# MAT 421 Complex Variables

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### Course Description

Complex numbers; point sets, sequences and mappings; analytic functions; elementary functions; integration; power series; the calculus of residues; and applications.

3 units credit.

### Prerequisites

MAT 211 and MAT 271 with a grade of "C" or better. MAT 331 and MAT 401 (may be taken concurrently) are recommended.

### Text

Texts are chosen by the instructor. For example:

Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, by Saff and Snider.

### Course Requirements, Tentative Schedule of Class Meetings and Topics, Readings, Assignments and Due dates, Exams

A schedule of class meetings, topics, assignments, due dates, exam dates, etc. will be provided by instructor. See your class syllabus.

Here is an example course outline, based on the above text.

• Chapter 1-Complex Numbers
• The Algebra of Complex Numbers
• Point Representation, Absolute Value, and Complex Conjugates
• Vectors, Powers, Roots, and Polar Forms
• Chapter 2-Analytic Functions
• Functions of a Complex Variable
• Limits, Continuity, and Analyticity
• The Cauchy-Reimann Equation and Harmonic Functions
• Chapter 3-Elementary Functions
• Exponential, Trigonometric, Hyperbolic, and Logarithmic Functions
• Complex Powers and Inverse Trigonometric Functions
• Chapter 4-Complex Integration
• Contours and Contour Integrals
• Independence of Path
• Cauchy's Integral Theorem including the Deformation of Contours Approach and the Vector Analysis Approach
• Cauchy's Integral Formula and Bounds for Analytic Functions
• Chapter 5-Series Representations for Analytic Functions
• Sequences and Series, including Taylor, Power, and Laurent Series
• Zeros, Singularities, and the Point at Infinity
• Chapter 6-Residue Theory
• The Residue Theorem
• Trigonometric Integrals, Improper Integrals, and Integrals involving Multiple-Valued Functions

The final exam is given at the date and time announced in the Schedule of Classes.

### Learning Objectives

After completing MAT 421 the student

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

### Computers and Calculators, Computer Literacy

Most instructors encourage the use of machines, calculators computers, phones etc., for analyzing data. The use of machines may be restricted during examinations or at certain other times. Ask your instructor for the policy in your class.

Students are not expected to be programmers or to know any particular computer language before starting this class. Some instructors may expect students to be able to access information on the internet, or to use calculators, or to learn to use particular software with instruction. Basic skill in algebra and the use of mathematical symbols, order of operations etc., and the willingness to read and follow instruction manuals and help files will suffice.

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in this course. The instructor determines the relative weights of these factors and the grading scale. See the syllabus for your particular class.

### Location of Class Meetings

Classes meet on the dates and room announced in the official Schedule of Classes. This is a traditional, face-to-face class.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates, Make-Up Work, Missed Exams, and Extra-Credit Assignments

Due dates and policy regarding make-up work and missed exams are set by the instructor. Instructors may, or may not, choose to offer extra credit assignments. If extra credit assignments are offered they will be available to all students.

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".)

### Accomodations 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 Disabled Student Services (DSS) and to talk with me about how we best can help you. All disclosures of disabilities will be kept strictly confidential. Please note: you must register with DSS to arrange an no accommodation. For information call (310) 243-3660 or send an email message to dss@csudh.edu or visit the DSS website http://www4.csudh.edu/dss/contact-us/index or visit their office WH D-180

### Behavioral Expectations

We all are adults so behavior rarely is an issue. Just follow the Golden Rule: "do unto others as you would have them do unto you" then everything will be fine.

The university must maintain a classroom environment that is suitable for learning, so anyone who insists on disrupting that environment will be expelled from the class.

Revision history:

Prepared by H. Anderson 1/10/00. Revised 4/28/01, 7/25/06, 1/10/16 (G. Jennings).