# 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; mappings by elementary functions; contour integration; power series; Taylor series; Laurent series; calculus of residues; conformal representation; applications. Credit for both MA 437 and MA 537 not allowed.

Prerequisites:   "C" or better in MA 238.

Textbook:   Complex Variables and Applications, 8th ed. by Brown, Churchill

Coverage

Chapter 1:  All sections (2 weeks)

Chapter 2:  All sections (2 weeks)

Chapter 3:  All sections (2 weeks)

Chapter 4:  All sections (3 weeks)

Chapter 5:  All sections (2 weeks)

Chapter 6:   All sections (2 weeks)

Chapter 7 sections:  Evaluation of Improper Integrals, 1 week

Improper Integrals from Fourier Analysis, Jordan’s Lemma

Note - time allotments are approximate and do not include exams.

Learning Objectives:

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

- Be able to state and apply the Cauchy-Riemann equations

- Harmonic functions

Understand properties of elementary functions including exponential,

trigonometric, and logarithmic functions

Understand and calculate contour integrals

- Path integration. Be able to compute complex contour integrals

using parameterization

- Cauchy theorem and Cauchy integral formula

Understand and compute Taylor and Laurent series expansions for

analytic functions

- Residues and isolated singular points

- Be able to classify of isolated singular points

- Residues; poles, residues at poles

- Cauchy’s residue theorem

- Understand a connection between zeros and poles of analytic functions

Be able to apply complex residue theory to integration of real valued

functions over the real line.

Last updated February 13, 2014