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!

• 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

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

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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	24 Aug	Chapter 1 1.1	Complex numbers.
	26 Aug	1.2, 1.3	Different ways to define and represent complex numbers. Inverse and square root of a complex number.	1.2: 1,3
	28 Aug	1.3, 1.4, 1.5	Cauchy sequences. Complete metric spaces. Conjugation. Complex solutions of real polynomial equations. Geometric properties of complex numbers.	1.3:1, 1.4:3,4
2	31 Aug	1.5, 2.1, 2.2	Cauchy inequality. Geometric Addition and multiplication. Binomial equation.	1.5:3, 2.1:3, 2.2:4
	2 Sep	2.3. Chapter 2 1.1	Analytic Geometry. Limits.
	4 Sep	1.1, 1.2	Continuity, Differentiability.
3	7 Sep		Labour Day.
	9 Sep	1.2	Cauchy-Riemann equations.	1.2:4,7
	11 Sep	1.3	Polynomials.	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	14 Sep	1.3	Lucas' Theorem.
	16 Sep	1.4, 2.1	Rational Functions. Sequences.	1.4:1,4
	18 Sep	2.2, 2.3	Uniform convergence. Series.	2.2:4
5	21 Sep	2.4	Abel's Theorem on power series.	2.4: 2,3
	23 Sep	2.4, 2.5, 3.1	Analiticity of power series within the radius of convergence. Second Abel theorem. Exponential function.	2.4: 5,7.
	25 Sep	3.2, 3.3, 3.4	Trigonometric and Logarithmic functions	3.2: 1. 3.3: 6,7
6	28 Sep	3.4 Chapter 3 1.1	Logarithmic functions. Point set topology.
	30 Sep	1.2	Metric Spaces	1.2: 1, 4
	2 Oct	1.3	Connectedness	1.3: 3, 7
7	5 Oct	1.3	Connectedness	1.3: 5
	7 Oct	1.4	Compactness	1.4: 4,5
	9 Oct	1.5, 1.6	Continuous Maps. Topological Spaces.	1.5: 1
8	12 Oct	2.1, 2.2	Piecewise smooth paths. Holomorphic maps.	2.2: 3
	14 Oct	2.3, 2.4	Conformal maps. Length and Area.
	16 Oct	3.1, 3.2	Projective space. Mobius trasnformations. Cross ratio.	3.1: 1, 4
9	19 Oct	Notes, p. 11	Goursat Theorem
	21 Oct	Notes, p. 2-3	Darboux Theorem
	23 Oct	Notes, p.4	Differential Forms in $\Bbb R^n$. Stokes Theorem.	1. Given the curve $\gamma(t)=(R\cos t,R\sin t), t\in[0,2\pi]$, evaluate the line integral over $\gamma$ of the 1-form $xdy-ydx$. After you get the result, try to understand its relation with the curve and explain why it is so (Stokes Thm...)
10	26 Oct	Notes, p. 5-7	Complex Differential 1-forms. Green Theorem. $C^1$ Cauchy Theorem	2. Let $f$ be a holomorphic map. Evaluate the external differential 2-form $d(f(z)dz)$. 3. Given the curve $\gamma(t)=Re^{it}, t\in[0,2\pi]$, evaluate the line integral over $\gamma$ of the 1-form $\bar z dz$. After you get the result, try to understand its relation with the curve and explain why it is so (Stokes Thm...).
	28 Oct	Notes, p. 8-9	Primitive of a closed 1-form over a path.	4. Find a primitive of $dz/z^2$ over the path $\gamma(t)=t+it^2, t\in[0,1]$.
	30 Oct	Notes, p. 10-12	Winding number. Homotopy. $C^0$ Cauchy Theorem	5. Write explicit homotpies for the following two pairs of homotopic paths: a. $\gamma_0(t)=t$, $\gamma_1(t)=t^2$; b. $\gamma_0(t)=e^{2\pi it}$, $\gamma_1(t)=e^{2\pi it}/2$. Notice that in the first case the homotopy must keep the endpoints fixed. In all cases the domain of the paths is $[0,1]$. In the second case, do find a primitive with respect to the homotopy of $dz/z$.
11	2 Nov	T. Tao Notes	Cauchy Integral Formula (Thm 15). Mean value property (Ex. 17). Holomorphic functions are analytic (Cor 18, Cor 20)	Ch 4, §2.2: 2,3
	4 Nov	T. Tao Notes	Factor Theorem (Cor 22). Analytic continuation (Cor 23). Non-constant analytic functions have only isolated zeros (Cor 24). Higher order Cauchy Integral formula (Thm 25). Cauchy inequalities (Cor 27).	Ch 4, §2.3: 1,2
	6 Nov	T. Tao Notes	Liouville Thm (Thm 27). Fundamental Thm of Algebra (Thm 30). Morera's Thm (Thm 33). Uniform limits of hol. functions are hol. (Thm 34). Removable singularities (Ex. 35). Schwartz reflection principle (Ex. 37).	Ch 4, §2.3: 6 (this is Ex. 36 in Tao's lectures)
12	9 Nov	T. Tao Notes	Isolated Singularities. Decomposition of a function in an annular region (Lemma 1).
	11 Nov		Veteran's Day
	13 Nov	T. Tao Notes	Meromorphic functions.
13	16 Nov	T. Tao Notes	Classification of isolated singularities. Laurent series.
	18 Nov	T. Tao Notes	The residue theorem. Using complex line integrals to evaluate real integrals.
	20 Nov	T. Tao Notes	The argument principle. Rouché Theorem.	Last Homework set: Determine the number of zeros of $2z^5-6z^2+z+1$ in the annulus $1\leq z\leq2$. Compute the integral $\displaystyle\int_0^\infty\frac{\log^2(x)}{1+x^2}dx$. Assume that $f$ and $g$ are entire, that $g(z)$ is never zero and that $|f(z)|\leq|g(z)|$ for all $z\in\Bbb C$. Prove that $f(z)=cg(z)$ for some complex constant $c$. Compute $\displaystyle\int_0^{2\pi}\frac{\cos\theta}{5-4\cos\theta}d\theta$. Calculate the Laurent series of $f(z)=\frac{z}{z^2-4z+3}$ in the following domains: $|z| < 1$, $1<|z|<3$, $3<|z|$.
14	23 Nov	T. Tao Notes F. Monard notes	Open mapping theorem. Maximum modulus principle. Inverse function theorem.
	25 Nov	T. Tao Notes	Midterm 2 review.
	27 Nov		Classes are over.