Research About The Residue

As we all know, algebraic function field theory is another manifestations of the theory of algebraic curves.An algebraic function field over K is an algebraic extension of finite degree over the rational functionfieldK (x). Residue Theory is one of the main content of the algebraic function field. In this paper, on thebasis of existing residue theory at home and abroad, given a application of the residue theorem.The full text is divided into two chapters.Chapter one We first give some basic concepts and the known result of the theory of algebraic functionfield. For example, valuations,places, derivations and differentials, P-adic expansion.Chapter tow We introduce the definition of residue and residue theorem,and apply the residue theoremgiven the residue of the z=(x-α)~m.q(x)，q (α)≠0， m∈ΖatPα in the rational functionfieldK (x)K.