Course overview
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.
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; Montel's Theorem and the Riemann Mapping Theorem.
Course learning outcomes
- Demonstrate understanding of the basic concepts underlying complex analyis.
- Demonstrate familiarity with a range of examples of these concepts.
- Prove basic results in complex analysis.
- Apply the methods of complex analysis to evaluate definite integrals and infinite series.
- Demonstrate understanding and appreciation of deeper aspects of complex analysis such as the Riemann Mapping theorem.
- Demonstrate skills in communicating mathematics orally and in writing.