Transition from complex numbers to complex function and series.

Abstract

Complex analysis, traditionally known as the theory of functions of a complex variable, is a branch of mathematical analysis that investigates functions of complex numbers. If n is odd, there is only one order n root, and if n is even, there are only two order n roots. Things change in complex numbers, though. Keep in mind that complex numbers have closed algebra. As a result, there exist n roots of order n always.Dating back to the 16th century when Italian mathematicians Girolamo Cardano and Raphael Bombelli first observed complex numbers when they were trying to solve a certain algebra and developed by Cauchy and Riemann in 19th century, complex analysis nowadays has played a more and more importance role on mathematics, physics and engineering, especially in the fields of algebraic geometry, fluid dynamics, quantum mechanics and so on. In this paper, the birth and discovery of complex number will be discussed. Starting from discovery of complex number, the properties of complex function and power series are elaborated which are foundation for mordern mathematical conjectures.