# Math 532 — Functions of a Complex Variable — Syllabus

Some details of the syllabus may change between semesters; consult your professor's syllabus!

## Stable Course Components

The course components described in this section are mostly professor independent, and should not greatly change from semester-to-semester.

### Catalog Description

Analytic functions, Cauchy-Riemann equations, theorem of Cauchy, Laurent series, calculus of residues, and applications.

### Prerequisites

Math 252 (Calculus III) with a grade of C (2.0) or better

### Overview

Complex Numbers: operations, algebraic properties, moduli, exponential forms, arguments, roots of complex numbers, regions in the complex plane. Analytic functions: functions of complex variables, mappings, limits, continuity, derivatives, differentiation rules, Cauchy-Riemann equation, harmonic functions. Elementary functions: exponential functions, logarithmic functions, identities for logarithmic functions, complex exponents, trigonometric functions, hyperbolic functions, inverse trigonometric functions. Integrals: integrals of complex valued functions, contour integrals, Cauchy-Goursat theorem, Cauchy integral formula, Liouville’s theorem, fundamental theorem of algebra, maximum modulus theorem. Series: convergence of sequences & series, Taylor series, Laurent series. Residues and poles: Cauchy residue theorem, residues at poles, three types of isolated singular points. Applications of residues: Argument principle, improper integrals, Rouches theorem. Conformal mappings, bilinear transforms.

### Student Learning Objectives

• Represent complex numbers algebraically and geometrically.
• Define and analyze limits and continuity for complex functions as well as consequences of continuity.
• Apply the concept and consequences of analyticity and the Cauchy-Riemann equations and of results on harmonic and entire functions including the fundamental theorem of algebra.
• Analyze sequences and series of analytic functions and types of convergence.
• Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem in its various versions, and the Cauchy integral formula.
• Represent functions as Taylor and Laurent series, classify singularities and poles, find residues and evaluate complex integrals using the residue theorem.
• Construct conformal mappings between specific domains.

## General Policies and Information

The information in this section applies to all courses offered by the department

### Students with Disabilities

If you are a student with a disability and believe you will need accommodations for this class, it is your responsibility to contact the Student Ability Success Center at (619) 594-6473. To avoid any delay in the receipt of your accommodations, you should contact Student Ability Success Center as soon as possible. Please note that accommodations are not retroactive, and that I cannot provide accommodations based upon disability until I have received an accommodation letter from Student Ability Success Center. Your cooperation is appreciated.

### Student Privacy and Intellectual Property

The Family Educational Rights and Privacy Act (FERPA) mandates the protection of student information, including contact information, grades, and graded assignments. I will not post grades or leave graded assignments in public places. Students will be notified at the time of an assignment if copies of student work will be retained beyond the end of the semester or used as examples for future students or the wider public. Students maintain intellectual property rights to work products they create as part of this course unless they are formally notified otherwise.

### Mathematics and Statistics Learning Center

The SDSU Math & Stat Learning Center is in the Love Library, Room LL-328. "The Math and Stats Learning Center is open to support students in all lower division math courses at SDSU. We have tutors available for walk-in help during all open hours. TAs for Math 141, 150, 151, and 252 also hold their office hours there. Please see the schedule of when the TAs for your class will be in the center by going to our website: mlc.sdsu.edu. The MLC is supported by your student success fee. We strongly encourage you to use this wonderful, free resource. Some students believe that they should not need to ask for help. But, research has shown that the average grade for students who attend the MLC is one half grade higher than those who don’t seek such support."

If you are enrolled in a class which does not have targeted support, the MLC can still serve as a great math study/meeting place; and if you are interested in becoming a tutor in the center, keep an eye on the center's webpage for hiring announcements.

### Cheating and Plagiarism

Students are generally encouraged to study together, and to work together to solve exercises. Finals, Midterms, Quizzes, Project, and other designated "individual work" activities must be completed without assistance. All violations will be reported to the Center for Student Rights and Responsibilities and will also result in score/grade reductions at the professor's discretion. Please review SDSU's full policy on academic honesty.

Examples of academic dishonesty include but are not limited to

• copying, in part or in whole, from another's test or other examination
• obtaining copies of a test, an examination, or other course material without the permission of the instructor
• collaborating with another or others in work to be presented without the permission of the instructor
• falsifying records, laboratory work, or other course data
• submitting work previously presented in another course, if contrary to the rules of the course
• altering or interfering with grading procedures
• assisting another student in any of the above
• using sources verbatim or paraphrasing without giving proper attribution (this can include phrases, sentences, paragraphs and/or pages of work)
• copying and pasting work from an online or offline source directly and calling it your own
• using information you find from an online or offline source without giving the author credit
• replacing words or phrases from another source and inserting your own words or phrases

### Religious Observances

According to the University Policy File, students should notify the instructors of affected courses of planned absences for religious observances by the end of the second week of classes.

### Medical-Related Absences

Students are instructed to contact their professor/instructor in the event they need to miss class, etc. due to an illness, injury or emergency. All decisions about the impact of an absence, as well as any arrangements for making up work, rest with the instructors. Student Health Services (SHS) does not provide medical excuses for short-term absences due to illness or injury. When a medical-related absence persists beyond five days, SHS will work with students to provide appropriate documentation. When a student is hospitalized or has a serious, ongoing illness or injury, SHS will, at the student's request and with the student’s consent, communicate with the student’s instructors via the Vice President for Student Affairs and may communicate with the student’s Assistant Dean and/or the Student Ability Success Center

### Sexual Violence / Title IX Mandated Reporting

All SDSU instructors are mandated reporters, and are thus required to share information regarding sexual violence on SDSU's campus with the Title IX Coordinator, Jessica Rentto 619-594-6017. The Title IX Coordinator has access to accommodations and support services at SDSU, and possibilities for holding accountable the person who harmed you. You are not required to share information you do not wish to disclose and your level of involvement will be at your discretion. If you do not want the Title IX Officer notified, instead of disclosing this information to your instructor, you can speak confidentially with the following people on campus and in the community: Sexual Violence Victim Advocate 619-594-0210 or Counseling and Psychological Services 619-594-5220, [email protected]. They can connect you with support services and discuss options for pursuing a University or criminal investigation. For more information regarding your university rights and options as a survivor of sexual misconduct or sexual violence, please visit titleix.sdsu.edu.

### Kumeyaay Land Acknowledgement

For millennia, the Kumeyaay people have been a part of this land. This land has nourished, healed, protected and embraced them for many generations in a relationship of balance andharmony. As members of the San Diego State community we acknowledge this legacy. We promote this balance and harmony. We find inspiration from this land; the land of theKumeyaay.