## Course Syllabus for "MA243: Complex Analysis"

This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable. Put differently, complex analysis is the theory of the differentiation and integration of functions that depend on one complex variable. Such functions, beautiful on their own, are immediately useful in Physics, Engineering, and Signal Processing. Because of the algebraic properties of the complex numbers and the inherently geometric flavor of complex analysis, this course will feel quite different from Real Analysis, although many of the same concepts, such as open sets, metrics, and limits will reappear. Simply put, you will be working with lines and sets and very specific functions on the complex plane—drawing pictures of them and teasing out all of their idiosyncrasies. You will again find yourself calculating line integrals, just as in multivariable calculus. However, the techniques you learn in this course will help you get past many of the seeming dead-ends you ran up against in calculus. Indeed, most of the definite integrals you will learn to evaluate in Unit 7 come directly from problems in physics and cannot be solved except through techniques from complex variables. We will begin by studying the minimal algebraically closed extension of real numbers: the complex numbers. The Fundamental Theorem of Algebra states that any non-constant polynomial with complex coefficients has a zero in the complex numbers. This makes life in the complex plane very interesting. We will also review a bit of the geometry of the complex plane and relevant topological concepts, such as connectedness. In Unit 2, we will study differential calculus in the complex domain. The concept of analytic or holomorphic function will be introduced as complex differentiability in an open subset of the complex numbers. The Cauchy-Riemann equations will establish a connection between analytic functions and differentiable functions depending on two real variables. In Unit 3, we will review power series, which will be the link between holomorphic and analytic functions. In Unit 4, we will introduce certain special functions, including exponentials and trigonometric and logarithmic functions. We will consider the Möbius Transformation in some detail. In Units 5, 6, and 7 we will study Cauchy Theory, as well as its most important applications, including the Residue Theorem. We will compute Laurent series, and we will use the Residue Theorem to evaluate certain integrals on the real line which cannot be dealt with through methods from real variables alone. Our final unit, Unit 8, will discuss harmonic functions of two real variables, which are functions with continuous second partial derivatives that satisfy the Laplace equation, conformal mappings, and the Open Mapping Theorem.

### Learning Outcomes

Upon successful completion of this course, the student will be able to:

- Fundamentals: Manipulate complex numbers in various representations, define fundamental topological concepts in the context of the complex plane, and define and calculate limits and derivatives of functions of a complex variable.
- Key Functions: Represent analytic functions as power series on their domains and verify that they are well-defined. Define a branch of the complex logarithm. Classify singularities and find Laurent series for meromorphic functions.
- Key Results: State and prove fundamental results, including: Cauchy’s Theorem and Cauchy’s Integral Formula, the Fundamental Theorem of Algebra, Morera’s Theorem and Liouville’s Theorem. Use them to prove related results.
- Key Application: Calculate contour integrals. Calculate definite integrals on the real line using the Residue Theorem.
- Mappings: Define linear fractional transformations and prove their essential characteristics. Find the image of a region under a conformal mapping. State, prove, and use the Open Mapping Theorem.

### Course Requirements

In order to take this course, you must:

√ Have access to a computer.

√ Have continuous broadband Internet access.

√ Have the ability/permission to install plug-ins or software (e.g. Adobe Reader or Flash).

√ Have the ability to download and save files and documents to a computer.

√ Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√ Have competency in the English language.

√ Have read the Saylor Student Handbook.

√ Have completed MA101, MA102, MA103, MA211, and MA221 or their equivalents. Completion of MA241 or its equivalent is recommended, but concurrent enrollment is acceptable.

### Course Information

Welcome to MA243. Below, please find general information on this course
and its requirements. This a very enjoyable course, with a distinctly
geometric flavor. While “analysis” is in the title, the course has
strong ties to both algebra and topology. Complex analysis is deep and
rich, and this course is merely an introduction, so peruse the tables of
contents of the resources listed below and delve a little deeper if
anything catches your attention.

**Course Designer:** Clare Wickman

**Primary Resources:**

San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis
Pixton, and Lucas Sabalka’s A First Course in Complex Analysis

Imperial College, London: W.W.L.Chen’s Introduction to Complex
Analysis

University of Illinois, Urbana-Champaign: Professor Robert Ash’s and
W.P. Novinger’s Complex Variables

**Requirements for Completion:** Completion of all readings,
assignments, and assessments.

**Time Commitment: 113 hours**