# Analytic functions and complex integration

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Ying Mei Lin (Creator)
Institution
The University of North Carolina at Greensboro (UNCG )
Web Site: http://library.uncg.edu/
E. E. Posey

Abstract: This paper presents some basic theory of analytic function. The class of analytic functions is formed by the complex functions of a complex variable which possess a derivative wherever the function is defined. Beginning with the definition of the derivative, the necessary and sufficient conditions for a function to be analytic at a point are developed. In order to prove that the derivative of an analytic function is itself analytic, the line integrals are defined. The exact differential and its relation to line integrals are discussed. Then Cauchy's theorem is stated. Since the proof of Cauchy's theorem is so complicated and can be found in most texts on complex variable, it is skipped. Lastly Cauchy's integral formula is developed. The formula is an ideal tool for the study of local properties of analytic functions. Particularly it helps to show that an analytic function has derivatives of all orders, which are then analytic. With this result, the converse of Cauchy's theorem, known as Morera's theorem, can be proved. Last, the paper discusses Liouville's theorem concerning entire function.