MATH 131 CALCULUS I Spring 2000 s1

MAT 349 – Introduction to Complex Analysis

Section 01: 12:00-1:00 p.m., MWThF, WIH 022B Instructor: Lucian Ionescu Contact Info: STV 312J, [email protected], 438-7167; web site: http://my.ilstu.edu/~lmiones/ Office Hours: TBA, or by appointment at other times. Text: Visual Complex Analysis, Tristan Needham, Oxford University Press, 1997. Content: MAT 349 is an introductory course in complex analysis, using a graphical approach. It also provides rigorous foundations for calculus and differential equations, while introducing new topics from topology and geometry, e.g. homotopy and groups of transformations of non-Euclidean geometry. Main topics presented include: differentiability, integrability, line integrals, Cauchy Theorems and analytic continuation, Cauchy- Riemann equations, Mobius transformations, simply connectedness, power series representations, vector fields and flows, harmonic functions and Dirichlet problem. Applications to other sciences will be presented, including applications to topology, vector fields and electromagnetism. Prerequisites: C or better in MAT 147, and MAT 247 or consent of instructor required. Objectives: The general goals of the course are 1) to introduce the basic ideas and concepts of complex analysis with a visual justification as well as rigorous, via proofs, 2) to help students develop a rigorous understanding of the theorems of calculus and differential equations in the context of two dimensions, where pictures can clarify the meaning. Course Format & Learning Cycle: You are responsible for reading assignments, and this is not a trivial task. Browsing assignments before the next scheduled meeting of the class is meant to prepare you to assimilate the explanations during the lecture (you need a map before going on the field trip). During the class, we will focus on understanding the concepts and methods. After the class read the assignment. After you are done and before attempting the homework problems, write one page outline of the corresponding section (include major concepts, methods and relations, with brief explanations). Check you have acquired the corresponding knowledge by reading the outline (keywords known? relations?) Before discussing the new assigned material, we will discuss your difficulties or questions regarding the theory and homework problems. You may choose to present solutions to the homework problems on transparencies, for “class participation” points, as time permits. Attendance and active participation are expected. For graduate credit the instructor will have students do additional work, such as, but not limited to, reading additional sections and completing additional homework problems; completing special projects is also a viable option (e.g. looking into methods for modeling electric circuits). The sections marked with * are for graduate credit. Homework: There will be weekly homework assignments. Homework will be collected on Wednesdays. Please show all the work (a list of answers is not a homework assignment). Exams: Attendance at exams is mandatory. It is highly recommended that you contact me before the exam, if an emergency has arisen. Evaluation: Grades will be based on the following points: Homework 100 Class activities (Attendance, outlines, and participation): 200 Midterm exam 100 Final exam 100 Total 500 points. The grading scale is based on: A [90-100%], B [80-90%), C [70-80%), D [60,70%), F [0,60%).

N.B.: Students who believe they may need accommodations in this class/program are encouraged to contact the Disability Access Center as soon as possible to better ensure that such accommodations are implemented in a timely fashion. “Warning: Plagiarism and cheating are serious offenses. Penalties can range from a minimum of a zero grade on the invalid instrument to expulsion from the University”

Topics Covered (Homework problems: TBA)

Week Section Week No. covered Topics No. Section covered Topics 1 1/12 Ch.1 Introduction, Euler’s formula 9 3/9 Geometry Some applications No classes Spring break and Complex Transformations and Euclidean Arithmetic Geometry 2 1/19 Ch.2 No class – M.L. King holyday 10 Complex integral Complex 3/16 Ch.8 Complex Power functions Functions as Polynomials, Power Series: integration: Transforma- exponentials, cos, sine Cauchy’s Fundamental theorem and Cauchy tions Multifunctions and logarithms Theorem formula 3 1/26 Ch.3 Mobius Introduction 11 Taylor series Trasforma- Inversion, Riemann sphere 3/23 Ch.9 Cauchy’s Calculus of residues tions and Mobius transformations Formula and Its Inversions Applications Laurent series Ch.4 Mappings in the plane 12 4 2/2 Differentia- Complex derivative 3/30 Ch.10 Vector I. Vector fields tion fields: Physics Conformal = Analytic and Topology II. Winding numbers and vector fields

5 2/9 Critical points and 13 III. Flows on closed surfaces Cauchy-Riemann Equations 4/6

6 2/16 Ch.5 Further Rules for differentiation 14 Ch.11 Vector I. Flux and work Geometry of Visual differentiation 4/13 Fields and II. Complex integration Differentia- Complex tion Integration III. The complex potential 7 2/23 Higher derivatives: curvature 15 Ch. 12 Flows Harmonic duals Celestial mechanics 4/20 and harmonic Flows Analytic continuation functions Riemann mapping theorem 8 3/2 R&p 16 R&p Review 4/27 Review Midterm Exam Review Discussions Review

The specific assignments will be announced in class, on a weekly basis.