Course Description
Differential and integral calculus of functions of a complex variable, with applications. Topics include the Cauchy integral formula, power series and Laurent series, and the residue theorem.
Additional Requirements for Graduate Students:
Extra problems on weekly homework.
Athena Title
COMPLEX VARIABLES
Prerequisite
(MATH 2270 and MATH 3100) or (MATH 2500 and MATH 3100) or MATH 3510 or MATH 3510H
Semester Course Offered
Offered every year.
Grading System
A - F (Traditional)
Course Objectives
Students will understand the complex derivative and its significance in complex function theory. Students will see why functions that have complex derivatives everywhere on an open set are analytic. Students will see the power of the Cauchy integral formula: its applications to power series expansions, the residue theorem and the evaluation of real integrals.
Topical Outline
1. Complex Numbers: algebra, geometry, deMoivre's formula. 2. Analytic Functions: Cauchy Riemann equations. 3. Elementary Functions: exponentials, trigonometric functions. 4. Integrals: Line integrals in the complex plane, Cauchy's Theorem, Cauchy Integral formula. 5. Power series and expansion of analytic functions in power series. 6. Poles, residues, and the residue theorem. Application to computation of definite integrals. 7. Conformal mappings.
Syllabus