Syllabus
1. Prerequisite: MATHS 267 or permission of the department chairperson.
2. Course Description: Algebra and geometric representation of complex numbers, properties of complex analytic functions, contour integration, power series and Laurent series, poles and residues, conformal mapping, and applications.
3. Course Objectives: This course gives a thorough treatment of complex analysis. Students are expected to become familiar with complex numbers, the complex extensions of elementary functions, and the multiple implications of complex analyticity. Emphasis is placed on understanding complex functions from multiple points of view, in terms of algebra, analysis, and geometry.
4. Course Rationale: The use of complex numbers and complex analytic functions has been central to the development of mathematics over the last three centuries. Complex analysis has played a part in such diverse areas as the study of prime numbers, the development of non-Euclidean geometry, and the discovery of fractals; it has furthermore been widely applied in technical disciplines such as electrical engineering and fluid dynamics. Thus, a solid background in complex analysis is essential for students who intend to continue their studies at the graduate level in mathematics or in the disciplines mentioned.
5. Course Content: The topics to be covered are: the algebra of complex numbers; properties of analytic functions of a complex variable; elementary functions; branches of the logarithm; path integrals and the Cauchy-Goursat theorem; Cauchy integral formula; Liouville’s theorem and the fundamental theorem of algebra; power series and Laurent series; absolute and uniform convergence; integration and differentiation; analytic continuation; residues; isolated singularities; zeros and poles of finite order; applications of residues, evaluation of improper integrals; linear and linear fractional transformations; mapping by elementary functions and square roots; Riemann surfaces; (if time permits) two-dimensional fluid flow.
6. Course Format: The modes of instruction may vary between lecture and student presentations.
7. Methods of Evaluating Student Performance: Course grades are determined primarily by student performance on examinations and quizzes, as well as homework and class participation. The evaluation and weight of these various components are at the discretion of the individual instructor.
8. Evaluation of the Course: The teaching of this course will be assessed by departmental student evaluations and peer evaluations. The course will be periodically reviewed, evaluated and revised by the department’s Undergraduate Programs Committee.
Written by Thomas Ivey, February 1999
Revised by Hanspeter Fischer, April 2001
Revised by Vania Mascioni, April 2004
Revised by M. Karls Spring 2006