MATH 185 - Introduction to Complex Variables

Gauss

Riemann

When & Where: MWF 2-3pm on Blackboard Collaborate Ultra

Email: roberto DOT deleo AT howard DOT edu

Textbook

M. Beck, G. Marchesi, D. Pixton, L. Sabalka, A First Course in Complex Analysis

Other Texts

• J.P. D'Angelo, An Introduction to Complex Analysis and Geometry
• J. Orloff, Lecture notes on Complex Variables with Applications, 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	Continuity, differentiability and holomorphicity.
3	3.1--3.3	Examples of functions 1.
4	3.3--3.5	Examples of funcitons 2.
5	4.1, 4.2	Integration 1.
6	4.3, 4.4	Integration 2.
7	Chapter 5	Applications of Cauchy's theorem.
8	Chapter 6	Harmonic functions.
9	7.1, 7.2	Sequences and series
10	7.3, 7.4	Sequences and series of functions.
11	Chapter 8	Taylor and Laurent series .
12	Chapter 9	Isolated singularities and the resideu theorem.
13	Chapter 10	Discrete applications of the residue theorem.
14		Review.

Lectures log

Week	Day	Sections	Topics	Homework
1	24 Aug	1.1	Complex numbers.
	26 Aug	1.2	Cartesian and polar representations of complex numbers.	1.1,1.2,1.4(a-c),1.5(a-b)
	28 Aug	1.3, 1.4	Geometric properties of complex numbers. Basic topological definitions.	1.9, 1.10, 1.13, 1.19, 1.20, 1.29
2	31 Aug	1.4, 2.1	Planar topology. Limits.	2.3
	2 Sep	2.1, 2.2	Continuity. Differentiability and Holomorphicity.	2.6, 2.15, 2.17
	4 Sep		University closed
3	7 Sep		Labour Day.
	9 Sep	2.3	Cauchy-Riemann equations.	2.20, 2.23
	11 Sep	2.3,2.4	Harmonic Functions, Constant Functions	2.21, 2.29
4	14 Sep	3.1	Mobius transformations	3.5
	16 Sep	3.2	Cross ratio and the Riemann sphere	3.14, 3.19
	18 Sep	3.4	Exponential and trigonometric functions	3.31, 3.52
5	21 Sep	3.5, 4.1	Complex logarithm and exponential. Integration.	3.31, 4.7
	23 Sep	4.2, 4.3	Antiderivatives. Homotopies. Cauchy's Theorem.	4.3, 4.5, 4.11, 4.15, 4.29, 4.30
	25 Sep	4.3	Cauchy's Theorem.	4.37
6	28 Sep	4.4	Cauchy's integral formula.	4.34, 4.39
	30 Sep	5.1, 5.2	For complex functions, C1 implies smooth. Morera's Theorem.	5.3,5.17,5.18
	2 Oct	5.3	The Fundamental Theorem of Algebra. Entire holomorphic functions are all constant.	Extra Credit: 5.13
7	5 Oct	5.3, 6.1	Evaluating real integrals via complex ones. Harmonic functions.	5.20, 6.1
	7 Oct	6.2	Harmonic functions, Mean value theorem.	6.8
	9 Oct	6.2, 7.1	Maximum/minimum principle. Converging sequences.	6.11
8	12 Oct	7.1	Sequences and Completeness.	7.2, 7.15
	14 Oct	7.2	Series. Absolute convergence.	7.18
	16 Oct	7.2, 7.3	Series convergence. Series of functions. Pointwise and uniform convergence.	7.21
9	19 Oct	7.3	Exchanging derivative with infinite sign symbols.	7.27
	21 Oct	7.4	Region of convergence of power series	7.34
	23 Oct	8.1	Holomorphic functions and Power series	8.3
10	26 Oct		Canceled for bad BB Ultra connectivity.
	28 Oct	8.1	Holomorphic functions and Power series	8.9
	30 Oct	8.2	Classification of zeros.
11	2 Nov	8.2	Identity Principle
	4 Nov	8.3	Maximum modulus Theorem
	6 Nov	8.3	Laurent series	8.34, 8.36
12	9 Nov	9.1	Classification of singularities	9.1, 9.3, 9.4
	11 Nov		Veteran's Day
	13 Nov	9.2	Essential singularities. Residues.	9.8b, 9.8e, 9.17
13	16 Nov	9.3	Argument principle and Rouch\'e's Theorem
	18 Nov		Infinite sums. Binomial coefficients.
	20 Nov	9.2	Fibonacci numbers. Coin-exchange problem. Dedekind sums.