Complex Variable Course

[Dawnell Sechler]

Thiscourse is for students who desire a rigorous introduction to the theory of functions of a complex variable. Topics include Cauchy's theorem, the residue theorem, the maximum modulus theorem, Laurent series, the fundamental theorem of algebra, and the argument principle.

Students with a Bachelor's degree will be assessed graduate level tuition rate for this course. However, one cannot receive graduate level credit for courses numbered below 400 at the University of Illinois.

Students currently registered in a University of Illinois Graduate Degree program will be restricted from registering in 16-week Academic Year-term NetMath courses. Matriculating UIUC Grad students will be allowed to register in Summer Session II NetMath courses.

This page has information regarding the self-paced, rolling enrollment course. If you are a UIUC student interested in taking a course during the summer, you may be interested in a Summer Session II course.

This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.

This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is known today. Each module consists of five video lectures with embedded quizzes, followed by an electronically graded homework assignment. Additionally, modules 1, 3, and 5 also contain a peer assessment.

Access to lectures and assignments depends on your type of enrollment. If you take a course in audit mode, you will be able to see most course materials for free. To access graded assignments and to earn a Certificate, you will need to purchase the Certificate experience, during or after your audit. If you don't see the audit option:

The course may offer 'Full Course, No Certificate' instead. This option lets you see all course materials, submit required assessments, and get a final grade. This also means that you will not be able to purchase a Certificate experience.

When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.

Complex Analysis, also known as Complex Variables, is a branch of mathematics that deals with the study of functions on the complex plane. It involves the investigation of complex numbers, which are numbers that can be expressed in the form of a + bi, where a and b are real numbers and i is the imaginary unit. Complex Analysis is used to study the properties and behavior of functions that are defined on the complex plane.

Complex Analysis has many practical applications in fields such as physics, engineering, and economics. It is used to model and analyze physical systems that involve electric currents, fluid flow, and heat transfer. In engineering, it is used to design and analyze electrical circuits, control systems, and signal processing algorithms. In economics, Complex Analysis is used to study economic models and financial markets.

Some key concepts in Complex Analysis include complex numbers, analytic functions, contour integration, and conformal mapping. Complex numbers are numbers that can be expressed in the form of a + bi, where a and b are real numbers and i is the imaginary unit. Analytic functions are functions that can be represented as a power series and have a derivative at each point in their domain. Contour integration involves integrating complex-valued functions along a path in the complex plane. Conformal mapping is a technique used to transform one complex plane to another while preserving angles and shapes.

Studying Complex Analysis can enhance problem-solving skills and critical thinking abilities. It also provides a deeper understanding of the behavior of functions and their properties. Additionally, it has numerous practical applications in various fields such as physics, engineering, and economics. Understanding Complex Analysis can also lead to advancements in other areas of mathematics, such as number theory and differential equations.

There are many resources available for learning Complex Analysis, including textbooks, online courses, and video lectures. Some popular textbooks include "Complex Analysis" by Joseph Bak and Donald J. Newman and "Complex Variables and Applications" by James Ward Brown and Ruel V. Churchill. Online resources such as Khan Academy and MIT OpenCourseWare offer free courses on Complex Analysis. Additionally, there are many video lectures on platforms like YouTube and Coursera that cover various topics in Complex Analysis.

I took Real Analysis 1 last semester, and it was challenging, but not as bad as I thought it would be. I am considering taking Function of a Complex Variable this semester, but I am torn. I don't know much about the complex number system, and I am afraid I'll struggle. Generally, should I expect Complex Variable to be more rigorous than Analysis?

Real Analysis 1:Properties of the real number system; point set theory for the real line including the Bolzano-Weierstrass theorem; sequences, functions of one variable: limits and continuity, differentiability, Riemann integrability

Function of a Complex Variable:Complex number system. Functions of a complex variable, their derivatives and integrals. Taylor and Laurent series expansions. Residue theory and applications, elementary functions, conformal mapping, and applications to physical problems.

I think this depends on what you're worried about, and what elements of the subject you think you might have trouble with. You mention that you thought real analysis was challenging, but you also ask specifically whether complex analysis is more rigorous than real analysis. So there are two possible aspects of the course one can look at: the level of mathematical rigor, which can differ depending on the style of the professor and/or textbook, and the difficulty of the mathematical concepts.

In terms of mathematical rigor, I would say it's rare that one would teach complex analysis more rigorously than one would teach real analysis, if it's being taught by the same professor (which, of course, is not given). That is not to say that complex analysis requires less rigor, but simply that because real analysis is in some sense more fundamental, in teaching it one often pays more attention to a lot of details.

In terms of mathematical concepts, I think it is typical for a student to say that complex analysis hearkens back to more familiar concepts learned in previous courses, e.g. basic calculus and geometry. In many respects, real analysis is a subject that has to deal with pathological issues: things like spaces that are path-connected but not connected, functions that are continuous everywhere but differentiable nowhere, metric spaces that are not complete, etc. This often makes it hard to visualize some of the concepts. Complex analysis, on the other hand, deals only with a nice space ($\mathbbC \cong \mathbbR^2$) and complex analytic functions, which turn out to be basically the nicest functions on the planet. You learn many geometric connections involving complex analytic functions, and often these connections can be visualized. I don't claim that complex analysis is an easier subject, but certainly the theory is much tighter and in many ways more elegant.

Students enroll in a course by selecting an open class below. Students progress at their own pace following course guidelines, with guidance from instructors who strive to meet individual needs. While the course provides students with independence and flexibility, students must manage their time to complete the course before the end date.

Note: You need an active CTY Account to complete registration through MyCTY.

Successful completion of AP︎ Calculus BC or equivalent required. Completion of both Multivariable Calculus and an introduction to proofs (such as Introduction to Abstract Mathematics) is strongly encouraged.

This course requires a computer with high-speed Internet access and an up-to-date web browser such as Chrome or Firefox. You must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.

This course uses a virtual classroom for instructor-student communication. The classroom works on standard computers with the Zoom desktop client, and on tablets or handhelds that support the Zoom Mobile app. Recorded meetings can only be viewed on a computer with the Zoom desktop client installed. The Zoom desktop client and Zoom Mobile App are both free to download.

Participate in the High School Math Club or enroll in 6 IP courses: Middle School Competitive Math I, Middle School Competitive Math II, Middle School Competitive Math III, Competitive Mathematics Prep, Competitive Mathematics I, Competitive Mathematics II.

I am so excited when students are able to persevere and decode a very challenging cipher in the course! I love that Cryptology teaches patience and dedication, and that mathematics is so much more than just the study of numbers and equations.

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