Difference between revisions of "Math 352: Introduction to Complex Analysis"

Title

Introduction to Complex Analysis.

(3:3:0)

F, W

Prerequisite

Math 290, and either Math 342 or concurrent enrollment.

Description

Complex algebra, analytic functions, integration in the complex plane, infinite series, theory of residues, conformal mapping.

Desired Learning Outcomes

This course is aimed at graduates majoring in mathematical and physical sciences and engineering. In addition to being an important branch of mathematics in its own right, complex analysis is an important tool for differential equations (ordinary and partial), algebraic geometry and number theory. Thus it is a core requirement for all mathematics majors. It contributes to all the expected learning outcomes of the Mathematics BS (see [1]).

Prerequisites

Students are expected to have completed and mastered Math 314 (Calculus of Several Variables) OR Math 342 (the second part of Theory of Analysis) to provide the necessary understanding of the modes of thought of mathematical analysis.

Minimal learning outcomes

Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, the full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.

1. Complex numbers, moduli, exponential form, arguments of products and quotients, roots of complex numbers, regions in the complex plane.
2. Limits, including those involving the point at infinity. Open, closed and connected sets. Continuity, derivatives.
3. Analytic functions, Cauchy-Riemann equations, harmonic functions, finding the harmonic conjugate.
4. Elementary functions in the complex plane: exponential and log functions, complex exponents, trigonometric and hyperbolic functions and their inverses.
5. Contour integrals, upper bounds for moduli, primitives, Cauchy-Goursat theorem, Cauchy integral formulae, Liouville theorem, maximum modulus theorem.
6. Taylor series, Laurent series, integration and differentiation of power series, uniqueness of series representation, multiplication and division of power series.
7. Isolated singularities, behavior near a singularity. Residue theorem, its application to improper integrals, Jordan's lemma. Argument principle, Rouche's theorem.
8. Conformal mappings. Moebius transformations.

Textbooks

Possible textbooks for this course include (but are not limited to):

These are at the instructor's discretion as time allows. Possibilities include applications of complex analysis in physics.

Courses for which this course is prerequisite

This course is required for Math 532 and 587. It is needed by anyone proceeding to graduate studies in mathematics. As a result it is essential that ALL required learning objectives be covered.