Math 437 syllabus

Complex Variables

Course Description:  Arithmetic of complex numbers; regions in the complex plane, limits, continuity and derivatives of complex functions; elementary complex functions; mapping by elementary functions; contour integration, power series, Taylor series, Laurent series, calculus or residues; conformal representation; applications. Credit for both MA 437 and MA 537 not allowed
Prerequisites:  C or better in MA 238.

Suggested Text:  Complex Variables and Applications, 9th ed. by Brown, Churchill, Published by McGraw-Hill, Inc.

Coverage: Chapters 1-6 (all sections), Chapter 7 (Evaluation of Improper Integrals, Improper Integrals from Fourier Analysis, Jordan’s Lemma)

Learning outcomes: Upon the successful completion of the course a student will:

Understand properties of complex numbers: algebraic operations (including powers and roots) with complex numbers; algebraic and exponential forms; geometric properties; regions in the complex plane; elementary mappings
Understand ideas of convergence, continuity and differentiation in the complex plane: complex functions, limits, derivatives, and analytic functions; Cauchy-Riemann equations; Harmonic functions
Understand properties of elementary functions including exponential, trigonometric, and logarithmic functions
Understand and calculate contour integrals: path integration, complex contour integrals using parameterization, the Cauchy theorem and the Cauchy integral formula
Understand and compute Taylor and Laurent series expansions for analytic functions: residues and isolated singular points; classify isolated singular points; Cauchy’s residue theorem; connection between zeros and poles of analytic functions
Apply complex residue theory to integration of real valued functions over the real line.