  We study the behaviour of the normal derivative of eigenfunctions of the
Helmholtz equation inside billiards with Dirichlet boundary condition. These
boundary functions are of particular importance because they uniquely determine
the eigenfunctions inside the billiard and also other physical quantities of
interest. Therefore they form a reduced representation of the quantum system,
analogous to the Poincar\'e section of the classical system. For the normal
derivatives we introduce an equivalent to the standard Green function and
derive an integral equation on the boundary. Based on this integral equation we
compute the first two terms of the mean asymptotic behaviour of the boundary
functions for large energies. The first term is universal and independent of
the shape of the billiard. The second one is proportional to the curvature of
the boundary. The asymptotic behaviour is compared with numerical results for
the stadium billiard, different limacon billiards and the circle billiard, and
good agreement is found. Furthermore we derive an asymptotic completeness
relation for the boundary functions.

Chaotic Dynamics