This paper presents a boundary element method for solving Dirichlet bou-ndary value problem of the Helmholtz equation in R~2.First

the existence andthe uniqueness of an extended solution for the problem are obtained.Then

thesolution is expressed in terms of simple layer potentials

and this expression

which is suitable to the interior as well as the exterior problem

leads to aboundary integral equation of the first kind.Finally

a finite element approachis applied to solve a variational form which is equivalent to the boundaryintegral equation.