Do consult these texts when you need more than you find in the textbook!

• Curtis McMullen's Complex Analysis notes
• Terence Tao's Complex Analysis notes.
• A First Course in Complex Analysis, a nice introductory undergraduate book with many examples and exercises.

Other Texts

• C. McMullen, Lecture notes on Advanced Complex Analysis
• W. Schlag, A Course in Complex Analysis and Riemann Surfaces, MIT Open Courseware

Noteworthy Lecture notes in Complex Analysis

• F. Monard, USCS.
• M. Stoll, U. of Bayreuth, Germany

Other Readings

• Useful elementary facts about poinwise and uniform convergence by S. Angenent (U. of Wisconsin - Madison)
• J.D. Gray and S.A. Morris, When is a function that satisfies the Cauchy-Riemann equations analytic?, The Am. Math. Monthly, 85, 1978
• A.F. Beardon and D. Minda, On the Pointwise Limit of Complex Analytic Functions, The Am. Math. Monthly, 110, 2003
• The hyperbolic metric and geometric functions theory, by A.F. Beardon (Imperial College) and D. Minda (U. of Cincinnati).
• H.P. Boas, JULIUS AND JULIA: MASTERING THE ART OF THE SCHWARZ LEMMA
• L. Zalcman, Normal Families: New Perspectives, Bull. of AMS, 35:3
• Some detailed notes on the proof of the Great Picard theomem by S. Friedl (U. of Regensburg, Germany)
• An introduction to Complex Dynamics by W. Bergweiler, U. of Kiel (Germany).
• Dynamics of Entire Functions by Dierk Schleicher, Université d'Aix-Marseille (France).
• Hyperbolic Art and the Poster Pattern, a nice art page about hyperbolic geometry with several interesting links.
• D.M. Campbell, Beauty and the beast: the strange case of Andre' Bloch
• H. Cartan and J. Ferrand, The case of Andre' Bloch
• The non-squeezing theorem in symplectic geometry
• The Symplectic Camel
• J.L. Walsh, History of the Riemann Mapping Theorem
• Lectures on moduli spaces of elliptic curves, by R. Hain (Duke U.)

Grades policy

Grades will be evaluated on the following bases:

1. (Almost) weekly homework: 30%;
2. Two take-home midterms: 2x20%;
3. Final exam: 30%.

Based on the total, you will get the following letter: A (90-100%), B (80-89%), C (70-79%), D (60-69%), F (0-59%).

Collaboration

Especially with online classes, teamwork is very important and will give you the occasion to interacct with your classmates. You are encouraged in general to work with study partners and also to collaborate on homework. On the other side, you must write your solutions yourself, in your own words, and you must list all collaborators and outside sources of information.
It is a serious violation of academic integrity to copy an answer without attribution.

Academic Code of Student Conduct

(please see Howard University handbook) No copying, unauthorized use of calculators, books, or other materials, or changing of answers or other academic dishonesty will be tolerated. Cheating will not be tolerated. Anyone caught cheating will receive an F for the course and may be expelled from the university.

Grievance Procedure

If you have any problems with the policies or rules of this course, discuss your concerns with your instructor. If you are still unable to come to a satisfactory arrangement, you may contact the Director of Undergraduate Studies, Dr. Jill McGowan, and, if still unsatisfied, the Chair of the Department, Dr. Bourama Toni.

Americans with Disabilities Act

Howard University is committed to providing an educational environment that is accessible to all students. In accordance with this policy, students in need of accommodations due to a disability should contact the Office of the Dean for Special Student Services (202-238-2420, bwilliams@howard.edu) for verification and determination of reasonable accommodations as soon as possible after admission and at the beginning of each semester as needed.

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Lectures plan:

Week	Content	Topics
1	Review	The Fundamental Theorem of Complex Analysis.
2	Review	The Fundamental Theorem of Complex Analysis.
3	Chapter 5	Entire functions. Jensen's formula. Hadamard's Theorem.
4	Chapter 5	Entire functions. Jensen's formula. Hadamard's Theorem.
5	Chapter 5	Entire functions. Jensen's formula. Hadamard's Theorem.
6	Chapter 5	Trigonometric functions. $\Gamma$ function.
7	Chapter 5	Normal Families. Riemann Mapping Theorem.
8	Chapter 8	Riemann Mapping Theorem. Picard's Theorem.
9	Chapter 6	Harmonic functions.
10	Chapter 7	Doubly periodic functions.
11	Chapter 7	The Weirstrass theory.
12	Chapter 8	Analytic continuation.
13	Chapter 8	Linear Differential Equations.
14		Review.

