MAT 421 Complex Analysis, Section 01, 21826, Spring 2018

Class meets MW 1:00 PM - 2:15 PM in NSM D129.

Instructor: Serban Raianu

Office: NSM E-108, office phone number: (310) 243-3139,

Office hours: Monday, Wednesday, Friday: 8:50 AM - 9:50 AM, Friday: in the Toro Learning Center: 11:30 AM – 12:30 PM, or by appointment.

Course Description: This course covers the algebra and geometry of the complex numbers; point sets, sequences and mappings; analytic functions; elementary functions; differentiation; integration; power series; the calculus of residues; and applications.

Text: Complex Analysis, by George Cain, available online at http://www.math.gatech.edu/~cain/winter99/complex.html

Objectives: After completing MAT 421 the student will

• understand complex numbers, the algebra and geometry of complex numbers and the complex plane
• understand and work with complex vectors, polar forms, powers, and roots
• understand limits and continuity, analyticity, and the Cauchy-Riemann equation
• understand complex exponential, trigonometric, hyperbolic, logarithmic and power functions
• understand complex integration, contour integrals, the Cauchy Integral Theorem and formula, and bounds for analytic functions
• understand sequences and series, including Taylor series, power series, and Laurent series and their use in representing analytic functions
• understand residue theory
• be able to prove basic theorems related to the above concepts
• apply mathematical reasoning and the theory of complex variables to solve theoretical and applied problems.

Prerequisites: MAT 211 and MAT 271  or equivalent with a grade of "C" or better.

Grades: Grades will be based on three in‑class full‑period examinations (60% total), a comprehensive final examination (25%), and quizzes, homework, and other assignments (15%) for the remainder. The exact grading system for your section is the following: each of the three full-period exams will be graded  on a 100 scale, then the sum of the scores is divided by 5 and denoted by E. Homework will be collected every Monday, and each homework is worth 5 points. No late homework will be accepted. The average of all homework scores is denoted by H.

5 to 10 minutes quizzes will be given in principle every Monday, with the exception of the review and exam days, and will be graded on a scale from 1 to 5. The average of the quizzes scores is denoted by Q. There are also 5 points awarded for attendance and class participation, this portion of the grade is denoted by A. The final exam will be graded out of a maximum possible 200, then the score is divided by 8 and denoted by F.

To determine your final grade  compute E+H+Q+A+F. The maximum is 100, and the grade will be given by the rule:

A: 93‑100;      A‑: 90‑92;      B+: 87‑89;      B: 83‑86;   B‑: 80‑82

C+: 77‑79;      C: 73‑76;        C‑: 70‑72;      D: 60‑69;   F: Less than 60.

Makeups: No makeup examinations or quizzes will be given. If you must miss an examination for a legitimate reason, let me know in advance, and I may then substitute the relevant score from your final examination for the missing grade.

Accommodations for Students with Disabilities: Cal State Dominguez Hills adheres to all applicable federal, state, and local laws, regulations, and guidelines with respect to providing reasonable accommodations for students with temporary and permanent disabilities. If you have a disability that may adversely affect your work in this class, I encourage you to register with Student disAbility Resource Center (SdRC) and to talk with me about how I can best help you. All disclosures of disabilities will be kept strictly confidential. Please note: no accommodation may be made until you register with the SdRC in WH D-180. For information call (310) 243-3660 or to use telecommunications Device for the Deaf, call (310) 243-2028.

Academic Integrity: The mathematics department does not tolerate cheating. Students who have questions or concerns about academic integrity should ask their professors or the counselors in the Student Development Office, or refer to the University Catalog for more information. (Look in the index under "academic integrity".)

Tentative schedule:

M 1/22: 1.1 Complex numbers. Introduction; 1.2 Geometry

W 1/24: 1.3 Polar coordinates; 2.1 Functions of a real variable

M 1/29: 2.2 Functions of a complex variable; 2.3 Derivatives

W 1/31: 3.1 Elementary functions. Introduction; 3.2 The exponential function

M 2/5: 3.2 The exponential function; 3.3 Trigonometric functions

W/2/7: 3.4 Logarithms and complex exponents

W 2/12: Review

M 2/14: Exam I

M 2/19: Presidents’ Day Holiday

W 2/21: 4.1 Integration. Introduction; 4.2 Evaluating integrals

M 2/26: 4.2 Evaluating integrals; 4.3 Antiderivatives

W 2/28: 5.1 Homotopy; 5.2 Cauchy’s Theorem

M 3/5: 5.2 Cauchy’s Theorem; 6.1 Cauchy’s Integral Formula

W 3/7: 6.2 Functions defined by integrals; 6.3 Liouville’s Theorem

M 3/12: 6.3 Liouville’s Theorem; 6.4  Maximum moduli

W 3/14: 7.1 The Laplace equation; 7.2 Harmonic functions

M 3/19: 7.2 Harmonic functions; 7.3 Poisson’s integral formula

W 3/21: Review

M 3/26: Spring Recess

W 3/28: Spring Recess

M 4/2: Exam II

W 4/4: 8.1 Sequences

M 4/9: 8.2 Series

W 4/11: 8.3 Power series

M 4/16: 8.4 Integration of power series; 8.5 Differentiation of power series

W 4/18: 8.5 Differentiation of power series; 9.1 Taylor series

M 4/23: 9.2 Laurent series

W 4/25: 10.1 Residues; 10.2 Poles and other singularities

M 4/30: 11.2 Argument principle; 11.2 Rouché’s Theorem

W 5/2: Review

M 5/7: Exam III

W 5/9: Review

Final exam: Monday, May 14, 1:00 PM – 3:00 PM

Important dates: