Complex Analysis III
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When the real numbers are replaced by the complex numbers in the definition of the derivative of a function, the resulting (complex-)differentiable functions turn out to have many remarkable properties not enjoyed by their real analogues. These functions, usually known as holomorphic functions, have numerous applications in areas such as engineering, physics, differential equations and number theory, to name just a few. The focus of this course is on the study of holomorphic functions and their most important basic properties.
To provide an introduction to the study of complex numbers, complex functions, and their applications. By the end of this course, a student should understand what it means for a complex-valued function to be complex- differentiable, and to be familiar with the basic examples of such functions. They should understand the statement and proof of the Cauchy integral formula and its immediate consequences, including Taylor's theorem, which states that holomorphic functions can be described by convergent power series. Students should also be able to use a version of the Cauchy integral formula to be able to evaluate certain definite integrals and infinite series. Students should be able to prove some of the key results on the zeros of holomorphic functions and on isolated singularities of holomorphic functions, including an understanding of Laurent series. Finally, students should hopefully have some appreciation of some of the deeper results of the theory, for example the Riemann Mapping theorem.
Topics covered are: Complex numbers and functions; complex limits and differentiability; elementary examples; 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; outlines of the Jordan Curve Theorem, Montel's Theorem and the Riemann Mapping Theorem.
This course is not a prerequisite or assumed knowledge for any other undergraduate courses. However, it is highly recommended for all students thinking of taking Honours in Pure Mathematics and is almost essential for any student wanting to go beyond Honours level in Pure Mathematics or Mathematical Physics.
Conway: Functions of One Complex Variable, 517.53 C767fMarsden: Basic Complex Analysis, 517.54 M363b Churchhill & Brown: Complex Variables and Applications,517.53 C563