#### Prerequisites & Bulletin Description

#### Course Objectives

The principal aim of Introduction to Functions of a Complex Variables is to introduce students to the field of complex numbers and to the algebra of complex valued functions of a complex variable. While this course does not emphasize rigorous proofs students are expected to solve a variety of problems involving complex differentiation, complex path integration, infinite series and conformal mapping. Students who successfully complete this course should be capable of:

- Determining whether complex-valued functions are analytic.
- Applying the Cauchy-Riemann Equations, Cauchy's Theorem, Cauchy's Integral Formula, Cauchy's Inequality, Liouville's Theorem and the Maximum Modulus Principle to complex valued functions.
- Applying Taylor's Theorem, Laurent's Theorem and the Residue Theorem. Students should be able to apply the Residue Theorem to evaluate improper integrals.
- Demonstrating the relationship between analytic functions and conformal mapping and use conformal mapping to determine heat conduction and fluid flow in various bounded regions.

#### Evaluation of Students

Students will be graded on their ability to devise, organize and present in correct mathematical English solutions to problems. While instructors may design their own methods of evaluating student performance, these methods must include in-class examination, graded homework assignments and a final exam.

#### Course Outline

Topics |
Number of Weeks |

The Field of Complex Numbers | 2 |

Analytic Functions: properties of analytic functions, Cauchy-Riemann equations. | 2 |

Elementary Functions: algebraic, trigonometric and logarithmic functions. | 2 |

Contour Integrals of Complex Functions: Cauchy's Theorem, Cauchy's Integral Formula, Cauchy's Inequalities, Liouville's Theorem, Morera's Theorem, Maximum Modulus Theorem and elementary properties of harmonic functions | 4 |

Complex Series: power series, Taylor's Theorem, Laurent's Theorem and Laurent series, residues. | 3 |

Conformal mapping with applications chosen from: Dirichlet and Neumann problems, heat conduction, fluid flow and electric potential. | 2 |

#### Textbooks & Software

Saff & Snider, *Fundamentals of Complex Analysis*, 3nd Edition

Brown and Churchill *Complex Variable and Applications*, 6th Edition

Submitted by: David Ellis

Date: January 31, 2003