Math 313 - Complex Variables, Fall 2023

Time and Location: Tue, Thur 12:30 - 1:45 pm, DSH-328
Office Hours: Tue, Thur, 11 - 11:50 am in SMLC 212 and also by appointment in Zoom
Professor Info: Anna Skripka, skripka [at] math [dot] unm [dot] edu
 
Description: Math 313 is the first course in complex analysis. The aim of this course is to introduce students to the classical theory of functions of a single complex variable, differential and integral calculus of complex functions. In this course we will discuss complex numbers, functions of a single complex variable, elementary functions and their transformations, limits, continuity, differentiability and the Cauchy-Riemann equations, contour integration, integral theorems, representation of analytic functions by power series, Taylor and Laurent expansions, zeros and poles, the theory of residues, applications to the evaluation of definite integrals, and, time permitting, conformal mapping. The topics will be interspaced with applications as appropriate.
Credit-hour statement: This is a 3 credit-hour course. In addition to two 75-minute sessions of direct instruction, please also plan 6-9 hours of out-of-class work (homework, study, assignment completion, class preparation) each week. If you lack necessary skills for this course, then out-of-class work can take more than 9 hours a week.
Course goals and basic study guide are posted on UNM Canvas in the category Syllabus.
Prerequisite: Calculus III and one 300-level or 400-level Math course.

Textbook: Fundamentals of complex analysis with applications, by E.B. Saff and A.D. Snider, Prentice Hall, 3rd ed., 2003, ISBN-10: 0139078746 or 0134689488
Extra examples and exercises: Schaum's outlines. Complex variables, by M. Spiegel, S. Lipschutz, J. Schiller, D. Spellman, McGraw-Hill, 2nd ed., 2009, ISBN 0071615695
Supplementary textbooks:
Visual Complex Analysis, by T. Needham, Oxford University Press, 1999, ISBN 0198534469
A First Course in Complex Analysis, by M.Beck, G.Marchesi, D.Pixton, L.Sabalka
Some video lectures on complex analysis: Brunton, Viacci
Some video courses on related topics: Proof Writing, MIT Calculus: 18.005, 18.006, 18.007, 18.008
 
Grades: 40% for homework+quiz, 60% for three exams (two midterms and one final).
Policies: You are expected to attend every classroom meeting and come prepared by reading lectures and doing exercises on old material as well as reading the book on new material ahead of lectures.
Your work will be graded on the clarity, completeness, correctness of your reasoning and presentation. On the homework assignments you are allowed to work in groups, but you must write up your own solutions in your own words. Submitting solutions obtained from third parties, including the internet, is strictly prohibited.
NO LATE ASSIGNMENT WILL BE ACCEPTED, but two lowest scores on homework/quizzes will be dropped to accommodate for unavoidable circumstances such as emergencies, illnesses, official UNM function. (You can start working on the assignment as soon as it is posted without waiting for the day it is due.) A replacement by a final exam score or make-up exam will be given for a missed midterm exam only in case of a documented absence prescribed by the university (family emergency, serious medical problem, official UNM function). All objections to grades should be made within one week since the assignments are graded.
Homework assignments and quiz dates are to be posted on UNM Canvas in the Moduli Week 1,2,3, etc.
 
Online calculators: Sage/Octave, Octave, Matlab
Free PDF scanners: CamScanner (android), TinyScanner (android), CamScanner (apple), TinyScanner (apple)
 
Tentative SCHEDULE (to be updated at the end of each week)
 
Week
Class dates
Book sections
Topics
1
Aug 22, 24
1.1, 1.2, 1.3; Schaum p. 9-32
 Complex numbers. Polar form.
2
Aug 29, 31
1.4, 1.5; home reading: 1.6
 Complex exponential. Roots. Planar sets.
3
Sep 5, 7
Drop with no grade: Sep 10
2.1, 2.2; Schaum p. 62-67
 Functions of complex variables. Limits and continuity.
4
Sep 12, 14
2.3, 2.4, 2.5; Schaum p. 85-96
 Analytic functions. Cauchy-Riemann equations. Harmonic functions.
5
Sep 19, 21
3.1, 3.2, 3.3
 Polynomials and rational functions. Exponential, trigonometric, and logarithmic functions.
6
Sep 26, 28
3.4, 3.5
 Application: boundary value problems. Complex powers & inverse trigonometric functions.
7
Oct 3, 5 (Exam 1)
covered material of 1.1-3.3
 Review.
8
Oct 10
4.1, 4.2, 4.3
 Complex integration.
9
Oct 17, 19
4.4, 4.5, 4.6; home reading: 5.1; Schaum p. 184-186
Cauchy's integral theorem. Cauchy's integral formula. Bounds for analytic functions.
10
Oct 24, 26
5.2, 5.3, 5.5; Schaum p. 188-190
 Review of series. Power, Taylor, and Laurent Series.
11
Oct 31, Nov 2
5.6, 5.7, 6.7
 Classification of singularities. Argument principle and Rouche's theorem. Uniqueness.
12
Nov 7, 9
Withdrawal deadline: Nov 10
6.1, 6.2, 6.3, 6.4; Schaum p. 214-226
 Residue theorem and its applications. Trigonometric integrals. Improper integrals.
13
Nov 14, 16 (Exam 2)
3.4,3.5,4.1-4.6,5.1-5.7,6.7
 Review.
14
Nov 21
6.5, 6.6
 Indented contours. Integrals involving multiple-valued functions.
15
Nov 28, 30
7.1, 7.2, 7.3
 Conformal mappings and their applications. Möbius transformations and boundary value problems.
16
Dec 5, 7
7.4, 7.6
 Boundary value problems. Review.
17
Dec 14 (Thursday), 10 am - noon
Final Exam
 
Academic Integrity: Each student is expected to maintain the highest standards of honesty and integrity in academic and professional matters. The University reserves the right to take disciplinary action, including dismissal, against any student who is found responsible for academic dishonesty. Any student who has been judged to have engaged in academic dishonesty in course work may receive a reduced or failing grade for the work in question and/or for the course. Academic dishonesty includes, but is not limited to, dishonesty on quizzes, tests or assignments; claiming credit for work not done or done by others; and hindering the academic work of other students. Learning the course material depends on completing and submitting your own work. UNM preserves and protects the integrity of the academic community through multiple policies including policies on student grievances (Faculty Handbook D175 and D176), academic dishonesty (FH D100), and respectful campus (FH CO9). These are in the Student Pathfinder and the Faculty Handbook.
Copyright Policy: All materials disseminated in class and on the course webpage are protected by copyright laws and are for personal use of students registered in the class. Redistribution or sale of any of these materials is strictly prohibited.
Accommodations: UNM is committed to providing equitable access to learning opportunities for students with documented disabilities. As your instructor, it is my objective to facilitate an inclusive classroom setting, in which students have full access and opportunity to participate. To engage in a confidential conversation about the process for requesting reasonable accommodations for this class and/or program, please contact Accessibility Resource Center at arcsrvs@unm.edu or by phone at 505-277-3506.
Disclaimer: It is your responsibility to know and understand the course policies. The professor reserves the right to change this syllabus should it become necessary. The changes will be announced and the up-to-date syllabus will posted on this webpage.