Lectures log

Week	Day	Sections	Topics
1	22 Jan		Review of CA I.
2	25 Jan	Book draft	The Fundamental Theorem of Complex Analysis.
	27 Jan	Book draft	The Fundamental Theorem of Complex Analysis.
	29 Jan	Notes, p. 18	Runge's Theorem.
3	1 Feb	Notes, p. 19	Characterization of power series of rational functions.
	3 Feb	Notes, p. 20	Locally every holomorphic map is $z^n$ for some integer $n\geq0$.
	5 Feb	Notes, p. 1-4; Chapter 5, Section 3.1	Entire Maps. Jensen's formula.
4	8 Feb	Notes, p. 4-5 Chapter 5, Section 2.2	The counting function. Structure of entire functions.
	10 Feb	Notes, p. 6 Chapter 5, Section 2.3	Weierstrass' Theorem.
	12 Feb	Notes, p. 7; Chapter 5, Section 2.4	The canonical product.
5	15 Feb		President Day.
	17 Feb		Canceled, to be recovered.
	19 Feb	Notes, p. 8-9; Chapter 5, Section 2.3	Hadamard's Theorem.
6	22 Feb	Notes, p. 10-12; Chapter 4, p. 135 The 'real' Schwarz Lemma	Two important theorems: Schwarz Lemma and Hadamard Theorem on polynomial growth.
	24 Feb	Notes, p. 14;	Trigonometric functions.
	26 Feb	Notes, p. 15-16; Chapter 5, Section 2.3 Intro to the $\Gamma$ function	The $\Gamma$ function.
7	1 Mar	Notes, p. 1-2; Chapter 4, p. 135	Last facts on $\Gamma$ function. Topology on the space of continuous functions.
	3 Mar	Notes, p. 3-4; Chapter 5, Section 4	Equicontinuity. Normal sets of maps. Montel Theorem.
	5 Mar	Notes, p. 5; Chapter 5, Section 1, Thm 2	Hurwitz Theorem.
Spring break
8	15 Mar	Notes, p. 1-2;	Hurwitz corollaries. Simply-connected subsets of $\mathbb C$.
	17 Mar	Notes, p. 6; Chapter 8, Section 1	Every simply-connected proper subset of $\mathbb C$ can be embedded in the unit open disc.
	19 Mar	Notes, p. 7; Chapter 8, Section 1	Riemann Mapping theorem.
9	22 Mar	Notes, p. 8-9;	Spherical Geometry. Spherical derivative of a meromorphic map.
	24 Mar	Notes, p. 10-11; Beardon & Minda	Automorphisms of simply connected domains and their relation with hyperbolic geometry.
	26 Mar	Notes, p. 11-13;	Marty's Theorem. Some Lemma.
10	29 Mar	Notes, p. 14-15;	"Twisted limit" of a sequence of entire maps. Little Picard Theorem.
	31 Mar	Notes, p. 16-18;	"Twisted limit" of a non-normal sequence of meromorphic maps. Montel Theorem. Great Picard Theorem
	2 Apr	Notes, p. 19; Notes, p. 1	Holomorphic maps of discs. Harmonic Maps.
11	5 Apr	Notes, p. 2-6;	General properties of harmonic maps. Maximum principle.
	7 Apr	Notes, p. 6-8;	Existence and uniqueness of Laplace eq. in simply-connected domains. Poisson Formula. Continuous maps satisfying the mean value property are harmonic.
	9 Apr	Notes, p. 9-11;	Schwarz Reflection Principle. Harnack inequalities.
12	12 Apr	Notes, p. 12-13; Notes, p. 1;	Harnack's Principle. Harmonic maps cannot be bounded below or above. Reminders about simply-periodic functions.
	14 Apr	Notes, p. 2-3;	Elliptic Functions. Schwarz-Christoffel formula.
	16 Apr	Notes, p. 4-5;	Weierstrass P function