Complex Analysis with Applications pp 95-138 | Cite as

# Analytic Functions

## Abstract

Complex analysis is concerned with the study of analytic functions. These are functions that have a complex derivative in an open planar set. The notion of analyticity is fundamental in complex analysis; it lays the foundational cornerstone and sets the stage for the development of the subject. Most of the theory of analytic functions is due to Augustin-Louis Cauchy (1789–1857) and was prepared for a course that he taught at the Institut de France in 1814 and later at the Ecole Polytechnique. Cauchy single-handedly defined the derivative and integral of complex functions and developed one of the most fruitful theories of mathematics. In the process of developing his theory, he defined for the first time the notion of limit for functions and gave rigorous definitions of continuity and differentiability for real-valued functions, as are known nowadays in calculus. He also developed a solid groundwork for the theory of definite integrals and series, and he established the theoretical aspects of complex analysis. In doing so, he paid great attention to rigorous mathematical proof, a trait that characterizes pure mathematics today.