MATH 8339 Advanced Complex Analysis
The focus of this course is on the study of holomorphic functions and their most important basic properties. Topics include: Complex numbers and functions; complex limits and differentiability; analytic functions; complex line integrals; Cauchy's theorem and the Cauchy integral formula; Taylor's theorem; zeros of holomorphic functions; Rouche's Theorem; the Open Mapping theorem and Inverse Function theorem; Schwarz' Lemma; automorphisms of the ball, the plane and the Riemann sphere; isolated singularities and their classification; Laurent series; the Residue Theorem; calculation of definite integrals and evaluation of infinite series using residues. An overview of theory of harmonic functions will be covered, including the Laplacian; relation to analytic functions; conjugate harmonic functions; Dirichlet problem; and applications. Additional topics such as the Gamma and Zeta functions and the prime number theorem may be included. Application of methods of complex analysis in the course include propagation of acoustic waves relevant for the design of jet engines, problems arising in solid and fluid mechanics, as well as conformal geometry in imaging, shape analysis and computer vision.
Prerequisite
Consent of Instructor.
Offered
As scheduled