MATH 229 - Complex Analysis I

Gauss

Riemann

When & Where: MWF 11-noon on Blackboard Collaborate Ultra

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!

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

Other Texts

• W. Schlag, A Course in Complex Analysis and Riemann Surfaces, MIT Open Courseware

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	Chapter 1	Complex numbers.
2	Chapter 2	Analytic functions, Power series.
3	Chapter 2	Power series, examples of functions.
4	Chapter 3	Elementary topology, conformality
5	Chapter 3	Linear transformations, Elementary conformal mappings.
6	Chapter 4	Integration, Cauchy's integral formula.
7	Chapter 4	Local properties of analytical functions.
8	Chapter 4	General form of Cauchy's thm, Calculus of residues.
9	Chapter 5	Power series expansions.
10	Chapter 5	Partial fractions.
11	Chapter 5	Entire functions, Riemann $\zeta$ function.
12	Chapter 5	Normal families.
13	Chapter 6	Riemann mapping theorem.
14		Review.

Lectures log

Week	Day	Sections	Topics	Homework
1	23 Aug	Chapter 1 1.1,1.3 Tao's Notes	The construction and algebra of the complex numbers.
	25 Aug	1.1,1.3 Tao's Notes	Different ways to define complex numbers. Operations with complex numbers.	Show that the multiplication by $i$ corresponds geometrically to a rotation by $\pi/2$ of the complex plane about the origin.
	27 Aug	1.2,1.4 Tao's Notes	Square roots of complex numbers. Conjugation and modulus of a complex numer.	1.2: 1(first two), 1.4: 3
2	30 Aug	1.5,2.1 Tao's Notes	Geometric Inequalities. Geometric representation of complex numbers.	1.5: 3
	1 Sep	2.2,2.3 Tao's Notes	Exponential notation of complex numbers. Complex equations of circles and straight lines.	2.3: 2
	3 Sep	Chapter 2 1.1	Continuous complex functions.
3	6 Sep		Labour Day.
	8 Sep	1.2	Complex plane as a metric and topological space.
	10 Sep	1.2	Complex differentiability. Cauchy-Riemann equations.	In the textbook (bottom of page 29) it is mentioned that the zeros of $p'(z)$ are contained in every convex polygon that contains the roots of $p(z)$. Prove directly this fact in case of a second degree polynomial.
4	13 Sep	1.2	The real and imaginary parts of a complex differentiable map are harmonic functions..	1.2: 3, 4, 5
	15 Sep	1.2	Differentiability of maps from $\Bbb R^n$ to $\Bbb R^m$. Recovering a complex differentialble map from its real or imaginary parts. Cauchy-Riemann conditions in $(z,\bar z)$ coordinates.
	17 Sep		Convocation
5	20 Sep	1.3	Polynomials in $z$ as complex differentiable maps.	Consider the polynomials $f(z)=2z$ and $g(z)=z^2$ and describe their action on the complex plane.
	22 Sep	Notes	Discussion on the real projective line.
	24 Sep		To be recovered
6	27 Sep	1.4	Riemann sphere. Rational functions as maps from the Riemann sphere to itself	p.33, #4
	29 Sep		${\Bbb C}P^1$	Let $R(z)=\frac{1}{z-2}$. 1. Write $R(z)$ as a function of $R([z:w])$ by replacing $z$ with $z/w$ and simplifying to get a new rational function. 2. Set $z=1$ to find a function $S(w)$. Which point corresponds, in terms of the coordinate $z$, to $w=0$? $w=1$? $w=\infty$? 3. Compare $\lim_{z\to\infty}R(z)$ with $\lim_{w\to0}S(w)$. Why do you get the same result?
	1 Oct		Mobius transformations. Partial fractions.
7	4 Oct	Chapter 2, 2.1, 2.2	General properties of Limits and Sequences	p.37, #1
	6 Oct	2.3	Uniform convergence.
	8 Oct	2,4	Series and Power series.	p.41, #1,3
8	11 Oct (18)	2.4	Any power series with positive radius of convergence is analytic.
	13 Oct (19)	2.5	Convergence of power series on the boundary of their disc of convergence. Exponential function.	p.44, #1
	15 Oct (20)	Chapter 3, 1.1, 2.2	Abel theorem. Exponential and logarithmic functions. Basic facts on open and closed sets. Maps holomorphic in a region.	p.47, #6 (recall that, by def, $a^b=\exp(b\log a)$).
9	18 Oct (21)	2.3	Conformal maps.
	20 Oct (22)	4.1, 4.2, 4.3	Visualization of conformal maps. Examples of Riemann surfaces. Differential forms.
	22 Oct (23)	Lecture Notes	Closed and exact forms. Stokes theorem.
10	25 Oct (24)	Lecture Notes	$f(z)$ is holomorphic if and only if $f(z)dz$ is closed. Integral of $dz/z$ on the unit circle.
	27 Oct (25)	Lecture Notes	Primitive of a closed 1-form on parametrized curves.
	29 Oct (26)	1.4	Definition and basic properties of Compact Sets.
11	1 Nov (27)	Lecture Notes	Winding number. Homotopy. Invariance of integrals of closed 1-forms under homotopy.
	3 Nov (28)	Lecture Notes	Every holomorphic map is analytic.
	5 Nov (29)	Lecture Notes	Cauchy Integral Formula.
Midterm, due Sunday Nov 7
12	8 Nov (30)	Lecture Notes	Cauchy Integral Formula.
	10 Nov (31)	Lecture Notes	Green, Goursat and Cohen theorems
	12 Nov (32)	Chapter 4, 2.3, 3.1	Morera's Thm, estimates on the derivatives, Liouville's Thm, Fund. Thm of Algebra, Removable singularities	p.133, #2,3,6; p.145, #2,3; p.147, #2.
13	15 Nov (33)	Lecture Notes	.
	17 Nov (34)	Tao's Lecture Notes	Laurent series expansion.
	19 Nov (35)	Mc Mullen's Annotated Notes	Residues Theorem.
14	22 Nov (36)	Mc Mullen's Annotated Notes	Residues Formula. Sum of residues over a Riemann surface. The argument principle.
	24 Nov		Thanksgiving, no lecture
	26 Nov		Thanksgiving, no lecture
15	29 Nov (37)	Mc Mullen's Annotated Notes	Open Map Theorem
	1 Dec (38)
	3 Dec (39)